text:
# A note on products of sequential fans

Two posts (the previous post and this post) are devoted to discussing the behavior of countable tightness in taking Cartesian products. The previous post shows that countable tightness behaves well in the product operation if the spaces are compact. In this post, we step away from the orderly setting of compact spaces. We examine the behavior of countable tightness in product of sequential fans. In this post, we show that countable tightness can easily be destroyed when taking products of sequential fans. Due to the combinatorial nature of sequential fans, the problem of determining the tightness of products of fans is often times a set-theoretic problem. In many instances, it is hard to determine the tightness of a product of two sequential fans without using extra set theory axioms beyond ZFC. The sequential fans is a class of spaces that have been studied extensively and are involved in the solutions of many problems that were seemingly unrelated. For one example, see [3].

For a basic discussion of countable tightness, see these previous post on the notion of tightness and its relation with free sequences. Also see chapter a-4 on page 15 of [4].

____________________________________________________________________

Sequential fans

Let $S$ be a non-trivial convergent sequence along with its limit point. For convenience, let $\displaystyle S=\left\{0 \right\} \cup \left\{1, 2^{-1}, 3^{-1}, 4^{-1}, \cdots \right\}$, considered as a subspace of the Euclidean real line. Let $\kappa$ be a cardinal number. The set $\kappa$ is usually taken as the set of all the ordinals that precede $\kappa$. The set $\omega$ is the first infinite ordinal, or equivalently the set of all non-negative integers. Let $\omega^\kappa$ be the set of all functions from $\kappa$ into $\omega$.

There are several ways to describe a sequential fan. One way is to describe it as a quotient space. The sequential fan $S(\kappa)$ is the topological sum of $\kappa$ many copies of the convergent sequence $S$ with all non-isolated points identified as one point that is called $\infty$. To make the discussion easier to follow, we also use the following formulation of $S(\kappa)$:

$\displaystyle S(\kappa)=\left\{\infty \right\} \cup (\kappa \times \omega)$

In this formulation, every point is $\kappa \times \omega$ is isolated and an open neighborhood of the point $\infty$ is of the form:

$\displaystyle B_f=\left\{\infty \right\} \cup \left\{(\alpha,n) \in \kappa \times \omega: n \ge f(\alpha) \right\}$ where $f \in \omega^\kappa$.

According to the definition of the open neighborhood $B_f$, the sequence $(\alpha,0), (\alpha,1), (\alpha,2),\cdots$ converges to the point $\infty$ for each $\alpha \in \kappa$. Thus the set $(\left\{\alpha \right\} \times \omega) \cup \left\{\infty \right\}$ is a homeomorphic copy of the convergent sequence $S$. The set $\left\{\alpha \right\} \times \omega$ is sometimes called a spine. Thus the space $S(\kappa)$ is said to be the sequential fan with $\kappa$ many spines.

The point $\infty$ is the only non-isolated point in the fan $S(\kappa)$. The set $\mathcal{B}=\left\{B_f: f \in \omega^\kappa \right\}$ is a local base at the point $\infty$. The base $\mathcal{B}$ is never countable except when $\kappa$ is finite. Thus if $\kappa$ is infinite, the fan $S(\kappa)$ can never be first countable. In particular, for the fan $S(\omega)$, the character at the point $\infty$ is the cardinal number $\mathfrak{d}$. See page 13 in chapter a-3 of [4]. This cardinal number is called the dominating number and is introduced below in the section “The bounding number”. This is one indication that the sequential fan is highly dependent on set theory. It is hard to pinpoint the character of $S(\omega)$ at the point $\infty$. For example, it is consistent with ZFC that $\mathfrak{d}=\omega_1$. It is also consistent that $\mathfrak{d}>\omega_1$.

Even though the sequential fan is not first countable, it has a relatively strong convergent property. If $\infty \in \overline{A}$ and $\infty \notin A$ where $A \subset S(\kappa)$, then infinitely many points of $A$ are present in at least one of the spine $\left\{\alpha \right\} \times \omega$ (if this is not true, then $\infty \notin \overline{A}$). This means that the sequential fan is always a Frechet space. Recall that the space $Y$ is a Frechet space if for each $A \subset Y$ and for each $x \in \overline{A}$, there exists a sequence $\left\{x_n \right\}$ of points of $A$ converging to $x$.

Some of the convergent properties weaker than being a first countable space are Frechet space, sequential space and countably tight space. Let's recall the definitions. A space $X$ is a sequential space if $A \subset X$ is a sequentially closed set in $X$, then $A$ is a closed set in $X$. The set $A$ is sequentially closed in $X$ if this condition is satisfied: if the sequence $\left\{x_n \in A: n \in \omega \right\}$ converges to $x \in X$, then $x \in A$. A space $X$ is countably tight (have countable tightness) if for each $A \subset X$ and for each $x \in \overline{A}$, there exists a countable $B \subset A$ such that $x \in \overline{B}$. See here for more information about these convergent properties. The following shows the relative strength of these properties. None of the implications can be reversed.

First countable space $\Longrightarrow$ Frechet space $\Longrightarrow$ Sequential space $\Longrightarrow$ Countably tight space

____________________________________________________________________

Examples

The relatively strong convergent property of being a Frechet space is not preserved in products or squares of sequential fans. We now look at some examples.

Example 1
Consider the product space $S(\omega) \times S$ where $S$ is the convergent sequence defined above. The first factor is Frechet and the second factor is a compact metric space. We show that $S(\omega) \times S$ is not sequential. To see this, consider the following subset $A$ of $S(\omega) \times S$:

$\displaystyle A_f=\left\{(x,n^{-1}) \in S(\omega) \times S: n \in \omega \text{ and } x=(n,f(n)) \right\} \ \forall \ f \in \omega^\omega$

$\displaystyle A=\bigcup_{f \in \omega^\omega} A_f$

It follows that $(\infty,0) \in \overline{A}$. Furthermore, no sequence of points of $A$ can converge to the point $(\infty,0)$. To see this, let $a_n \in A$ for each $n$. Consider two cases. One is that some spine $\left\{m \right\} \times \omega$ contains the first coordinate of $a_n$ for infinitely many $n \in \omega$. The second is the opposite of the first – each spine $\left\{m \right\} \times \omega$ contains the first coordinate of $a_n$ for at most finitely many $n$. Either case means that there is an open set containing $(\infty,0)$ that misses infinitely many $a_n$. Thus the sequence $a_n$ cannot converge to $(\infty,0)$.

Let $A_1$ be the set of all sequential limits of convergent sequences of points of $A$. With $A \subset A_1$, we know that $(\infty,0) \in \overline{A_1}$ but $(\infty,0) \notin A_1$. Thus $A_1$ is a sequentially closed subset of $S(\omega) \times S$ that is not closed. This shows that $S(\omega) \times S$ is not a sequential space.

The space $S(\omega) \times S$ is an example of a space that is countably tight but not sequential. The example shows that the product of two Frechet spaces does not even have to be sequential even when one of the factors is a compact metric space. The next example shows that the product of two sequential fans does not even have to be countably tight.

Example 2
Consider the product space $S(\omega) \times S(\omega^\omega)$. We show that it is not countably tight. To this end, consider the following subset $A$ of $S(\omega) \times S(\omega^\omega)$.

$\displaystyle S(\omega)=\left\{\infty \right\} \cup (\omega \times \omega)$

$\displaystyle S(\omega^\omega)=\left\{\infty \right\} \cup (\omega^\omega \times \omega)$

$\displaystyle A_f=\left\{(x,y) \in S(\omega) \times S(\omega^\omega): x=(n,f(n)) \text{ and } y=(f,j) \right\} \ \forall \ f \in \omega^\omega$

$\displaystyle A=\bigcup_{f \in \omega^\omega} A_f$

It follows that $(\infty,\infty) \in \overline{A}$. We show that for any countable $C \subset A$, the point $(\infty,\infty) \notin \overline{C}$. Fix a countable $C \subset A$. We can assume that $C=\bigcup_{i=1}^\infty A_{f_i}$. Now define a function $g \in \omega^\omega$ by a diagonal argument as follows.

Define $g(0)$ such that $g(0)>f_0(0)$. For each integer $n>0$, define $g(n)$ such that $g(n)>\text{max} \{ \ f_n(0),f_n(1),\cdots,f_n(n) \ \}$ and $g(n)>g(n-1)$. Let $O=B_g \times S(\omega^\omega)$. The diagonal definition of $g$ ensures that $O$ is an open set containing $(\infty,\infty)$ such that $O \cap C=\varnothing$. This shows that the space $S(\omega) \times S(\omega^\omega)$ is not countably tight.

Example 3
The space $S(\omega_1) \times S(\omega_1)$ is not countably tight. In fact its tightness character is $\omega_1$. This fact follows from Theorem 1.1 in [2].

____________________________________________________________________

The set-theoretic angle

Example 2 shows that $S(\omega) \times S(\omega^\omega)$ is not countably tight even though each factor has the strong property of a Frechet space with the first factor being a countable space. The example shows that Frechetness behaves very badly with respect to the product operation. Is there an example of $\kappa>\omega$ such that $S(\omega) \times S(\kappa)$ is countably tight? In particular, is $S(\omega) \times S(\omega_1)$ countably tight?

First off, if Continuum Hypothesis (CH) holds, then Example 2 shows that $S(\omega) \times S(\omega_1)$ is not countably tight since the cardinality of $\omega^{\omega}$ is $\omega_1$ under CH. So for $S(\omega) \times S(\omega_1)$ to be countably tight, extra set theory assumptions beyond ZFC will have to be used (in fact the extra axioms will have to be compatible with the negation of CH). In fact, it is consistent with ZFC for $S(\omega) \times S(\omega_1)$ to be countably tight. It is also consistent with ZFC for $t(S(\omega) \times S(\omega_1))=\omega_1$. We point out some facts from the literature to support these observations.

Consider $S(\omega) \times S(\kappa)$ where $\kappa>\omega_1$. For any regular cardinal $\kappa>\omega_1$, it is possible that $S(\omega) \times S(\kappa)$ is countably tight. It is also possible for the tightness character of $S(\omega) \times S(\kappa)$ to be $\kappa$ (of course in a different model of set theory). Thus it is hard to pin down the tightness character of the product $S(\omega) \times S(\kappa)$. It all depends on your set theory. In the next section, we point out some facts from the literature to support these observations.

Example 3 points out that the tightness character of $S(\omega_1) \times S(\omega_1)$ is $\omega_1$, i.e. $t(S(\omega_1) \times S(\omega_1))=\omega_1$ (this is a fact on the basis of ZFC only). What is $t(S(\omega_2) \times S(\omega_2))$ or $t(S(\kappa) \times S(\kappa))$ for any $\kappa>\omega_1$? The tightness character of $S(\kappa) \times S(\kappa)$ for $\kappa>\omega_1$ also depends on set theory. We also give a brief explanation by pointing out some basic information from the literature.

____________________________________________________________________

The bounding number

The tightness of the product $S(\omega) \times S(\kappa)$ is related to the cardinal number called the bounding number denoted by $\mathfrak{b}$.

Recall that $\omega^{\omega}$ is the set of all functions from $\omega$ into $\omega$. For $f,g \in \omega^{\omega}$, define $f \le^* g$ by the condition: $f(n) \le g(n)$ for all but finitely many $n \in \omega$. A set $F \subset \omega^{\omega}$ is said to be a bounded set if $F$ has an upper bound according to $\le^*$, i.e. there exists some $f \in \omega^{\omega}$ such that $g \le^* f$ for all $g \in F$. Then $F \subset \omega^{\omega}$ is an unbounded set if it is not bounded. To spell it out, $F \subset \omega^{\omega}$ is an unbounded set if for each $f \in \omega^{\omega}$, there exists some $g \in F$ such that $g \not \le^* f$.

Furthermore, $F \subset \omega^{\omega}$ is a dominating set if for each $f \in \omega^{\omega}$, there exists some $g \in F$ such that $f \le^* g$. Define the cardinal numbers $\mathfrak{b}$ and $\mathfrak{d}$ as follows:

$\displaystyle \mathfrak{b}=\text{min} \left\{\lvert F \lvert: F \subset \omega^{\omega} \text{ is an unbounded set} \right\}$

$\displaystyle \mathfrak{d}=\text{min} \left\{\lvert F \lvert: F \subset \omega^{\omega} \text{ is a dominating set} \right\}$

The cardinal number $\mathfrak{b}$ is called the bounding number. The cardinal number $\mathfrak{d}$ is called the dominating number. Note that continuum $\mathfrak{c}$, the cardinality of $\omega^{\omega}$, is an upper bound of both $\mathfrak{b}$ and $\mathfrak{d}$, i.e. $\mathfrak{b} \le \mathfrak{c}$ and $\mathfrak{d} \le \mathfrak{c}$. How do $\mathfrak{b}$ and $\mathfrak{d}$ relate? We have $\mathfrak{b} \le \mathfrak{d}$ since any dominating set is also an unbounded set.

A diagonal argument (similar to the one in Example 2) shows that no countable $F \subset \omega^{\omega}$ can be unbounded. Thus we have $\omega < \mathfrak{b} \le \mathfrak{d} \le \mathfrak{c}$. If CH holds, then we have $\omega_1 = \mathfrak{b} = \mathfrak{d} = \mathfrak{c}$. On the other hand, it is also consistent that $\omega < \mathfrak{b} < \mathfrak{d} \le \mathfrak{c}$.

We now relate the bounding number to the tightness of $S(\omega) \times S(\kappa)$. The following theorem is from Theorem 1.3 in [3].

Theorem 1 – Theorem 1.3 in [3]
The following conditions hold:

• For $\omega \le \kappa <\mathfrak{b}$, the space $S(\omega) \times S(\kappa)$ is countably tight.
• The tightness character of $S(\omega) \times S(\mathfrak{b})$ is $\mathfrak{b}$, i.e. $t(S(\omega) \times S(\mathfrak{b}))=\mathfrak{b}$.

Thus $S(\omega) \times S(\kappa)$ is countably tight for any uncountable $\kappa <\mathfrak{b}$. In particular if $\omega_1 <\mathfrak{b}$, then $S(\omega) \times S(\omega_1)$ is countably tight. According to Theorem 5.1 in [6], this is possible.

Theorem 2 – Theorem 5.1 in [6]
Let $\tau$ and $\lambda$ be regular cardinal numbers such that $\omega_1 \le \tau \le \lambda$. It is consistent with ZFC that $\mathfrak{b}=\mathfrak{d}=\tau$ and $\mathfrak{c}=\lambda$.

Theorem 2 indicates that it is consistent with ZFC that the bounding number $\mathfrak{b}$ can be made to equal any regular cardinal number. In the model of set theory in which $\omega_1 <\mathfrak{b}$, $S(\omega) \times S(\omega_1)$ is countably tight. Likewise, in the model of set theory in which $\omega_1 < \kappa <\mathfrak{b}$, $S(\omega) \times S(\kappa)$ is countably tight.

On the other hand, if the bounding number is made to equal an uncountable regular cardinal $\kappa$, then $t(S(\omega) \times S(\kappa))=\kappa$. In particular, $t(S(\omega) \times S(\omega_1))=\omega_1$ if $\mathfrak{b}=\omega_1$.

The above discussion shows that the tightness of $S(\omega) \times S(\kappa)$ is set-theoretic sensitive. Theorem 2 indicates that it is hard to pin down the location of the bounding number $\mathfrak{b}$. Choose your favorite uncountable regular cardinal, there is always a model of set theory in which $\mathfrak{b}$ is your favorite uncountable cardinal. Then Theorem 1 ties the bounding number to the tightness of $S(\omega) \times S(\kappa)$. Thus the exact value of the tightness character of $S(\omega) \times S(\kappa)$ depends on your set theory. If your favorite uncountable regular cardinal is $\omega_1$, then in one model of set theory consistent with ZFC, $t(S(\omega) \times S(\omega_1))=\omega$ (when $\omega_1 <\mathfrak{b}$). In another model of set theory, $t(S(\omega) \times S(\omega_1))=\omega_1$ (when $\omega_1 =\mathfrak{b}$).

One comment about the character of the fan $S(\omega)$ at the point $\infty$. As indicated earlier, the character at $\infty$ is the dominating number $\mathfrak{d}$. Theorem 2 tells us that it is consistent that $\mathfrak{d}$ can be any uncountable regular cardinal. So for the fan $S(\omega)$, it is quite difficult to pinpoint the status of a basic topological property such as character of a space. This is another indication that the sequential fan is highly dependent on additional axioms beyond ZFC.

____________________________________________________________________

The collectionwise Hausdorff property

Now we briefly discuss the tightness of $t(S(\kappa) \times S(\kappa))$ for any $\kappa>\omega_1$. The following is Theorem 1.1 in [2].

Theorem 3 – Theorem 1.1 in [2]
Let $\kappa$ be any infinite regular cardinal. The following conditions are equivalent.

• There exists a first countable $< \kappa$-collectionwise Hausdorff space which fails to be a $\kappa$-collectionwise Hausdorff space.
• $t(S(\kappa) \times S(\kappa))=\kappa$.

The existence of the space in the first condition, on the surface, does not seem to relate to the tightness character of the square of a sequential fan. Yet the two conditions were proved to be equivalent [2]. The existence of the space in the first condition is highly set-theory sensitive. Thus so is the tightness of the square of a sequential fan. It is consistent that a space in the first condition exists for $\kappa=\omega_2$. Thus in that model of set theory $t(S(\omega_2) \times S(\omega_2))=\omega_2$. It is also consistent that there does not exist a space in the first condition for $\kappa=\omega_2$. Thus in that model, $t(S(\omega_2) \times S(\omega_2))<\omega_2$. For more information, see [3].

____________________________________________________________________

Remarks

Sequential fans and their products are highly set-theoretic in nature and are objects that had been studied extensively. This is only meant to be a short introduction. Any interested readers can refer to the small list of articles listed in the reference section and other articles in the literature.

____________________________________________________________________

Exercise

Use Theorem 3 to show that $t(S(\omega_1) \times S(\omega_1))=\omega_1$ by finding a space $X$ that is a first countable $< \omega_1$-collectionwise Hausdorff space which fails to be a $\omega_1$-collectionwise Hausdorff space.

For any cardinal $\kappa$, a space $X$ is $\kappa$-collectionwise Hausdorff (respectively $< \kappa$-collectionwise Hausdorff) if for any closed and discrete set $A \subset X$ with $\lvert A \lvert \le \kappa$ (repectively $\lvert A \lvert < \kappa$), the points in $A$ can be separated by a pairwise disjoint family of open sets.

____________________________________________________________________

Reference

1. Bella A., van Mill J., Tight points and countable fan-tightness, Topology Appl., 76, (1997), 1-27.
2. Eda K., Gruenhage G., Koszmider P., Tamano K., Todorčeviće S., Sequential fans in topology, Topology Appl., 67, (1995), 189-220.
3. Eda K., Kada M., Yuasa Y., Tamano K., The tightness about sequential fans and combinatorial properties, J. Math. Soc. Japan, 49 (1), (1997), 181-187.
4. Hart, K. P., Nagata J. I., Vaughan, J. E., editors, Encyclopedia of General Topology, First Edition, Elsevier Science Publishers B. V, Amsterdam, 2003.
5. LaBerge T., Landver A., Tightness in products of fans and psuedo-fans, Topology Appl., 65, (1995), 237-255.
6. Van Douwen, E. K., The Integers and Topology, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, (1984) 111-167.

.

____________________________________________________________________
$\copyright \ 2015 \text{ by Dan Ma}$

# A note about the Arens’ space

The Arens’ space is a canonical example of a sequential space that is not a Frechet space. It also has a subspace that is not sequential (thus the notion of being a sequential is not hereditary). We show that any space that is sequential but not Frechet contains a copy of the Arens’ space. For previous discussion on sequential spaces and Frechet spaces, see the links at the end of this post. Also see [1] and [2].

Let $\omega$ be the set of all nonnegative integers. Let $\mathbb{N}$ be the set of all positive integers. In one formulation, the Arens’ space is the set $X=\left\{\infty\right\} \cup \mathbb{N} \cup (\mathbb{N} \times \mathbb{N})$ with the open neighborhoods defined by:

• The points in $\mathbb{N} \times \mathbb{N}$ are isolated;
• The neighborhoods at each $n \in \mathbb{N}$ are of the form $B_{n,m}=\left\{n\right\} \cup \left\{(n,j) \in \mathbb{N} \times \mathbb{N}:j \ge m\right\}$ for some $m \in \mathbb{N}$;
• The neighborhoods at $\infty$ are obtained by removing from $X$ finitely many $B_{n,1}$ and by removing finitely many isolated points in each of the remaining $B_{n,1}$.

Another formulation is that of a quotient space. For each $n \in \omega$, let $K_n=\left\{x_{n,j}:j \in \mathbb{N}\right\} \cup \left\{y_n\right\}$ be a convergent sequence such that $y_n$ is the limit. Let $G$ be a topological sum of the convergent sequences $K_n$. We then identify $\left\{x_{0,j},y_j\right\}$ for each $j \in \mathbb{N}$. The Arens’ space is the resulting quotient space and let $Y$ denote this space (in the literature $S_2$ is used). Note that the Arens’ space has been previously defined in this blog (see An example of a quotient space, II). Note that the quotient space $Y$ is topologically identical to $X$. In the remainder of this note, we work with $X$ in discussing the Arens’ space.

The Arens’ space is sequential since it is a quotient space of a first countable space. The subspace $\left\{\infty\right\} \cup (\mathbb{N} \times \mathbb{N})$ is not sequential, proving that the Arens’ space is not a Frechet space.

We now show that any sequential space that is not Frechet contains a copy of the Arens’ space. We have the following theorem.

Theorem
Let $W$ be a sequential space. Then $W$ is Frechet if and only $W$ does not contain a copy of the Arens’ space.

Proof
$\Longrightarrow$ This direction is clear since the Frechet property is hereditary.

$\Longleftarrow$ For any $T \subset W$, let $T^s$ be the set of limits of sequences of points of $T$. Suppose $W$ is not Frechet. Then for some $A \subset W$, there exists $x \in \overline{A}$ such that $x \notin A^s$. Since $A^s$ is non-closed in $W$ and since $W$ is sequential, there is a sequence $w_n$ of points of $A^s$ converging to $z_0 \notin A^s$. We can assume that $w_n \notin A$ for all but finitely many $n$ (otherwise $z_0 \in A^s$). Thus without loss of generality, assume $w_n \notin A$ for all $n$.

For each $n \in \mathbb{N}$, there is a sequence $z_{n,j}$ of points of $A$ converging to $w_n$. It is OK to assume that all $w_n$ are distinct and all $z_{n,j}$ are distinct across the two indexes. Let $W_0=\left\{z_0\right\} \cup W_1 \cup W_2$ where $W_1=\left\{w_n: n \in \mathbb{N}\right\}$ and $W_2=\left\{z_{n,j}:n,j \in \mathbb{N}\right\}$. Then $W_0$ is a homeomorphic copy of the Arens’ space. $\blacksquare$

Remark
The above theorem is not valid outside of sequential spaces. Let $Z$ be a countable space with only one non-isolated point where $Z$ is not sequential (for example, the subspace $Z=\left\{\infty\right\} \cup (\mathbb{N} \times \mathbb{N})$ of the Arens’ space). Clearly $Z$ contains no copy of the Arens’ space. Yet $Z$ is not Frechet (it is not even sequential).

Previous posts on sequential spaces and Frechet spaces:
Sequential spaces, I
Sequential spaces, II
Sequential spaces, III
Sequential spaces, IV
Sequential spaces, V
k-spaces, I
k-spaces, II

Reference

1. Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
2. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.

# k-spaces, I

There are many examples in general topology of defining a new topology on a set based on a given topology already defined on the set. For example, given a topology $\tau$ on $X$, one can define a finer topology $\tau_s$ consisting of all sequentially open subsets of $X$ (based on the original topology $\tau$). Sequential spaces are precisely those spaces for which the original topology coincides with $\tau_s$ (see the post Sequential spaces, II). A related concept is the notion of k-spaces. We show that the compactly generated open sets form a finer topology and that k-spaces are precisely those spaces for which the compactly generated topology coincides with the original topology. We also give an external characterization of k-spaces, namely, those spaces that are quotient images of locally compact spaces. All spaces under consideration are Hausdorff.

Let $X$ be a space. We say $A \subset X$ is a compactly generated closed set in $X$ if $A \cap K$ is closed in $K$ for any compact $K \subset X$. We say $B \subset X$ is a compactly generated open set in $X$ if $X-B$ is a compactly generated closed set in $X$. The space $X$ is said to be a k-space if

$A \subset X$ is a compactly generated closed set in $X$ if and only if $A$ is a closed set.

The direction $\Leftarrow$ of the above statement always holds. So a space is a k-space if it satisfies the direction $\Rightarrow$ in the above statement. We also want to mention that in the above definition, “closed” can be replaced by “open”.

Suppose $\tau$ is the topology of the space $X$. Then define $\tau_k$ as the set of all compactly generated open sets in $X$. It can be easily verified that $\tau_k$ is a topology defined on the set $X$ and that $\tau_k$ is a finer topology than the original topology $\tau$, i.e. $\tau \subset \tau_k$. It follows that $X$ is a k-space if and only if $\tau=\tau_k$.

Much of the discussion here mirrors the one in Sequential spaces, II. In a sequential space, the topology coincides with $\tau_s$, the open sets generated by convergent sequences (a particular type of compact sets). In a k-space, the topology conincides with $\tau_k$, the open sets generated by the compact sets. A sequential space is the quotient space of a topological sum of disjoint convergent sequences. Any k-space is the quotient space of a topological sum of disjoint compact sets.

We have the following theorem.

Theorem
For any space $X$, the following conditions are equivalent:

1. $X$ is a k-space.
2. $X$ is a quotient space of a locally compact space.

Proof. $1 \Rightarrow 2$ Let $X$ be a k-space. Let $\mathcal{K}$ be the set of all compact subsets of $X$. Let $Y=\oplus_{K \in \mathcal{K}}K$ be the topological sum of all $K \in \mathcal{K}$ where each $K \in \mathcal{K}$ has the relative topology inherited from the space $X$. Then $Y$ is a locally compact space. There is a natural mapping we can define on $Y$ onto $X$. The space $Y$ is a disjoint union of all compact subsets of $X$. We can map each compact set $K \in \mathcal{K}$ onto the corresponding compact subset $K$ of $X$ by the identity map. Call this mapping $f$ where $f:Y \mapsto X$. We claim that the quotient topology generated by this mapping coincides with the original topology on $X$.

Let $\tau$ be the given topology on $X$ and let $\tau_f$ be the quotient topology generated on the set $X$. Clearly, $\tau \subset \tau_f$. We need to show that $\tau_f \subset \tau$. Let $O \in \tau_f$. Since $X$ is a k-space, if we can show that $O$ is a compactly generated open set, then $O \in \tau$.

Let $K \subset X$ be compact. We need to show that $O \cap K$ is open in $K$. Since $O \in \tau_f$, $f^{-1}(O)$ is open in $Y$. Since $K$ is open in $Y$, $f^{-1}(O) \cap K$ is open in $Y$. It is also the case that $f^{-1}(O) \cap K$ is open in $K$. We can consider $f^{-1}(O) \cap K$ as a subset of $K \subset Y$ and as a subset of $K \subset X$. As a subset of $K \subset X$, we have $f^{-1}(O) \cap K=O \cap K$. Thus $O \cap K$ is open in $K$ and $O \in \tau$.

$2 \Rightarrow 1$ Let $Y$ be locally compact and let $f:Y \mapsto X$ be a quotient map. We show that $X$ is a k-space. To this end, we show that if $A \subset X$ is a compactly generated closed set in $X$, $A$ is closed in $X$. Or equivalently, if $A$ is not closed in $X$, then $A$ is not a compactly generated closed set in $X$. Under the quotient map $f$, $A$ is closed in $X$ if and only if $f^{-1}(A)$ is closed in $Y$.

Suppose $A$ is not closed in $X$. Then $f^{-1}(A)$ is not closed in $Y$. Then there is $y \in \overline{f^{-1}(A)}-f^{-1}(A)$. Let $U \subset Y$ be open in $Y$ such that $y \in U$ and $\overline{U}$ is compact. Then $f(\overline{U})$ is compact. It follows that $A \cap f(\overline{U})$ is not closed in $f(\overline{U})$. Note that $f(y) \in f(\overline{U})$ and $f(y) \notin A \cap f(\overline{U})$. However, $f(y) \in \overline{A \cap f(\overline{U})}$. Thus $A$ is not a compactly generated closed set in $X$. $\blacksquare$

Reference

1. Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
2. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.

# Sequential spaces, I

Any topological space where there is a countable base at every point is said to satisfy the first axiom of countability or to be first countable. In this post we discuss several properties weaker than the first axiom of countability. All spaces under consideration are Hausdorff. Any first countable space $X$ satisfies each of the following conditions:

1. If $A \subset X$ and $x \in \overline{A}$ then there is a sequence $\left\{x_n\right\}$ of points in $A$ such that the sequence converges to $x$.
2. The set $A \subset X$ is closed in $X$ if $A$ is sequentially closed in $X$, which means that: if $\left\{x_n\right\}$ is a sequence of points in $A$ such that $\left\{x_n\right\}$ converges to $x \in X$, then $x \in A$.
3. The set $A \subset X$ is closed in $X$ if this condition holds: if $K \subset X$ is compact, then $A \cap K$ is closed in $K$.

Spaces satisfying:

• condition 1 are called Frechet spaces,
• condition 2 are called sequential spaces,
• condition 3 are called k-spaces.

All three of these conditions hold in first countable spaces. We have the following implications:

First countable $\Rightarrow \$ Frechet $\Rightarrow \$ Sequential $\Rightarrow \$ k-space

This post is an introductory discussion of these notions. Each of the above implications is not reversible (see the section below on examples). After we discuss sequential spaces, we take a look at the behavior of these four classes of spaces in terms of whether the property can be passed onto subspaces (the property being hereditary) and in terms of images under quotient maps. For previous posts in this blog on first countable spaces and quotient spaces, see the links at the end of the post. Excellent texts on general topology are [1] and [2].

For a subsequent discussion on sequential space, see Sequential spaces, II.

The Forward Implications

First countable $\Rightarrow \$ Frechet
Suppose $x \in \overline{A}$ and $A \subset X$. Let $U_1,U_2,\cdots$ be a local base at $x$. Then choose $x_n \in A \cap U_n$ and we have $x_n \mapsto x$.

Frechet $\Rightarrow \$ Sequential
Let $A \subset X$ be sequentially closed in $X$. Suppose $A$ is not closed in $X$. Then there is $x \in \overline{A}$ such that $x \notin A$. By Frechet, there is a sequence $x_n \in A$ such that $x_n \mapsto x$. Since $A$ is sequentially closed, $x \in A$, a contradiction. So any sequentially closed set in a Frechet space must be a closed set.

Sequential $\Rightarrow \$ k-space
Suppose $A \subset X$ is not closed in $X$. Since $X$ is sequential, there is a sequence $x_n \in A$ such that $x_n \mapsto x$ and $x \notin A$. The set $K=\left\{x_n:n=1,2,3,\cdots\right\} \cup \left\{x\right\}$ is a compact set. Note that $A \cap K$ is not closed in $K$. This shows that $X$ is a k-space.

Sequential Spaces

Sequential spaces are ones in which the topology can be completely described by convergent sequences. Let $X$ be a space. Let $A \subset X$. The set $A$ is said to be sequentially closed in $X$ if whenever we have a convergent sequence of points in $A$, the sequential limit must be in $A$. In other words, $A$ contains all the limits of the convergent sequences of points in $A$. For $U \subset X$, $U$ is sequentially open in $X$ if this condition holds: if $\left\{x_n \in X: n=1,2,3,\cdots\right\}$ is a sequence of points converging to some $x \in U$, then $x_n \in U$ for all but finitely many $n$. It can be verified that:

$U$ is sequentially open in $X$ if and only if $X-U$ is sequentially closed in $X$.

Theorem 1
A space $X$ is Frechet if and only if every subspace of $X$ is sequential.

Proof
$\Rightarrow$ Suppose $W \subset X$ is not a sequential space. Then there is $A \subset W$ such that $A$ is sequentially closed in $W$ but $A$ is not closed in $W$. We show that $X$ is not Frechet. There is a point $w \in W$ such that $w$ is a limit point of $A$ (in the subspace $W$) and $w \notin A$. Since $A$ is sequentially closed, no sequence of points in $A$ can converge to $w$ (otherwise $w \in A$).

Clearly, the point $w$ is also a limit point of $A$ with respect to the toplology of $X$, i.e. $w \in \overline{A}$ with respect to $X$. Since no sequence of points in $A$ can converge to $w$, $X$ is not Frechet.

$\Leftarrow$ Suppose $X$ is not Frechet. Then there is $x \in \overline{A}$ such that $A \subset X$ and no sequence of points in $A$ can converge to $x$. Consider the subspace $Y=A \cup \left\{x\right\}$. The set $Y-\left\{x\right\}$ is sequentially closed in $Y$ but is not closed in $Y$. $\blacksquare$

Theorem 2
Every quotient space of a sequential space is always a sequential space.

Proof. Let $X$ be a sequential space. Let $f:X \mapsto Y$ be a quotient map. We show that $Y$ is a sequential space. Suppose that $B \subset Y$ is sequentially closed. We need to show that $B$ is closed in $Y$. Because $f$ is a quotient map, $B$ is closed in $Y$ if and only if $f^{-1}(B)$ is closed in $X$. So we need to show $f^{-1}(B)$ is closed in $X$.

Suppose $x_n \in f^{-1}(B)$ for each $n=1,2,3,\cdots$ and the sequence $x_n$ converges to $x \in X$. Then $f(x_n) \in B$. Since the map $f$ is continuous, $f(x_n) \mapsto f(x)$. Since $B$ is sequentially closed, $f(x) \in B$. This means $x \in B$. Thus $f^{-1}(B)$ is sequentially closed in $X$. Since $X$ is sequential, $f^{-1}(B)$ is closed in $X$. $\blacksquare$

Corollary 3
Every quotient space of a first countable space is sequential. Every quotient space of a Frechet space is sequential.

Some Examples

Example 1. First countable $\nLeftarrow \$ Frechet
The example is defined in the post An example of a quotient space, I. This is a non-first countable example. There is only one non-isolated point $p$ in the space. It is easy to verify it is a Frechet space.

Example 2. Frechet $\nLeftarrow \$ Sequential
We consider the space $Y$ defined in the post An example of a quotient space, II. Note that the space $Y$ is the quotient image of a first countable space. Thus $Y$ is sequential by Corollary 3. Consider the subspace $Z=\left\{(0,0)\right\} \cup V$. Within $Z$, no sequence of points in $V$ can converge to the point $(0,0)$. However, $(0,0)$ is a limit point of $V$. Thus $Z$ is not sequential. By Theorem 1, $Y$ is not Frechet.

Example 3. Sequential $\nLeftarrow \$ k-space
Any compact space is a k-space. Let $\omega_1$ be the first uncountable ordinal. Then $\omega_1+1=[0,\omega_1]$ with the ordered topology is compact. Note that $[0,\omega_1)$ is sequentially closed but not closed. Thus $[0,\omega_1]$ is not sequential.

Example 4. A space that is not a k-space
This example is also defined in the post An example of a quotient space, II. Consider the subspace $Z=\left\{(0,0)\right\} \cup V$. Every compact subset of $Z$ is finite. So $V \cap K$ is closed in $K$ for every compact $K \subset Z$. But $V$ is not closed.

Which of the four properties discussed here are preserved in subspaces? Or which of them are hereditary? It is fairly straightforward to verify that first countability is hereditary and so is the property of being Frechet. By Theorem 1, for any sequential space that is not Frechet has a subspace that is not sequential. Thus the property of being a sequential space is not hereditary. However, closed subspaces and open subspaces of a sequential space are sequential.

The property of being a k-space is also not hereditary. The space $Y$ defined in An example of a quotient space, II is a sequential space (thus a k-space). Yet the subspace $Z=\left\{(0,0)\right\} \cup V$ is not a k-space.

Continuous image of a first countable space needs not be first countable. The other three properties (Frechet, sequential and k-space) are also not necessarily preserved by continuous mappings. A quick example is to consider any space $X$ that does not have any one of the four properties. Then consider $D=X$ with the discrete topology. Then the indentity map from $D$ onto $X$ is continuous.

Example 1 shows that the property of being first countable is not preserved by quotient map. Example 2 shows that the Frechet property is not preserved by quotient map. Theorem 2 shows that the property of being sequential space is preserved by quotient map. We have the following theorem about k-spaces under quotient map.

k-spaces

The spaces that are k-spaces are called compactly generated spaces. In a k-space, the closed sets and open sets are generated by compact sets. For example, for a k-space $X$, $A \subset X$ is closed in $X$ if and only if $A \cap K$ is closed in $K$ for every compact $K \subset X$. Let’s take another look at sequential spaces. The following definition is equivalent to the definition of sequential space given above:

$A \subset X$ is closed in $X$ if and only if $A \cap K$ is closed in $K$ for every compact $K \subset X$ of the form $\left\{x\right\} \cup \left\{x_1,x_2,x_3,\cdots\right\}$ where the $x_n$ are a convergent sequence and $x$ is the sequential limit.

Thus the sequential spaces are compactly generated by a special type of compact sets, namely the convergent sequences.

Theorem 4
Quotient images of k-spaces are always k-spaces.

Proof. Let $X$ be a k-space. Let $f:X \mapsto Y$ be a quotient map. We wish to show that $Y$ is a k-space. Suppose $B \subset Y$ is not closed in $Y$. Since $f$ is a quotient mapping, $f^{-1}(B)$ is not closed in $X$. Since $X$ is a k-space, there is a compact $K \subset X$ such that $f^{-1}(B) \cap K$ is not closed in $K$. Let $x \in K$ such that $x \in \overline{f^{-1}(B) \cap K}-(f^{-1}(B) \cap K)$. We have just produced a compact set $f(K)$ in $Y$ such that $B \cap f(K)$ is not closed in $f(K)$. Note that $f(x) \in f(K)$ and $f(x)$ is a limit point of $B \cap f(K)$. This implies that if $B \cap C$ is closed in $C$ for every compact $C \subset Y$, then $B$ must be closed in $Y$ (i.e. $Y$ is a k-space). $\blacksquare$

The discussion on sequential space continues with the post Sequential spaces, II.

Reference

1. Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
2. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.

# An example of a quotient space, II

This post presents another example of a quotient space of a first countable space. The resulting quotient space is not first countable. After we first define the space without referring to the concepts of quotient space, we show that this example is a quotient space. This example will be further discussed when we discuss sequential spaces.

For a previous discussion on quotient space in this blog, see An example of a quotient space, I. For more information on quotient spaces in general, see [2].

Let $\mathbb{N}$ be the set of all positive integers. For each $i \in \mathbb{N}$ and $j \in \mathbb{N}$, let $V_{i,j}=\left\{(\frac{1}{i},\frac{1}{k}):k \ge j\right\}$. Let $V=\bigcup \limits_{i=1}^{\infty} V_{i,1}$. Let $H=\left\{(\frac{1}{i},0):i \in \mathbb{N}\right\}$. For each $i \in \mathbb{N}$, let $V_i=V_{i,1} \cup \left\{(\frac{1}{i},0)\right\}$. Define the space $Y$ as:

$Y=\left\{(0,0)\right\} \cup H \cup V$.

The set $H$ is the horizontal part of the space and the set $V$ is the vertical part of the space. The origin $(0,0)$ is an additional point. In the topology for the space $Y$, each point in $V$ is isolated. Each point $(\frac{1}{i},0) \in H$ has an open neighborhood of the form

$\left\{(\frac{1}{i},0)\right\} \cup V_{i,j}$ for some $j \in \mathbb{N}$

The open neighborhoods at $(0,0)$ are obtained by removing finitely many $V_i$ from $Y$ and by removing finitely many isolated points in the $V_i$ that remain. The open neighborhoods just defined form a base for a topology on the set $Y$, i.e. by taking unions of these open neighborhoods, we obtain all the open sets for this space.

We wish to discuss $Y$ and its subspace $Z=\left\{(0,0)\right\} \cup V$. If we consider $Z$ as a topological space in its own right (open sets in $Z$ are of the form $U \cap Z$ where $U$ is open in $Y$), then in $Z$ there are no infinite compact sets (all compact subsets of $Z$ are finite). In particular, no sequence of points $\left\{x_n \in V:n \in \mathbb{N}\right\}$ can converge to the point $(0,0)$. However, with respect to $Z$, $(0,0)$ is a limit point of $V$. This implies that the space $Y$ is not first countable.

A space $W$ is said to be a Frechet space if $w \in \overline{A}$ where $A \subset W$, then there is some sequence $\left\{w_n \in A:n \in \mathbb{N}\right\}$ that converges to the point $w$. Any first countable space is a Frechet space. If there is a local base $\left\{U_n: n \in \mathbb{N}\right\}$ at $w$ and if $w \in \overline{A}$, then whenever we pick $w_n \in A \cap U_n$, $w_n$ would converge to $w$.

The space $Y$ defined above is not a Frechet space. Note that $(0,0) \in \overline{V}$ and no sequence of points in $V$ can converge to $(0,0)$.

Though the space $Y$ defined above is not a first countable space, $Y$ is a quotient space of a first countable space $X$ (in fact, a subspace of the Euclidean plane). Let $E=H \cup V$ where $H$ and $V$ are defined as in the definition of the space $Y$ above. Let $H_{-1}$ be the set $H_{-1}=\left\{(\frac{1}{i},-1):i \in \mathbb{N}\right\}$. Let $X=E \cup H_{-1} \cup \left\{(0,-1)\right\}$ with the Euclidean topology inherited from the Euclidean plane.

Define a quotient space from $X$ by collapsing each pair of points $\left\{(\frac{1}{i},0),(\frac{1}{i},-1)\right\}$ into one point $\left\{(\frac{1}{i},0)\right\}$. This is done for each $i \in \mathbb{N}$. For convenience, the point $\left\{(0,-1)\right\}$ is shifted to $\left\{(0,0)\right\}$. The resulting quotient space is the same as the space $Y$ defined above.

The quotient space we just describe can also be described by the quotient map $f:X \mapsto Y$:

$f((0,-1))=(0,0)$,

$f((\frac{1}{i},-1))=(\frac{1}{i},0)$ for each $(\frac{1}{i},-1) \in H_{-1}$,

$f((\frac{1}{i},0))=(\frac{1}{i},0)$ for each $(\frac{1}{i},0) \in H$,

$f(x)=x$ for each $x \in V$.

Then the following topology coincides with the topology on $Y$ that we define earlier in the post:

$\tau_f=\left\{U \subset Y: f^{-1}(U) \text{ is open in } X\right\}$

Reference

1. Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
2. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.

# An example of a quotient space, I

It is well known that in a compact Hausdorff space, if every point in the space is a $G_\delta$ point, then the space is first countable (see The cardinality of compact first countable spaces, II). Outside of compact spaces, this is not true. We present an example of a space where every point is a $G_\delta$ point but has no countable local base at any one point. This is an opportunity to introduce the notion of quotient space. Our example can be viewed as the image of a first countable space under a quotient map. We first define the example and then present a brief discussion of quotient spaces.

Let $\mathbb{N}$ be the set of all positive integers. Let $W=\left\{\frac{1}{n}: n \in \mathbb{N}\right\}$. Let $S_1=\mathbb{N} \times W$ and $S_2=\mathbb{N} \times \left\{0\right\}$. Consider $X=S_1 \cup S_2$ as a subspace of the Eucliean plane. In the Euclidean topology, each point is $S_1$ is isolated and an open neighborhood at each point $(i,0)$ in $S_2$ is of the form:

$\left\{(i,0)\right\} \cup \left\{(i,\frac{1}{k}):k \ge n\right\}$ for some $n \in \mathbb{N}$

We use $X$ to define a space $Y$ by considering $S_2$ as one point (call this point $p$). Thus $Y=S_1 \cup \left\{p\right\}$. Points in $S_1$ remain isolated. An open neighborhood of $p$ is of the form $\left\{p\right\} \cup U^-$ where $U^-=U-S_2$ and $U$ is a Euclidean open subset of $X$ such that $S_2 \subset U$.

To facilitate the discussion, we describe the open sets at the point $p$ in more details. Let $\mathbb{N}^{\mathbb{N}}$ be the set of all functions $f:\mathbb{N} \mapsto \mathbb{N}$. For each $i \in \mathbb{N}$ and $j \in \mathbb{N}$, let $V_{i,j}=\left\{(i,\frac{1}{k}): k \ge j\right\}$. An open neighborhood of $p$ is of the following form:

$B_f=\left\{p\right\} \cup \bigcup \limits_{i=1}^{\infty} V_{i,f(i)}$

To describe in words, each open neighborhood at $p$ is obtained by removing finitely many points in each vertical segment $V_{i,1}$.

The resulting space $Y$ is not first countable. We show that any countable set of open neighborhoods at $p$ cannot be a local base at $p$. To this end, we consider $B_{f_1},B_{f_2},B_{f_3},\cdots$ where each $f_i \in \mathbb{N}^{\mathbb{N}}$. We wish to find an open set $O$ such that $p \in O$ and each $B_{f_i}$ is not a subset of $O$.

For each $i \in \mathbb{N}$, pick some $g(i) \in \mathbb{N}$ such that $g(i)>f_i(i)$. Consider $O=B_g$. Note that for each $i \in \mathbb{N}$, $(i,\frac{1}{f_i(i)}) \in B_{f_i}$ and $(i,\frac{1}{f_i(i)}) \notin O=B_g$. This shows that no countable number of open neighborhoods can form a base at the point $p$.

However, every point in the space $Y$ is a $G_\delta$ point. For the point $p$, we have $\left\{p\right\}=\bigcap \limits_{i=1}^\infty B_{h_i}$ where $h_i \in \mathbb{N}^{\mathbb{N}}$ is a constant function mapping to $i$.

Quotient Spaces
We now present one definition of quotient spaces. Let $(X,\tau)$ be a topological space. Let $q:X \mapsto Y$ be a surjection, i.e. $q(X)=Y$. Consider the following collection of subsets of $Y$:

$\tau_q=\left\{O \subset Y: q^{-1}(O) \text{ is open in }X\right\}$

The set $\tau_q$ is a topology for the set $Y$. With this topology $\tau_q$ on $Y$, the function $q$ is continuous. In fact, $\tau_q$ is the finest (largest or strongest) topology on $Y$ that makes the function $q$ continuous, i.e. if $\tau_1$ is another topology on $Y$ making $q$ a continuous function, then $\tau_1 \subset \tau_q$.

The topology $\tau_q$ is called the quotient topology induced on $Y$ by the mapping $q$. When $Y$ is given such a quotient topology, it is called a quotient space of $X$ (most of the times just called quotient space). The induced map $q$ is called a quotient map. If $Y$ has a quotient topology defined by a quotient map, then $Y$ is said to be the quotient image of $X$. We plan to discuss quotient space and quotient topology in greater details in this blog. For further information, see [2].

In the example discussed in this post, the quotient map is to map each point in $S_1$ to itself and to map each point in $S_2$ to a single point $p$. In essence, we collapse the whole x-axis $S_2$ into one single point. The open sets for the point $p$ are simply the Euclidean open sets containing $S_2$. This example shows that the quotient image of a first countable space needs not be first countable. Though the quotient image of a first countable space may not be first countable, it has the property that sequences suffice to define the topology (these are called sequential spaces, see [1]).

Reference

1. Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
2. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.


text:
# Beamer ignores citecolor from hyperref

How do I make citation links green in Beamer? It seems that Beamer ignores the hyperref option citecolor. I tried using the linkcolor option instead, but that affects all links (such as the link to the next slide, see example below). I want to change only the citation color.

\documentclass{beamer}

\hypersetup{
citecolor = green
}

\begin{document}

\begin{frame}
This is a great result~\cite{Paper}.
But this is a link to slide~\ref{Bib}.
\end{frame}

\begin{frame}{Bibliography}\label{Bib}
\begin{thebibliography}{}
\bibitem[Author, 1942]{Paper}
Author. B.
\newblock The great result.
\end{thebibliography}
\end{frame}

\end{document}

-
Weird: citecolor=green works in article, but not in Beamer. Someone else might know a work around. –  Jubobs Mar 26 '13 at 0:02

You could redefine \@cite to locally change the link color:

\documentclass{beamer}

\hypersetup{
}

\makeatletter
\let\@mycite\@cite
\makeatother

\begin{document}

\begin{frame}
This is a great result~\cite{Paper}.
But this is a link to slide~\ref{Bib}.
\end{frame}

\begin{frame}{Bibliography}\label{Bib}
\begin{thebibliography}{}
\bibitem[Author, 1942]{Paper}
Author. B.
\newblock The great result.
\end{thebibliography}
\end{frame}

\end{document}

-

The above solution did not work for me. But I was successful by putting the following line in the preamble:

\let\oldcite=\cite
\renewcommand{\cite}[1]{\textcolor[rgb]{.7,.7,.7}{\oldcite{#1}}}

-
Welcome to TeX.SX! You can have a look at our starter guide to familiarize yourself further with our format. –  Paul Gessler Mar 11 '14 at 22:10

format: plain


text:
# Ubuntu – How to scp files of size less than x

command linescp

I am developing something on my laptop and scp'ing the files to a high performance PC (a server with no monitor). Out of laziness I just do this currently (on the laptop):

scp -r ~/project/* server:~/project/

However some of the files in the project directory are some log files and out files which I don't need to send, which can sometimes be a gig or so. What I would like is a nice clean command which basically does the same thing (transmits all the files in project and its subdirectories) but only sends files under a certain filesize threshold, such as 100 mb. I'm only mildly experienced with bash and not at all with regexp so I don't really have any idea how to do this without some convoluted method such as importing sys into python and using a bunch of ifs and comparisons.

#### Best Answer

you can use 'rsync' command which has an option to controlling the file size to be excluding.

rsync -rv --max-size=100m /path/in/local/ server:~/project/

from 'man rsync',

--max-size=SIZE    don't transfer any file larger than SIZE

Updated answer : simply exclude the log files with 'rsync' --exclude option as below

rsync -rv --exclude='*.log' --max-size=100m /path/in/local/ server:~/project/

or you can use the 'find' command combination with 'scp' and specify the file size limit with its -size option. use ! -name '*.log' to exclude the certain files end with 'log'.

find /path/in/local -type f ! -name '*.log' -size -100m -exec scp '{}' server:~/project/ \;


text:
# Changes between Version 287 and Version 288 of ReviewHeuristicLab3.3.0Application

Ignore:
Timestamp:
05/07/10 01:01:22 (12 years ago)
Comment:

--

### Legend:

Unmodified
 v287 * swagner: Setting parameters automatically does not work in user defined algorithms. In a user defined algorithm all parameter settings (values, actual names, etc.) have to be set manually by the user. * The name of data analysis solutions is not set or displayed correctly. * Read-only parameters would be nice (similar to the data items of HeuristicLab.Data) === Priority: MEDIUM ===


text:
## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$(2y+3)(6y-5)$
The given expression, $8y-15+12y^2 ,$ can be re-written as \begin{array}{l} 12y^2+8y-15 .\end{array} The two numbers whose product is $ac= 12(-15)=-180$ and whose sum is $b= 8$ are $\{ 18,-10 \}$. Using these two numbers to decompose the middle term of the expression, $8t^2+14t-15 ,$ then the factored form is \begin{array}{l} 12y^2+18y-10y-15 \\\\= (12y^2+18y)-(10y+15) \\\\= 6y(2y+3)-5(2y+3) \\\\= (2y+3)(6y-5) .\end{array}


text:
# Plot Calculated Errors in an existing Histogram

I plotted the histogram given below:

``````
h221=(TH1D*)f->Get("hrate_all_rule0");
``````

I calculated the errors (not the one produced by GetBinError, default by ROOT) for the above histogram and stored them in:

``````
TH1D *errACP0NH_295 =(TH1D*)sumACP0NH_295->Clone("errACP0NH_295");
``````

Now I want to call the BinContent from ** errACP0NH_295* and plot as vertical error bars for the histogram h221 on the same pad.

How should I do it?
ROOT Version: 6.24/02
Platform: Focal Release 20.04
Compiler: C++

Hi @Ankur,

In this case, you have to explicitly set the errors of `h221` to be the content of `errACP0NH_2095`:

``````for(int i=0 ; i <= h221->GetNbinsX() ; i++){
h221->SetBinError(i, errACP0NH_2095->GetBinContent(i));
}
``````

If you don’t want to modify the original histogram, you can create a `Clone()` of it.

Hope this helps!
Jonas

Thank you @jonas for your reply. Before proceeding, let me confirm if you mean ` errACP0NH_2095->GetBinContent` instead of `errACP0NH_2095->GetBinError(i)`? My idea is that the errors for h221 are calculated and stored in errACP0NH_2095.

Yes, sorry that’s what I meant! I have edited my post.

1 Like

Thank you, Get back to you shortly.

@jonas Sorry for the late reply. I was adding few others into my code. The technique works perfectly fine. Thank You.


text:
## Monday, October 25, 2004

I'm still getting used to doing this, and I think I prefer the blogger interface/experiance to livejournal. It's still not a mathML enabled blog, but I really don't have the time to set up a proper server, etc, yet. So for now, I'm sticking to this.

In the last post I talked about determining the strong coupling constant from lattice QCD. You'll recall that I talked about having to use a special coupling, determined from the perturbative static quark potential. I also mentioned that the perturbative scale for the average plaquette was not $\pi/a$, but was instead some scale $q^{*}$. The next four posts will explain those cryptic comments. They will basically be a summery of a famous 1992 paper by Lepage and Mackenzie, in which they saved lattice perturbation theory.

We'll start by introducing the problem.

The procedure I outlined, for determining the strong coupling constant, could also be run in reverse. That is, I'll start out knowing the coupling constant at $M_{Z}$ run it down to the lattice scale $\pi/a$ and convert it to the bare lattice coupling. Then I use that to compute the second order value of the average plaquette. I get some number from this, X. Now I fire up my Monte-Carlo simulation, and I compute the average plaquette that way, and I get some number Y. The trouble is X and Y are not equal. Even if I allow for an error coming from third order perturbation theory, it's way, way off.

This is a disaster. It means that peturbation theory on the lattice is suspect. You'll recall that the coefficients in improved actions were to be determined perturbativly, and using improved actions was crucial in error reduction. If lattice perturbation theory fails, we can no longer do this, and we'll be forced to wait many years until computers get faster. We need to figure out what is wrong with lattice perturbation theory, and, if possible, fix it.

The next post will talk about problem #1, a bad choice of coupling constant.


text:
# closed subset of locally compact

A space $X$ is said locally compact if for any $x\in X$ and for any neighbourhood $U$ of $x$ there is a compact neighbourhood $V$ such that $V\subseteq U$.

Does closed subset of locally compact is locally compact?

My friend said that $\{1/n : n \in\mathbb{N}\} \cup \{0\}$ subset of $\mathbb{R}$ is not locally compact. Is it true?

• What are your thoughts so far? What have you tried? The more you can tell us about your thoughts and efforts, the easier it will be for us to tailor our answers to your needs. We don't want to tell you a bunch of things you already know, nor do we want to try to explain using a bunch of concepts with which you're unfamiliar. Apr 5, 2015 at 15:13
• I think that $\{1/n: n \in \mathbb{N}\} \cup \{0\}$ is not locally compact because there is no a compact neighbourhood of $0$. But it contradicts to this property [math.stackexchange.com/questions/698029/… Apr 5, 2015 at 15:25
• Who says there's no compact neighborhood of $0$? Doesn't $[-1,1]$ qualify? Apr 5, 2015 at 15:33
• In fact, $\{1/n:n\in\Bbb N\}\cup\{0\}$ is compact, and so locally compact. You can also see that it's closed and contained in $[-1,1],$ which is compact, so by the result in the link you posted, $\{1/n:n\in\Bbb N\}\cup\{0\}$ is compact Apr 5, 2015 at 15:34

If $F$ is a closed subset of $X$, with relative topology, then $\forall x\in F$, $\forall U$ is a neighborhood of $x$ in $X$, such that $\exists V\subset U$ is a neighborhood of $x$ in $X$, and $\overline V$ is compact. Hence, for every neighborhood $U_0$ in $F$, since $U_0=U\cap F$ for some $U$, we know $V\cap F$ satisfies the condition. The last assertion follows from the fact that $V\cap F$ is a closed subset of a compact set.


text:
# Biblatex + biber - Sort citations by report issue

I'm trying to sort some tech report citations from the same year in the bibliography by report issue number, but have got no chance so far, i've even tried adding the month in order to make them sort chronologically but it does not have any effect.

myreferences.bib

@techreport{VanBuskirk2014a,
author = {{Van Buskirk}, Joe and Roxburgh, Amanda and Bruno, Raimondo and Burns, Lucinda},
institution = {National Drug and Alcohol Research Centre},
month = {mar},
number = {2},
pages = {1--14},
title = {{Drugs and The Internet}},
year = {2014}
}

@techreport{VanBuskirk2014b,
author = {{Van Buskirk}, Joe and Roxburgh, Amanda and Bruno, Raimondo and Burns, Lucinda},
institution = {National Drug and Alcohol Research Centre},
month = {sep},
number = {3},
pages = {1--5},
title = {{Drugs and The Internet}},
year = {2014}
}

As i cite the VanBuskirk2014b before the VanBuskirk2014a it's ordered and numbered the wrong way, the september report gets the "a" key and is before in the bibliography whereas the march report gets the "b" key and goes after in the bibliography.

Here a reduced version to illustrate the problem:

test.tex

\documentclass[12pt,a4paper,twoside]{report}
\usepackage[bindingoffset=0.5cm, left=3cm, right=3cm, top=2.5cm, bottom=2.5cm]{geometry}
\usepackage[french]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}

\usepackage{csquotes}
\usepackage[backend=biber, style=apa, sortcites=true]{biblatex}
\DeclareLanguageMapping{french}{french-apa}
\setlength\bibitemsep{1.5\itemsep}

\begin{document}

\autocite{VanBuskirk2014b}\\

\autocite{VanBuskirk2014a}

\printbibliography

\end{document}

• Adding the month doesn't help as all the standard sort schemes only consider the year when sorting, see biblatex sorting by date for sorting by the entire date. – moewe May 6 '16 at 15:17

You can use the sortyear field to control the sorting manually, use it as sortyear = {<year>-<number>}

@techreport{VanBuskirk2014a,
author = {{Van Buskirk}, Joe and Roxburgh, Amanda and Bruno, Raimondo and Burns, Lucinda},
institution = {National Drug and Alcohol Research Centre},
month = {mar},
number = {2},
pages = {1--14},
title = {{Drugs and The Internet}},
year = {2014},
sortyear = {2014-2},
}

@techreport{VanBuskirk2014b,
author = {{Van Buskirk}, Joe and Roxburgh, Amanda and Bruno, Raimondo and Burns, Lucinda},
institution = {National Drug and Alcohol Research Centre},
month = {sep},
number = {3},
pages = {1--5},
title = {{Drugs and The Internet}},
year = {2014},
sortyear = {2014-3},
}

Technically, we could really sort by number as well, but since there is no way to restrict sorting to one type (see the recent Different sorting schemes for different entry types? and how to adapt the sorting of a custom type @standard in using biblatex/biber) that could mess up the sorting for other types that use number.

This is a version of the standard nyvt sorting, where the volume is also considered.

\DeclareSortingScheme{nynot}{
\sort{
\field{presort}
}
\sort[final]{
\field{sortkey}
}
\sort{
\field{sortname}
\field{author}
\field{editor}
\field{translator}
\field{sorttitle}
\field{title}
}
\sort{
\field{sortyear}
\field{year}
}
\sort{

• Hi, would you mind taking a look at this? You have far more experience with biblatex than me, and even i can see, that this is Bäh. I would hate if that gets distributed even further. Thanks. – Johannes_B May 10 '16 at 5:05
• @Johannes_B First off, the call to biblatex contains quite a few options, seeing that the drivers are rewritten from scratch et least the doi, isbn, url and eprint options are practically useless. The comment for the backref option is wrong (of course backref=false turns backrefs off). The \AtBeginEnvironment{thebibliography} block looks a bit fishy to me, but I don't know about that (I wouldn't change penalties unless I have to). The \DeclareFieldFormats could be combined into one \DeclareFieldFormat*{title}{#1\par} and I'm not too fond of the \par in there. – moewe May 10 '16 at 5:14
• ... (cont'd) The newly rewritten drivers are of course much simpler than the standard drivers. I don't like the \pars and the font commands, but OK. The \par\vspace{0.3\baselineskip} at the end looks fishy, one could better use \bibitemsep or similar for that. The \DeclareNameFormat{author} will of course explode with biblatex >= 3.3. The use of refsections for type filtering is also a bit naff, one could just use the same refsection everywhere and type=... in the option to \printbibliography. – moewe May 10 '16 at 5:19
• The original friggery-cv template already went havoc :-) I said i'll fix when 3.4 is around. – Johannes_B May 10 '16 at 5:20


text: Last 7 days     Results 61-80 of 314.   1 2 3 4 5 6 7 8 9   Thermotropic Liquid Crystal-Assisted Chemical and Biological SensorsPopov, Nicolai; Honaker, Lawrence William ; Popova, Maia et alin Materials (2017), 11(20), In this review article, we analyze recent progress in the application of liquid crystal-assisted advanced functional materials for sensing biological and chemical analytes. Multiple research groups ... [more ▼]In this review article, we analyze recent progress in the application of liquid crystal-assisted advanced functional materials for sensing biological and chemical analytes. Multiple research groups demonstrate substantial interest in liquid crystal (LC) sensing platforms, generating an increasing number of scientific articles. We review trends in implementing LC sensing techniques and identify common problems related to the stability and reliability of the sensing materials as well as to experimental set-ups. Finally, we suggest possible means of bridging scientific findings to viable and attractive LC sensor platforms. [less ▲]Detailed reference viewed: 11 (1 UL) Computational Sciences and the Transition to Data-Driven Modelling and Simulation: Case studies in Engineering & Personalised Medicine - Presentation for the 50 years of Inria and the 10 years of ERCBordas, Stéphane Presentation (2017, December 20)Detailed reference viewed: 80 (1 UL) Preparing Future Teachers for Strategic Uses of Educational Technology: A Project-Based ApproachReuter, Bob ; Busana, Gilbert Scientific Conference (2017, December 01)Given (1) the need for teachers who are willing and able to design, implement and reflect meaningful and successful educational technology practices in schools and (2) the current rarity of strategic uses ... [more ▼]Given (1) the need for teachers who are willing and able to design, implement and reflect meaningful and successful educational technology practices in schools and (2) the current rarity of strategic uses of ICT in education in Luxembourgish schools, the self-declared lack of competencies and the low confidence of teachers when it comes to teaching with ICT (Reuter, Busana & Linckels, 2016), we have designed, implemented and tested a project-based approach to the development of practice- oriented TPACK in pre-service teachers. Future teacher students are instructed to define an educational problem they want to solve, to explore existing pedagogical and technological solutions and to design an original solution that involves the strategic use of ICT (theoretically grounded and practically meaningful). They then have to implement the designed learning & teaching scenario in a classroom, to document the teaching & learning processes, to analyse, evaluate and to critically reflect them. Finally, they are required to write a structured report about their project and to present it to their peers and to interested teachers from schools. Overall, we observe that while most students become able to design and implement relatively meaningful educational technology practices, they still tend to feel that they are not ready to teach with technology in a more general sense, because their work was focused on one or two tools and does not give them access to a broader range of meaningful ICT-enriched teaching scenarios for their later professional career. [less ▲]Detailed reference viewed: 28 (2 UL) Fast Discretized Gaussian Sampling and Post-quantum TLS CiphersuiteXinwei, Gao; Li, Lin; Jintai, Ding et alin The 13th International Conference on Information Security Practice and Experience - ISPEC 2017 (2017, December 01)Detailed reference viewed: 5 (0 UL) Evaluating the Impact of Juice Filming Charging Attack in Practical EnvironmentsMeng, Weizhi; Lee, Wanghao; Liu, Zhe et alin The 20th Annual International Conference on Information Security and Cryptology - ICISC2017 (2017, December 01)Detailed reference viewed: 4 (0 UL) Facing the Safety-Security Gap in RTES: the Challenge of TimelinessVolp, Marcus ; Kozhaya, David ; Verissimo, Paulo Scientific Conference (2017, December)Safety-critical real-time systems, including real-time cyber-physical and industrial control systems, need not be solely correct but also timely. Untimely (stale) results may have severe consequences that ... [more ▼]Safety-critical real-time systems, including real-time cyber-physical and industrial control systems, need not be solely correct but also timely. Untimely (stale) results may have severe consequences that could render the control system’s behaviour hazardous to the physical world. To ensure predictability and timeliness, developers follow a rigorous process, which essentially ensures real-time properties a priori, in all but the most unlikely combinations of circumstances. However, we have seen the complexity of both real-time applications, and the environments they run on, increase. If this is matched with the also increasing sophistication of attacks mounted to RTES systems, the case for ensuring both safety and security through aprioristic predictability loses traction, and presents an opportunity, which we take in this paper, for discussing current practices of critical realtime system design. To this end, with a slant on low-level task scheduling, we first investigate the challenges and opportunities for anticipating successful attacks on real-time systems. Then, we propose ways for adapting traditional fault- and intrusiontolerant mechanisms to tolerate such hazards. We found that tasks which typically execute as analyzed under accidental faults, may exhibit fundamentally different behavior when compromised by malicious attacks, even with interference enforcement in place. [less ▲]Detailed reference viewed: 14 (0 UL) Symbol-level Precoding for the Non-linear Multiuser MISO Downlink ChannelSpano, Danilo ; Alodeh, Maha; Chatzinotas, Symeon et alin IEEE Transactions on Signal Processing (2017)This paper investigates the problem of the interference among multiple simultaneous transmissions in the downlink channel of a multi-antenna wireless system. A symbol-level precoding scheme is considered ... [more ▼]This paper investigates the problem of the interference among multiple simultaneous transmissions in the downlink channel of a multi-antenna wireless system. A symbol-level precoding scheme is considered, in order to exploit the multi-user interference and transform it into useful power at the receiver side, through a joint utilization of the data information and the channel state information. In this context, this paper presents novel strategies which exploit the potential of symbol-level precoding to control the per-antenna instantaneous transmit power. In particular, the power peaks amongst the transmitting antennas and the instantaneous power imbalances across the different transmitted streams are minimized. These objectives are particularly relevant with respect to the non-linear amplitude and phase distortions induced by the per-antenna amplifiers, which are important sources of performance degradation in practical systems. More specifically, this work proposes two different symbol-level precoding approaches. A first approach performs a weighted per-antenna power minimization, under Quality-of-Service constraints and under a lower bound constraint on the per-antenna transmit power. A second strategy performs a minimization of the spatial peak-to-average power ratio, evaluated amongst the transmitting antennas. Numerical results are presented in a comparative fashion to show the effectiveness of the proposed techniques, which outperform the state of the art symbol-level precoding schemes in terms of spatial peak-to-average power ratio, spatial dynamic range, and symbol-error-rate over non-linear channels. [less ▲]Detailed reference viewed: 17 (5 UL) Systembasierte Analyse von Wirkstoffresistenzen bei MelanomLucarelli, Philippe ; De Landtsheer, Sébastien ; Sauter, Thomas E-print/Working paper (2017)Detailed reference viewed: 9 (3 UL) Bei Innovationen geht es nicht um Wachstum oder Upscaling, sondern um Ubiquität und ExemplaritätReckinger, Rachel Article for general public (2017)Detailed reference viewed: 10 (1 UL) Isolation of nucleic acids from low biomass samples: detection and removal of sRNA contaminantsHeintz-Buschart, Anna; Yusuf, Dilmurat; Kaysen, Anne et alE-print/Working paper (2017)Sequencing-based analyses of low-biomass samples are known to be prone to misinterpretation due to the potential presence of contaminating molecules derived from laboratory reagents and environments. Due ... [more ▼]Sequencing-based analyses of low-biomass samples are known to be prone to misinterpretation due to the potential presence of contaminating molecules derived from laboratory reagents and environments. Due to its inherent instability, contamination with RNA is usually considered to be unlikely. Here we report the presence of small RNA (sRNA) contaminants in widely used microRNA extraction kits and means for their depletion. Sequencing of sRNAs extracted from human plasma samples was performed and significant levels of non-human (exogenous) sequences were detected. The source of the most abundant of these sequences could be traced to the microRNA extraction columns by qPCR-based analysis of laboratory reagents. The presence of artefactual sequences originating from the confirmed contaminants were furthermore replicated in a range of published datasets. To avoid artefacts in future experiments, several protocols for the removal of the contaminants were elaborated, minimal amounts of starting material for artefact-free analyses were defined, and the reduction of contaminant levels for identification of bona fide sequences using 'ultra-clean' extraction kits was confirmed. In conclusion, this is the first report of the presence of RNA molecules as contaminants in laboratory reagents. The described protocols should be applied in the future to avoid confounding sRNA studies. [less ▲]Detailed reference viewed: 7 (0 UL) 100 Jaar Prandtl-Wig: De draagkrachtfactorenVan Baars, Stefan in Geotechniek (2017), DecemberIn 1920 publiseerde Prandtl een artikel over het bezwijken van een materiaal onder een strookbelasting. De bezweken grondmoot is hierbij verdeeld in drie zones, die tezamen de zogenoemde Prandtl-wig ... [more ▼]In 1920 publiseerde Prandtl een artikel over het bezwijken van een materiaal onder een strookbelasting. De bezweken grondmoot is hierbij verdeeld in drie zones, die tezamen de zogenoemde Prandtl-wig vormen. Prandtl heeft nooit aan de funderingstechnische toepassing gedacht, maar geotechnici vertalen deze oplossing nu als een analytische oplossing voor de draagkracht van de grond onder een strookfundering. Deze oplossing is uitgebreid door Reissner met een bovenbelasting naast de funderingsstrook en door Keverling Buisman en Terzaghi voor het grondgewicht. Terzaghi schreef dit als eerste met drie gescheiden draagkrachtfactoren en Meyerhof als eerste met “inclinatie-” en “shape” factoren, voor de drie afzonderlijke draagkrachtcomponenten; cohesie, bovenbelasting en het grondgewicht. Deze schrijfwijze werd later door Brinch Hansen overgenomen. In dit artikel zijn een groot aantal numerieke berekeningen gemaakt om het bezwijkmechanisme van de Prandtl-wig en de bijbehorende draagkrachtfactoren te controleren. Het blijkt dat het Prandtl-wig bezwijkmechanisme juist is, maar niet in alle gevallen. Ook blijkt dat de analytische oplossingen voor de draagkrachtfactoren die tegenwoordig worden gebruikt, alleen juist zijn voor grond met een onrealistisch hoge dilatantiehoek. [less ▲]Detailed reference viewed: 6 (0 UL) Modular classes of Q-manifolds: a review and some applicationsBruce, Andrew in Archivum Mathematicum (2017)A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold – which is viewed as the obstruction to the existence of a Q ... [more ▼]A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold – which is viewed as the obstruction to the existence of a Q-invariant Berezin volume – is not well know. We review the basic ideas and then apply this technology to various examples, including $L_{\infty}$-algebroids and higher Poisson manifolds. [less ▲]Detailed reference viewed: 7 (1 UL) Quantum thermodynamics of the resonant-level model with driven system-bath couplingSchmidt, Thomas ; Esposito, Massimiliano ; Haughian, Patrick E-print/Working paper (2017)Detailed reference viewed: 12 (0 UL) Multiagent Deontic Logic and its Challenges from a Normative Systems PerspectivePigozzi, Gabriella; van der Torre, Leon in The IfCoLog Journal of Logics and their Applications (2017)This article gives an overview of several challenges studied in deontic logic, with an emphasis on challenges involving agents. We start with traditional modal deontic logic using preferences to address ... [more ▼]This article gives an overview of several challenges studied in deontic logic, with an emphasis on challenges involving agents. We start with traditional modal deontic logic using preferences to address the challenge of contrary-toduty reasoning, and STIT theory addressing the challenges of non-deterministic actions, moral luck and procrastination. Then we turn to alternative normbased deontic logics detaching obligations from norms to address the challenge of Jørgensen’s dilemma, including the question how to derive obligations from a normative system when agents cannot assume that other agents comply with their norms. We discuss also some traditional challenges from the viewpoint of normative systems: when a set of norms may be termed ‘coherent’, how to deal with normative conflicts, how to combine normative systems and traditional deontic logic, how various kinds of permission can be accommodated, how meaning postulates and counts-as conditionals can be taken into account,how sets of norms may be revised and merged, and how normative systems can be combined with game theory. The normative systems perspective means that norms, not ideality or preference, should take the central position in deontic semantics, and that a semantics that represents norms explicitly provides a helpful tool for analysing, clarifying and solving the problems of deontic logic. We focus on the challenges rather than trying to give full coverage of related work, for which we refer to the handbook of deontic logic and normative systems. [less ▲]Detailed reference viewed: 8 (2 UL) Germany, Greece and Poland: The Complicated story of Reparations and European IntegrationParavantis, Spero Presentation (2017, November 28)Detailed reference viewed: 12 (0 UL) Law and the software development life cycleBartolini, Cesare ; Lenzini, Gabriele Scientific Conference (2017, November 25)The increasing demand of reliable software services and the dependability that our daily personal and professional life have on them is bringing significant changes in the domain of software service ... [more ▼]The increasing demand of reliable software services and the dependability that our daily personal and professional life have on them is bringing significant changes in the domain of software service engineering. One of the most revolutionary is the introduction of regulations, repeating what in the past has concerned the product market. Regulations need to find a balance between the interests of several roles and reduce the inevitable tensions that would otherwise arise among them, as well as to defend the right of the weakest parties (normally the end users). There are multiple interests to balance: the interests of end users, the protection of intellectual property, a fair competition against other enterprises, just to name a few. While some of these requirements concern the structure and organization of the enterprise, some of them are fit to penetrate into the software development life cycle. This would serve multiple purposes: allow the enterprise to design services which already take the legal requirements into account; visually represent the requirements and their interaction with the functionality of the system; develop the software components using tools and methodologies that are able to deal with those requirements; define metrics to measure the degree to which such requirements are met; measure the impact of the requirements on the functionality of the service and on other parameters of the service (such as performance or storage occupation); verify and monitor whether the legal requirements are met; and, last but not least, to have an argument to be used in case of a complaint in a court or at a competent authority. Before being considered in the software service life cycle, legal requirements must undergo a preprocessing phase in which they are translated into some form which is compatible with the tools and methodologies proper of the software engineering, for instance being modelled into a formalism that makes them processable by a machine. There is a significant amount of interdisciplinary topics that need to be combined together to reach an integration between regulation and software life cycle. In particular, at least from three complementary perspectives are needed. One perspective requires the analysis of the provisions of the law, the extraction of the legal requirements classified according to the stakeholders affected, and the translation of those requirements into some formal model that can be processed using appropriate software tools. A second perspective requires a study of the legal requirements from the point of view of requirements engineering techniques, also defining metrics to measure them. The third concerns the models used in the various stages of software engineering (design, modeling, development, validation and testing), which need to be extended to accommodate the legal requirements in their formal representation. Only by putting together these perspectives a comprehensive approach to deal with legal requirements in software engineering is possible. [less ▲]Detailed reference viewed: 7 (0 UL) Loss of Myosin Vb in colorectal cancer is a strong prognostic factor for disease recurrenceLetellier, Elisabeth ; Schmitz, Martine ; Ginolhac, Aurélien et alin British Journal of Cancer (2017), 117(11), 1689-1701Background: Selecting the most beneficial treatment regimens for colorectal cancer (CRC) patients remains challenging due to a lack of prognostic markers. Members of the Myosin family, proteins recognized ... [more ▼]Background: Selecting the most beneficial treatment regimens for colorectal cancer (CRC) patients remains challenging due to a lack of prognostic markers. Members of the Myosin family, proteins recognized to play a major role in trafficking and polarization of cells, have recently been reported to be closely associated with several types of cancer and might thus serve as potential prognostic markers in the context of CRC. Methods: We used a previously established meta-analysis of publicly available gene expression data to analyse the expression of different members of the Myosin V family, namely MYO5A, 5B, and 5C, in CRC. Using laser-microdissected material as well as tissue microarrays from paired human CRC samples, we validated both RNA and protein expression of MYO5B and its known adapter proteins (RAB8A and RAB25) in an independent patient cohort. Finally, we assessed the prognostic value of both MYO5B and its adapter-coupled combinatorial gene expression signatures. Results: The meta-analysis as well as an independent patient cohort study revealed a methylation-independent loss of MYO5B expression in CRC that matched disease progression. Although MYO5B mutations were identified in a small number of patients, these cannot be solely responsible for the common down-regulation observed in CRC patients. Significantly, CRC patients with low MYO5B expression displayed shorter overall, disease- and metastasis-free survival, a trend that was further reinforced when RAB8A expression was also taken into account. Conclusions: Our data identifies MYO5B as a powerful prognostic biomarker in CRC, especially in early stages (stages I and II), which might help stratifying patients with stage II for adjuvant chemotherapy. [less ▲]Detailed reference viewed: 55 (13 UL) Elucidating the fine details of cholesteric liquid crystal shell reflection patternsGeng, Yong; Noh, Junghyun ; Drevensek-Olenik, Irena et alin Liquid Crystals (2017), 44(12-13), Clusters of planar-aligned short-pitch cholesteric liquid crystal spheres generate dynamic colourful patterns due to multiple selective reflections from the radially oriented cholesteric helices in ... [more ▼]Clusters of planar-aligned short-pitch cholesteric liquid crystal spheres generate dynamic colourful patterns due to multiple selective reflections from the radially oriented cholesteric helices in neighbour shells at varying distances. These photonic communication patterns were widely investigated for the cases of both droplets and shells, demonstrating not only intriguing optical phenomena but also potential for applications as new optical elements for photonics, sensing or security pattern generation. However, the optics of these clusters is truly complex and until now only the strongest and most fundamental reflections have been analysed and explained. In this report, we elucidate the origin of a number of more subtle reflections and we explain the extension in space of various spots as well as their internal colour variations. [less ▲]Detailed reference viewed: 9 (0 UL) Alternative Paths Towards Sustainable Localized Food Sovereignty. Convergence between Prosumers and Ethical Entrepreneurs over TimeReckinger, Rachel Scientific Conference (2017, November 14)Recently, a number of resourceful community-driven initiatives for local food production and retail have arisen in Luxembourg, in a context of particularly low organic agricultural rates paradoxically ... [more ▼]Recently, a number of resourceful community-driven initiatives for local food production and retail have arisen in Luxembourg, in a context of particularly low organic agricultural rates paradoxically paired with high consumer demands for organic produce, leading to a specific market of largely imported organic goods. As an encompassing reaction to this situation, a niche of social innovators are combining agro-ecological land use and food production with locavoracity and circular economy. Based on qualitative in-depth interviews and participant observation, we would like to expand on four micro-case studies of circular-economy-type fruit and vegetable production as well as unpackaged and/or socially responsible food retail in today’s Luxembourg. One has been established since the 1980s with over 150 employees, partly in social insertion measures, producing and importing organic fruits and vegetables, delivered via a classical box scheme system. Additionally, over the last three years three significantly smaller initiatives with higher citizen and/or community involvement have emerged. They are characterized by a cooperative governance structure, a claimed community-supported agricultural outlook, a more dynamic presence on social media and regular hands-on workshops and activities. These more recent initiatives are also more radical in their agro-ecological and/or permaculture practices, focusing on local production without relying on imports, as a politicized step further than mere (possibly industrialized) organic production, which is an altogether renewed enacting of circular economy precepts. Yet their position on the market is for the moment more fragile and marginal. Particularly, the retailers among them have to build creative consensus – according to specific priorities and stances – between their standards and the adjustment to consumers who, albeit sensitized, are in search for a certain variety and convenience. By focusing on heterodox experiences of more or less established alternative actors in diverse yet complementary food production and retail niches, we will explore topics such as emotional collective commitment and consensus-building, ethical entrepreneurship in relation to possibly reframed standards over time, governmentality, political enabling or disabling structures and regulations, as well as commodification and upscaling issues. Therefore, this paper touches on political processes and strategies, urban agro-ecological practices as well as post-capitalist economics. [less ▲]Detailed reference viewed: 6 (0 UL) Die Bedeutung des Tieres in Landwirtschaft und Ernährung (Podiumsdiskussion mit Sarah Wiener, Fernand Etgen, Martin von Mackensen, Camille Müller und Rachel Reckinger)Reckinger, Rachel Speeches/Talks (2017)Detailed reference viewed: 6 (1 UL)


text:
# Homotopy Type Theory module > history (changes)

Showing changes from revision #6 to #7: Added | Removed | Changed

## Definiton

Let

$A$ be an abelian group, let $R$ be a commutative ring. $A$ is an $R$-module if it comes with an abelian group homomorphism $\alpha:R \to (A \to A)$ such that

• $\alpha(1) = id_A$

• for all $a:R$ and $b:R$, $\alpha(a) \circ \alpha(b) = \alpha(a \cdot b)$

## Properties

Every abelian group is a $\mathbb{Z}$-module.


text:
An observer counts 240 veh/h at a specific highway location. Assume that the vehicle arrival at the location is Poisson distributed, the probability of having one vehicle arriving over a 30-second time interval is ____________

Free Practice With Testbook Mock Tests

Options:

1. 0.27

2. 0.31

3. 0.29

4. 0.30

Correct Answer: Option 1 (Solution Below)

This question was previously asked in

GATE CE 2014 Official Paper: Set 1

GATE CE 2014 Official Paper: Set 2

GATE CE 2014 Official Paper: Shift 1

GATE CE 2014 Official Paper: Shift 2

Solution:

Concept:

Poisson Distribution

The Poisson distribution can be used to model the vehicle arrival

$${\rm{P}}\left( {{\rm{n}},{\rm{t}}} \right) = \frac{{{{\rm{e}}^{ - {\rm{λ t}}}}{{\left( {{\rm{λ t}}} \right)}^{\rm{n}}}}}{{{\rm{n}}!}}$$

Where,

λ = Number of vehicles per second

t = Time interval

n = Number of vehicles

Calculation:

Given,

λ = 240 veh/hr = 240/3600 veh/sec = 0.0666 veh/sec

n = 1, t = 30sec

$${\rm{P}}\left( {{\rm{n}},{\rm{t}}} \right) = \frac{{{{\rm{e}}^{ - {\rm{λ t}}}}{{\left( {{\rm{λ t}}} \right)}^{\rm{n}}}}}{{{\rm{n}}!}}$$

$${\rm{P}}\left( {{\rm{1}},{\rm{30}}} \right) = \frac{{{{\rm{e}}^{ - {\rm{0.0666× 30}}}}{{\left( {{\rm{0.0666× 30}}} \right)}^{\rm{1}}}}}{{{\rm{1}}!}}$$

P (1, 30) = e-2 × 2 = 0.27


text:
# A Biochemically Accurate Model of Bacterial Chemotaxis

## Transducing an extracellular signal to a cell’s interior

We now turn to the question of how the cell conveys the extracellular signal it has detected via the process of signal transduction to the cell’s interior and produces an action. When E. coli senses an increase in the concentration of glucose, meaning that more ligand-receptor binding is taking place at the receptor that recognizes glucose, how does the bacterium change its behavior?

The engine of signal transduction is phosphorylation, a chemical reaction that attaches a phosphoryl group (PO3-) to an organic molecule. Phosphoryl modifications serve as an information exchange of sorts because, as we will see, they activate or deactivate certain enzymes.

A phosphoryl group usually comes from one of two sources. First, the phosphoryl can be broken off of an adenosine triphosphate (ATP) molecule, the “energy currency” of the cell, producing adenosine diphosphate (ADP). Second, the phosphoryl can be exchanged from a phosphorylated molecule that loses its phosphoryl group in a dephosphorylation reaction.

For many cellular responses, including bacterial chemotaxis, a sequence of phosphorylation and dephosphorylation events called a phosphorylation cascade serves to transmit information within the cell about the amount of ligand binding being detected on the cell’s exterior. In this lesson, we discuss how this cascade of chemical reactions leads to a change in bacterial movement.

A high-level view of the transduction pathway for chemotaxis is shown in the figure below. The cell membrane receptors that we have been working with are called methyl-accepting chemotaxis proteins (MCPs), and they bridge the cellular membrane, binding both to ligand stimuli in the cell exterior and to other proteins on the inside of the cell. The pathway includes a number of additional proteins, which all start with the prefix “Che” (short for “chemotaxis”).

A summary of the chemotaxis transduction pathway. A ligand binding signal is propagated through CheA and CheY phosphorylation, which leads to a response of clockwise flagellar rotation. The blue curved arrow denotes phosphorylation, the grey curved arrow denotes dephosphorylation, and the blue dashed arrow denotes a chemical interaction. Our figure is a simplified view of Parkinson Lab illustrations.

On the interior of the cellular membrane, MCPs form complexes with two proteins called CheW and CheA. In the absence of MCP-ligand binding, this complex is more stable, and the CheA molecule autophosphorylates, meaning that it adds a phosphoryl group taken from ATP to itself — a concept that might seem mystical if you have not already followed our discussion of autoregulation in the previous module.

Phosphorylated CheA can pass on its phosphoryl group to a molecule called CheY, which interacts with the flagellum in the following way. Each flagellum has a protein complex called the flagellar motor switch that is responsible for controlling the direction of flagellar rotation. The interaction of this protein complex with phosphorylated CheY induces a change of flagellar rotation from counter-clockwise to clockwise. As we discussed earlier in the module, this change in flagellar rotation causes the bacterium to tumble, which in the absence of an increase in attractant occurs every 1 to 1.5 seconds.

Yet when a ligand binds to the MCP, the MCP undergoes conformation changes, which reduce the stability of the complex with CheW and CheA. As a result, CheA is less readily able to autophosphorylate, which means that it does not phosphorylate CheY, which cannot change the flagellar rotation to clockwise, and so the bacterium is less likely to tumble.

In short, attractant ligand binding causes more phosphorylated CheA and CheY, which means that it causes fewer flagellar interactions and therefore less tumbling, so that the bacterium will run for a longer period of time.

Note: A critical part of this process is that if a ligand is detected, and the cell has a high concentration of CheY, then it needs to decrease the CheY concentration quickly. Otherwise, the cell will not be able to change its tumbling frequency. To this end, the cell is able to dephosphorylate CheY using an enzyme called CheZ.

## Adding phosphorylation events to our model of chemotaxis

We would like to use the Gillespie algorithm that we introduced in the previous lesson to simulate the reactions driving chemotaxis signal transduction and see what happens if the bacterium “senses an attractant”, meaning that the attractant ligand’s concentration increases and leads to more receptor-ligand binding.

This model will be more complicated than any we have introduced thus far. We will need to account for both bound and unbound MCP molecules, as well as phosphorylated and unphosphorylated CheA and CheY enzymes. We will also need to model phosphorylation reactions of CheA, which depend on the current concentrations of bound and unbound MCP molecules. We will at least make the simplifying assumption that the MCP receptor is permanently bound to CheA and CheW, so that we do not need to represent these molecules individually. In other words, rather than thinking about CheA autophosphorylating, we will think about the receptor that includes CheA autophosphorylating.

In the previous lesson, we very briefly introduced BioNetGen as a way to convert reactions into a software package applying the Gillespie algorithm. However, BioNetGen is useful not only for running particle-free simulations, but also because it implements its own language for rule-based modeling.

Say that we wanted to specify all particles and the reactions involving them in the manner used up to this point in the book. We would need one particle type to represent MCP molecules, another particle type to represent ligands, and a third to represent bound complexes. A bound complex molecule binds with CheA and CheW and can be either phosphorylated or unphosphorylated, necessitating two different molecule types. In turn, CheY can be phosphorylated or unphosphorylated as well, requiring two more particles.

Instead, the BioNetGen language will allow us to conceptualize this system much more concisely using rules that apply to particles that are in a variety of states. The BioNetGen representation of the four particles in our model is shown below. The notation Phos~U~P indicates that a given molecule type can be either phosphorylated or unphosphorylated, so that we do not need multiple different expressions to represent the molecule.

L(t)             #ligand molecule
T(l,Phos~U~P)    #receptor complex
CheY(Phos~U~P)
CheZ()

The conciseness of BioNetGen’s molecule representation helps us represent our reactions concisely as well. We first reproduce the reversible binding and dissociation reaction from the previous lesson.

LR: L(t) + T(l) <-> L(t!1).T(l!1) k_lr_bind, k_lr_dis

Next, we represent the phosphorylation of the MCP complex. Recall that the phosphorylation of CheA can occur at different rates depending on whether the MCP is bound, and so we will need two different reactions to model these different rates. In our model, the phosphorylation of the MCP will occur at one fifth the rate when it is bound to the attractant ligand.

FreeTP: T(l,Phos~U) -> T(l,Phos~P) k_T_phos
BoundTP: L(t!1).T(l!1,Phos~U) -> L(t!1).T(l!1,Phos~P) k_T_phos*0.2

Finally, we represent the phosphorylation and dephosphorylation of CheY. The former requires a phosphorylated MCP receptor, while the latter is done with the help of a CheZ molecule that can be in any state.

YP: T(Phos~P) + CheY(Phos~U) -> T(Phos~U) + CheY(Phos~P) k_Y_phos
YDep: CheZ() + CheY(Phos~P) -> CheZ() + CheY(Phos~U) k_Y_dephos

Now that we have converted the reactions from the chemotaxis signal transduction pathway into BioNetGen’s rule-based language, we would like to see what happens when we change the concentrations of the ligand. Ideally, the bacterium should be able to distinguish different ligand concentrations. That is, the higher the concentration of an attractant ligand, the lower the concentration of phosphorylated CheY, and the lower the tumbling frequency of the bacterium.

But does higher attractant concentration in our model really lead to a lower concentration of CheY? Let’s find out by incorporating the phosphorylation pathway into our ligand-receptor model in the following tutorial.

## Changing ligand concentrations leads to a change in internal molecular concentrations

The following figure shows the concentrations of phosphorylated CheA and CheY in a system at equilibrium in the absence of ligand. As we might expect, these concentrations remain at steady state (with some healthy noise), and so the cell stays at its background tumbling frequency.

Molecular concentrations (in number of molecules in the cell) over time (in seconds) in a BioNetGen chemotaxis simulation in which no ligand is present.

The sudden addition of 5,000 attractant ligand molecules increases the concentration of bound receptors, therefore leading to less CheA autophosphorylation, and less phosphorylated CheY.

Molecular concentrations (in number of molecules in the cell) over time (in seconds) in a BioNetGen chemotaxis simulation with 5,000 initial attractant ligand particles.

If we instead add 100,000 attractant molecules, then we see an even more drastic decrease in phosphorylated CheA and CheY.

Molecular concentrations (in number of molecules in the cell) over time (in seconds) in a BioNetGen chemotaxis simulation with 100,000 initial attractant ligand particles.

This Gillespie model confirms the biological observations that an increase in attractant reduces the concentration of phosphorylated CheY. This reduction takes place remarkably quickly, with the cell attaining a new equilibrium in a fraction of a second.

And yet you may remain skeptical of our model. After all, the biochemistry powering chemotaxis may be elegant, but it is also simple, and perhaps you are not surprised that the model’s particle concentrations reproduced the response of E. coli to an attractant ligand.

But what we have shown in this lesson is just part of the story. In the next lesson, we will see that the biochemical realities of chemotaxis are even more complicated, and for good reason — this added complexity will allow E. coli, and our model of it, to react to a dynamic world with surprising sophistication.