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# General topological space with closure operation as in Russian translation of Hausdorff's 1914 and 1927 Mengenlehre

In the Russian translation (by P. Alexandroff and A. Kolmogorov) of chapter "Point Sets in General Spaces" Hausdorff (1914), the notion of a general topological space is defined as set $$R$$ with closure operation $$\overline{\circ}$$ defined for each subset of $$R$$.

I.e. a subset $$M$$ of $$R$$ does not necessarily belongs to its closure. Moreover, mostly no more constraints are put on $$\overline{\circ}$$. As 6th example, he provides:

6) $$R$$ is an infinite set. Let $$\overline{M} = R \setminus M$$.

My two questions:

1. How such topological structures are called in the modern setting?
2. What fields are they used in?

UPD: Probably, the example was introduced not by Hausdorff but by P. Alexadroff or A. Kolmogorov. This "general" closure operation is used in the russian merged version of 1914 and 1927 books to axiomatize general topological spaces and show the correspondence between metric, neighbourhoods, and topological (without "general") spaces. Still, the questions persist.

• I use Russian translation that is a merged version (by P. Alexandrov and A. Kolmogorov) of Hausdorff 1914 and 1927 books. I tried to find the corresponding pages in the original without success (archive.org/details/grundzgedermen00hausuoft/page/212). But I cannot read Deutsch. So, probably, the example was introduced by P. Alexandroff or A. Kolmogorov (I can send a scan if it helps). Still, how such `general' t.s. are called these days? – Leonid Dworzanski Sep 22 '19 at 22:50
• @BjørnKjos-Hanssen, thank you - I've refined the question. – Leonid Dworzanski Sep 22 '19 at 23:23
• @ToddTrimble, you've got it right. "But I don't see how the complementation operator deserves to be called topological" - me either, but I leaving room for doubt and trying to clear it up. – Leonid Dworzanski Sep 24 '19 at 3:45
• I don't know about Hausdorff's book. I'm fairly familiar with the 3rd edition, because an English translation exists (a copy of which I've had for several decades), but not the 1st edition, which includes a lot of topics omitted in the 2nd and 3rd editions. However, Introduction to General Topology by Zlatko P Mamuzić (1963; review on pp. 127-128 here) begins by considering "closure operators" satisfying NO axioms (continued) – Dave L Renfro Sep 24 '19 at 8:12
• (the complement of a set is the 3rd example on p. 13, lines 7-9 from bottom) and then adds various axioms (e.g., empty set maps to empty set, each set is a subset of its "closure", the "closure" of the union of two sets is a subset of the union of the closures of the two sets, etc.) and discusses the relationships among these axioms. A better known text, Topological Spaces by Eduard Čech (1966), is a very complete study spaces in which idempotency is not assumed for the closure operation. See also this. – Dave L Renfro Sep 24 '19 at 8:24

# 1. Introductory Comments and Contents/Summary

First, to repeat a point that came up in the comments, we’re not talking about alternative methods of defining a topological space, such as using the interior operator or the frontier operator. Rather, we’re talking about removing some, or even all of the axioms for the topological closure operator and asking what such generalized notions of a topological space are called and where they arise. Regarding what they are called, the most common terms are probably generalized topology and extended topology and pretopology.

In what follows I’ll give a progression of generalized closure notions, beginning with arbitrary operators from $${\mathcal P}(X)$$ to $${\mathcal P}(X)$$ (where $$X$$ is a set and $${\mathcal P}(X)$$ is the set of subsets of $$X)$$ and ending with topological closure operators. Many closure operators of interest are not directly comparable to all those in this progression, but I think the notions I’ve chosen provide an appealing path to the topological closure operator. Because the literature is filled with terms having multiple meanings $$-$$ depending on the author and the field $$-$$ and a seemingly endless parade of terms for every kind of nuance, I’ve tried to pick terms that are unlikely to have multiple meanings and I've tried to minimize the introduction of terms (other than sometimes pointing out useful search terms). For example, I’ll try to be consistent in using “operator” (rather than mapping, function, operation, transformation, etc.) when mentioning a closure operator.

Let $$T$$ be a function from $${\mathcal P}(X)$$ to $${\mathcal P}(X).$$ The Kuratowski closure operator axioms for $$T$$ are (K1) through (K4) below, where their ordering is in accordance with that of the progression of generalized closure operators I'll give.

(semi-K1) $$\;$$For each $$A,\,B \in {\mathcal P}(X),$$ we have $$T(A) \cup T(B) \subseteq T(A \cup B).$$

(K1) $$\;$$For each $$A,\,B \in {\mathcal P}(X),$$ we have $$T(A \cup B) = T(A) \cup T(B).$$

(K2) $$\;T(\emptyset) = \emptyset.$$

(K3) $$\;$$For each $$A \in {\mathcal P}(X),$$ we have $$A \subseteq T(A).$$

(K4) $$\;$$For each $$A \in {\mathcal P}(X),$$ we have $$T(T(A)) = T(A).$$

CONTENTS/SUMMARY

2. Unrestricted Closure Operators

Notions such as interior, neighborhood, limit point, etc. can be defined. Definition of a partial order on closure operators.

3. Monotone Closure Operators $$\;$$ [satisfy (semi-K1)]

Two ways to define a certain “natural” monotone closure operator from an unrestricted closure operator. Each monotone closure operator has a least fixed point and a greatest fixed point, and there are two standard ways of obtaining them $$-$$ impredicative or “from above”, iterative or “from below”. Many common mathematical objects can be realized as such fixed points $$-$$ closed sets, open sets, the linear span of a set, the convex hull of a set, the deductive closure of a set of wffs in logic, the $$\sigma$$-algebra generated by a collection of sets, etc. A certain monotone closure operator can even be used to prove the Cantor-Bernstein theorem for cardinal numbers.

4. Union-Preserving Closure Operators $$\;$$ [satisfy (K1)]

There exists a greatest union-preserving closure operator that is less than or equal to a given monotone closure operator (in fact, for a given unrestricted closure operator).

5. Base Closure Operators $$\;$$ [satisfy (K1), (K2)]

The prototypical example is the derived set operator in topology and its various generalizations obtained by using different notions of “limit point”. There is a “natural” topological space associated with each base closure operator. These so-called fine topologies are important in abstract potential theory.

6. Čech Closure Operators $$\;$$ [satisfy (K1), (K2), (K3)]

There is a “natural way” to obtain a Čech closure operator from any base closure operator. A relatively large part of general topology can be carried out in the context of Čech closure spaces, but most of the discussion here will involve the interrelationships between a base closure operator and its associated Čech and topological closure operators.

7. Topological Closure Operators $$\;$$ [satisfy (K1), (K2), (K3), (K4)]

Each Čech closure operator can be transfinitely iterated to obtain a topological closure operator, and the resulting topological space is the same as that obtained from the collection of closed sets in the Čech closure space.

8. Selected References

References are mostly restricted to those that specifically involve, entirely or partly, various generalized closure operator notions.

# 2. Unrestricted Closure Operators

Definition: An unrestricted closure operator on the set $$X$$ is a function from $${\mathcal P}(X)$$ to $${\mathcal P}(X).$$

I’m using “unrestricted” rather than “arbitrary” because “arbitrary” is too pervasive and useful as a general descriptive term.

Stadler/Stadler [62] make the following observation in the first sentence of their manuscript: “$$\dots$$ we explore the surprising fact that some meaningful topological concepts can already be defined on a set $$X$$ endowed with an arbitrary set-valued set-function, which we will interpret as a generalized closure operator.”

To illustrate this observation, note that given an restricted closure operator $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$$ and $$A \subseteq X,$$ we can define the $$T$$-interior of $$A$$ by $${\text {int}}_T(A) = X – T(X – A).$$ Now we can define a $$T$$-neighborhood of a point $$x \in X$$ to be any set $$N \subseteq X$$ such that $$x \in {\text {int}}_T(N),$$ and we can define $$x \in X$$ to be a $$T$$-limit point of $$A \subseteq X$$ if each $$T$$-neighborhood of $$x$$ intersects $$A$$ in at least one point other than $$x.$$ These and other notions for unrestricted closure operators are discussed in Stadler/Stadler [62] (pp. 1-6).

The only topology text I'm aware of that at least briefly discusses unrestricted closure operators is Mamuzić [44] (mostly in §1$$-$$§3, pp. 13-22), but I haven't put forth much effort in looking for such texts. Mamuzić's discussion includes notions such as accumulation point, derived set, interior and exterior boundary of a set, etc. in the setting of an unrestricted closure operator. However, the nature of his treatment makes it difficult to use the first few pages of his book as a reference for this topic $$-$$ results involving topological spaces and his more general notion of a neighborhood space are mixed together in such a way that close reading for context is often needed to determine just what assumptions are being made for those results that specifically pertain to generalized topological spaces.

Incidentally, Sierpiński [60] begins with an essentially unrestricted notion of a derived set (set of limit points of a set), but his approach does not appear to provide a natural setting for arbitrary unrestricted closure operators. Sierpiński begins by assigning to each point $$x \in X$$ a (nonempty?) set $${\mathcal N}_x \subseteq {\mathcal P}(X)$$ of "neighborhoods" of $$x$$ (I believe he requires $$x$$ to belong to each neighborhood of $$x,$$ but he is not explicit about this) and then the notion of a limit point of a subset of $$X$$ is defined as above. Now define a closed set to be a set that contains all its limit points, and finally define the closure of $$A \subseteq X$$ to be the intersection of all closed sets containing $$A.$$ However, it is easy to see that not every unrestricted closure operator on $$X$$ will arise by taking closures of sets in this manner, because all of Sierpiński's generalized closure operators $$cl$$ obviously satisfy $$A \subseteq B \implies cl(A) \subseteq cl(B).$$ In fact, Sierpiński's closure operators satisfy (semi-K1), (K2), (K3), and (K4) $$-$$ see bottom of p. 7 of Sierpiński [60].

It will be useful to define a partial order on the set of unrestricted closure operators on a fixed set.

Definition: Let $$T_1$$ and $$T_2$$ be unrestricted closure operators on $$X.$$ We say that $$T_1 \leq T_2$$ if, for each $$A \in {\mathcal P}(X),$$ we have $$T_1(A) \subseteq T_2(A).$$

It is easy to see that, for all unrestricted closure operators $$T_1,$$ $$T_2,$$ and $$T_3$$ on $$X,$$ we have (i) $$T_1 \leq T_2;$$ (ii) $$T_1 \leq T_2$$ and $$T_2 \leq T_1$$ implies $$T_1 = T_2;$$ (iii) $$T_1 \leq T_2$$ and $$T_2 \leq T_3$$ implies $$T_1 \leq T_3.$$

# 3. Monotone Closure Operators $$\;$$ [satisfy (semi-K1)]

Definition: We say that $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$$ is a monotone closure operator on $$X$$ if for each $$A,\,B \in {\mathcal P}(X)$$ we have $$T(A) \cup T(B) \subseteq T(A \cup B).$$

In the literature these are also called isotone and increasing and order-preserving.

Lemma: Let $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$$ be an unrestricted closure operator on $$X.$$ Then $$T$$ is a monotone closure operator on $$X$$ if and only if for all $$A,\,B \in {\mathcal P}(X)$$ we have $$A \subseteq B$$ implies $$T(A) \subseteq T(B).$$

Monotone Modification of an Unrestricted Closure Operator: Given an unrestricted closure operator $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X),$$ there is a natural way to associate with $$T$$ a monotone closure operator $${\overline{T}}:{\mathcal P}(X) \rightarrow {\mathcal P}(X).$$ (Here I’m using “natural” in an informal descriptive sense, not in a category-theoretic sense.) For each $$A \in {\mathcal P}(X),$$ either of the following two equivalent formulations can be used to define $$\overline{T}(A)$$:

$$\begin{array}{l} \overline{T}(A) & = & \bigcap\,\{T(B):A \subseteq B\} \\ \\ \overline{T}(A) & = & \{x \in X: A \cap N \neq \emptyset \;\; {\text {for each}} \; T {\text {-neighborhood}}\; N \; {\text {of}} \; x \}\end{array}$$

Recall that “$$T$$-neighborhood of $$x$$” was defined earlier for unrestricted closure operators.

For the following results, see Day [14] (p. 186) and Stadler/Stadler [62] (p. 7; note that the union in equation (11) on p. 7 should be an intersection).

(a) $$\;\overline{T}$$ is a monotone closure operator.

(b) $$\;\overline{T} \leq T$$

(c) $$\;T_1 \leq T_2\;$$ implies $$\;\overline{T_1} \leq \overline{T_2}$$

(d) $$\;$$If $$T'$$ is a monotone closure operator on $$X$$ and $$\;T' \leq T,\;$$ then $$\;T' \leq \overline{T}.$$

Remarks: (1) Note that (a), (b), and (d) can be summarized by saying $$\overline{T}$$ is the greatest monotone closure operator that is less than or equal to $$T.$$ (2) If $$T$$ itself is a monotone closure operator, then $$T = \overline{T}.$$ This follows from (a), (b), (d), and antisymmetry of $$\leq.$$ (3) If $$T$$ is any unrestricted closure operator, then $$\overline{T} = \overline{\overline{T}}.$$ This follows from (a) and (2). (Just to keep matters straight, I'll mention that this last assertion says the "monotone modification process" is idempotent, not that $$\overline{T}$$ is an idempotent closure operator.) (4) An immediate consequence of (2) is that every monotone closure operator on $$X$$ arises from the monotone modification of some unrestricted closure operator on $$X.$$ Thus, the monotone modification process does not necessarily introduce any additional properties, such as (K2) or (K3).

Example (complement operator): If $$T^c$$ is the complement operator, defined by $$T^c(A) = X – A,$$ then the monotone modification of $$T^c$$ is such that $$\overline{T^c}(A) = \emptyset$$ for each $$A \in {\mathcal P}(X).$$ I suppose this can be understood by recognizing that the monotone modification of the complement operator has to be maximally trivial to counteract the maximal way in which the complement operator fails to be monotone. Of course, more generally, this maximal trivialness of $$\overline{T}$$ will also occur for any unrestricted closure operator $$T$$ that satisfies $$T(X) = \emptyset.$$

The most significant application of monotone closure operators is the fact that each monotone closure operator has at least one fixed point, along with the two contrasting methods for obtaining a fixed point $$-$$ “from above” and “from below”. Note that an unrestricted closure operator can have no fixed points. A rather trivial example is $$X = \{a\}$$ with $$T$$ defined by $$T(\emptyset) = \{a\}$$ and $$T(\{a\}) = \emptyset.$$

Fixed Points of Monotone Closure Operators (impredicative or “from above” approach)

Let $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$$ be a monotone closure operator on $$X,$$ and define subsets $$E_{*}$$ and $$E^{*}$$ of $$X$$ by

$$\begin{array}{l} E_{*} & \triangleq & \bigcap\,\{A \in {\mathcal P}(X): \; T(A) \subseteq A\} \\ E^{*} & \triangleq & \bigcup\,\{A \in {\mathcal P}(X): \; A \subseteq T(A) \} \end{array}$$

The following results hold:

(a) $$\;T(E_{*}) = E_{*}$$

(b) $$\;T(E^{*}) = E^{*}$$

(c) $$\; T(E) = E\;$$ implies $$\;E_{*} \subseteq E \subseteq E^{*}$$

Some to all of the results above can be found in several standard undergraduate level set theory texts, such as Dalen/Doets/Swart [13] (Theorem 4.1 on p. 262), Enderton [18] (Exercise 30 on p. 54 $$-$$ see this and this), Hrbacek/Jech [33] (Exercises 1.10 & 1.12 on p. 69), and Moschovakis [50] (Problem 6.11 on p. 85). Although a complete statement of the results above, with proof, is given in Devidé [15] (1962), the less precise result that a monotone closure operator has a fixed point was observed in Knaster [38] (p. 133, lines −5 to −3), which was joint work with Tarski presented on 9 December 1927, and the results above (and more) were obtained in 1939 by Tarski in the more general context of a non-decreasing function from a complete lattice to itself (see footnotes on pp. 285, 286 in Tarski [63]).

Fixed Points of Monotone Closure Operators (iterative or “from below” approach)

Let $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$$ be a monotone closure operator on $$X.$$ For each ordinal $$\alpha \geq 1,$$ we define (by transfinite induction) $$T^{\alpha}:{\mathcal P}(X) \rightarrow {\mathcal P}(X),\,$$ the $$\alpha$$-th iterate of $$T,$$ by

$$\begin{array}{l} T^1 (A) & = & T(A) \\ T^{\alpha}(A) & = & T\left(\,\bigcup \,\{T^{\gamma}(A): \; \gamma < \alpha \} \,\right) \;\; {\text{if}} \; \alpha > 1 \end{array}$$

For those concerned about statements involving “all ordinals”, it is enough to define $$T^{\alpha}$$ for all ordinals less than or equal to the least infinite ordinal whose cardinality is greater than the cardinality of $$X.$$ In (c) below, $$|X|^{+}$$ denotes the least cardinal number greater than $$|X| \,=\, {\text {card}}\,(X).$$

The following results hold:

(a) $$\;T^{\alpha}$$ is a monotone closure operator for each $$\alpha.$$

(b) $$\; \alpha \leq \beta\;$$ implies $$\; T^{\alpha} \leq T^{\beta}.$$

(c) $$\;$$For each $$A \in {\mathcal P}(X)$$ there exists an ordinal $$\;\lambda(A) < |X|^{+}\;$$ such that, for each $$\;\alpha \geq \lambda(A), \;$$ we have $$\;T^{\alpha}(A) = T^{\lambda(A)}(A).$$

(d) $$\;T^{\lambda(\emptyset)}(\emptyset) = E_{*},\;$$ where $$E_{*}$$ is from the earlier “from above” approach.

(e) $$\;T^{\lambda(X)}(X) = E^{*},\;$$ where $$E^{*}$$ is from the earlier “from above” approach.

The least upper bound of the (minimal) ordinals $$\lambda(A)$$ in (c) is often called the closure ordinal of $$T,$$ which we will denote by $$|T|.$$ (Hammer uses the term projecting order of $$T$$ on p. 61 of [23].) Note that $$|T| \leq |X|^{+}$$ and, for each $$\alpha \geq |T|,$$ we have $$T^{\alpha} = T^{\lambda}.$$ (For completeness, I suppose we would want to define the closure ordinal of the identity closure operator to be $$0.)$$ For example, $$|T| = 1$$ for the usual closure and interior operators of a topological space, as well as for the linear span, the convex hull, the deductive closure, and the $$\sigma$$-algebra examples below. Also, $$|T| = 2$$ for the usual boundary [= frontier] operator in a topological space (or even the boundary operator in certain more generalized spaces $$-$$ see Theorem C.6 on p. 77 of Hammer [20]). However, it is easy to see that $$|T|$$ can be arbitrarily large (even more, $$|T|$$ can have any specified ordinal value) by considering the “limit point” operation when $$X$$ is an ordinal (see also this). Incidentally, since $$|T|$$ is the least ordinal at which the iterates of $$T$$ “stabilize” for all subsets of $$X,$$ a natural question is whether $$E_{*}$$ and/or $$E^{*}$$ can be reached in fewer than (and possibly also different from each other) $$|T|$$ many steps from $$\emptyset$$ and $$X.$$ Again, ordinals come to our rescue. If $$X$$ is the ordinal $${\omega}^{\omega} + {\omega}^2$$ and $$T$$ is the limit point operator, then I believe $$|T| = {\omega}^{\omega},$$ while $$E_{*} = \emptyset$$ is reached in $$0$$ steps and $$E^{*} = {\omega}^{\omega}$$ is reached in $$3$$ steps.

If $$T$$ is continuous in the sense that $$\;T\left(\bigcup\limits_{n=1}^{\infty}A_n\right) = \bigcup\limits_{n=1}^{\infty}T(A_n)\;$$ for every non-decreasing sequence $$\;A_1 \subseteq A_2 \subseteq \cdots\;$$ of subsets of $$X,$$ then $$E_{*}$$ can be reached in at most $${\omega}_0$$ many steps and, in fact, we have $$\;E_{*} = \bigcup\limits_{n=1}^{\infty}T^{n}(\emptyset).$$

These results about iterating monotone closure operators and their connections with monotone inductive definitions can be found in Aczel [1], Barwise [2] (Chapter VI), Barwise/Moss [3] (Chapter 15), Dalen/Doets/Swart [13] (Section III.4 on pp. 260-266), Devlin [16] (Section 7.4 on pp. 159-163), Enderton [18] (Exercise 9 on p. 78), Hinman [32] (pp. 22-26), Hrbacek/Jech [33] (Exercises 1.13 & 1.14 on p. 69), Moschovakis [48], Moschovakis [49] (pp. 404-405), Moschovakis [50] (Chapter 6), and Pohlers [52] (pp. 109-114). For generalizations of the notion of $$T$$ being continuous and their consequences on restricting the size of $$|T|,$$ see Let $$\Gamma$$ be a $$\kappa$$-based monotone operator where $$\kappa$$ is regular. Then the closure ordinal of $$\Gamma$$ is $$\kappa$$, Aczel [1] (pp. 746-747), and Schwarz [58].

Some Monotone Closure Operators Whose Fixed Points are Well Known

(1) The topological closure, interior, derived, and other operators of a topological space.

Also, various refinements of some of these operators, such as the sequential convergence operator and the analogous operators in various generalized topological spaces (e.g. see Lei/Zhang [41]).

(2) The set of linear combinations of a subset of a vector space.

(3) The convex hull of a subset of a normed vector space.

In 1955 Preston Clarence Hammer (1913-1986), motivated by certain issues in convex geometry, published the first of his several papers on an extensive study of generalized topological notions that he called extended topology. In the references below I’ve included Hammer’s papers that seem most relevant to closure operator notions and which I happen to have a copy of, ordered by submission date. Hammer’s Ph.D. students Don Arthur Mattson (1965) and George Clifford Gastl (1966) also did some of this work (both at University of Wisconsin-Madison), but I have not seen their dissertations. Before these two Ph.D. dissertations, there was a Ph.D. dissertation at Oregon State University in 1959, Rio [54], that was largely based Hammer’s 1955 paper [19]. Although Rio does not mention Hammer in his Acknowledgments, Rio’s supervisor was Bradford Henry Arnold, who came to Oregon State University in 1947, which was about the same time that Hammer left Oregon State University (having been there since 1940). I do not know whether Hammer and Arnold knew each other sufficiently to have inspired the choice of Rio's topic, or whether the choice of Rio's topic was simply a coincidence (i.e. Arnold/Rio were simply interested in the topic after seeing Hammer's 1955 paper). Incidentally, Hammer [23] (pp. 71-74) discusses some results in Rio’s dissertation.

For results more specific to viewing the convex hull as a monotone closure operator, see How to characterize the convex hull/closure operator, Bennett [4], Jamison [34], Koenen [39], and van de Vel [66].

(4) The deductive closure of a set of well formed formulas (i.e. the consequence operator) in systems of logic.

For more about this example, see Properties of the deductive closure, Brown/Suszko [7], Malinowski [43], Martin/Pollard [45], Pohlers [52], and Wójcicki [68].

(5) sigma-algebras

Let $$Y$$ be a set, and let $$X = {\mathcal P}(Y),$$ and define $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$$ by

$$T(\mathcal C) \; = \; \left\{\bigcup_{n=1}^{\infty}A_n:\; A_n \in \mathcal C \; \text{for each} \; n \right\} \; \cup \; \{Y-A: \; A \in \mathcal C\} \; \cup \; \{\emptyset, \, Y\}.$$

Then given a collection $$\mathcal C$$ of subsets of $$Y,$$ we have that $$E_*$$ is the $$\sigma$$-algebra generated by $${\mathcal C}.$$

The following sentence in Rayburn [53] (middle of p. 756) caught my interest, but I don’t know whether what he does in this paper is of much significance here: “By analogy with topology, we ask about operators from $$P(X)$$ to $$P(X)$$ which generate $$\sigma$$-algebras.”

(6) Cantor-Bernstein Theorem

Let $$f:X \rightarrow Y$$ and $$g:Y \rightarrow X$$ be injective functions and define $$\;T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)\;$$ by $$\;T(A) = X - g[Y - f[A]].$$ (Here, $$h[B]$$ denotes the image of the set $$B$$ under the function $$h.)$$ Then $$T$$ is a monotone closure operator on $$X,$$ and $$E_{*}$$ can be used to give a proof of the Cantor-Bernstin theorem for cardinal numbers. Indeed, $$T$$ is continuous, and hence the iterative method produces $$E_{*}$$ as the countable union $$\;\bigcup_{n=1}^{\infty}T^n(\emptyset),\;$$ which can be identified with one of the iterative methods of proving the Cantor-Bernstein Theorem.

This example originated with Knaster [38] (joint work with Tarski, presented 9 December 1927), which also probably gives the first published statement that a monotone closure operator has a fixed point. See also Wikipedia: Knaster–Tarski theorem, Mathworld: Tarski's Fixed Point Theorem, Devlin [16] (pp. 77-78), Hinkis [31] (Chapters 31 and 35, on pp. 317-322 and 343-355), Hrbacek/Jech [33] (Exercises 1.10−1.14 on p. 69, where the continuity of $$T$$ and its implications are also discussed), Mendelson [46] (Appendix D: A Lattice-Theoretic Proof of the Schröder-Bernstein Theorem on p. 200), Sierpiński [59] (pp. 145-147), Tarski [63] (footnotes on pp. 285, 286), and Willard [67] (Problem 1J on p. 15).

Regarding the above applications of monotone closure operators, see also Arturo Magidin’s 28 July 2011 essay.

SELECTED REFERENCES

THIS CONTINUES MY ANSWER THAT BEGINS HERE

SELECTED REFERENCES

# 4. Union-Preserving Closure Operators $$\;$$ [satisfy (K1)]

Definition: We say that $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$$ is a union-preserving closure operator on $$X$$ if $$T$$ is a monotone closure operator on $$X$$ such that for each $$A,\,B \in {\mathcal P}(X)$$ we have $$T(A \cup B) = T(A) \cup T(B).$$

The term additive operator is often used, but I thought “additive” creates too much mental interference with unintended meanings from other areas of mathematics. Also, closure spaces arising from closure operators that are only required to satisfy (K1) do not seem to be of enough interest for the use of “union-preserving” here to cause problems. An example of a monotone closure operator that is not a union-preserving closure operator is the interior operator on $$X = {\mathbb R}.$$ Note that the interior operator satisfies (semi-K1) $$-$$ and also (K2) and (K4), for those keeping track $$-$$ but not (K2) (consider $$A = \mathbb Q$$ and $$B = {\mathbb R} – {\mathbb Q}).$$

I do not know of a naturally occurring example of a union-preserving closure operator that doesn’t also satisfy at least one of the remaining axioms for a topological operator, let alone an example from which useful insight is provided by considering it in the context of a closure operator. I also do not know how to obtain a “minimally differing” union-preserving closure operator from a monotone closure operator that doesn't explicitly make use of existing union-preserving closure operators, but the following at least shows that there exists a greatest union-preserving closure operator that is less than or equal to a given monotone closure operator. In fact, the same method works for an unrestricted closure operator $$-$$ see Hammer [23] (definition of $$f_6$$ at the bottom of p. 70).

Union-Preserving Modification of a Monotone Closure Operator: Given a monotone closure operator $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X),$$ the greatest union-preserving closure operator $$T^{\text {u-p}}$$ on $$X$$ that is less than or equal to $$T$$ can be defined by letting

$$T^{\text {u-p}}(A) \; = \; \bigcup\,\{T'(A): \; T' \leq T \; {\text {and}} \; T' \; {\text {is a union-preserving closure operator on}} \; X \}$$

Regarding the assumption (K1), the following may be of interest:

(from Hammer [23], top of p. 65) The classical topologists have persisted in requiring additivity of the closure function in the definition of topological spaces. From one standpoint the additivity axiom might be called the sterility axiom. That is to say, it requires that two sets cannot produce anything (a limit point) by union that one of them alone cannot produce. On the other hand, of course, as with the analogous independence of events in probability theory, the known presence of additivity produces special benefits.

# 5. Base Closure Operators $$\;$$ [satisfy (K1), (K2)]

Definition: We say that $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$$ is a base closure operator on $$X$$ if $$T$$ is a union-preserving closure operator such that $$T(\emptyset) = \emptyset.$$

Sometimes (but not here) the requirement $$T(X) = X$$ is included (e.g. see bottom of p. 4700 of Bliedtner/Loeb [6]). I believe the name originated in Lukeš/Malý/Zajíček [42], but I don’t know why they used the specific word “base”. A rather trivial example of a union-preserving closure operator that is not a base closure operator is given by $$T(A) = X$$ for all $$A \in {\mathcal P}(X),$$ with $$X \neq \emptyset.$$ Most of the results I give here concerning base closure operators, and many other results not included here, can be found in Chapter 1: Base Operator Spaces (pp. 5-37) of Lukeš/Malý/Zajíček [42].

Base Modification of a Union-Preserving Closure Operator: Given a union-preserving closure operator $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X),$$ we can obtain a base closure operator $$T^{\text {base}}$$ on $$X$$ by redefining $$T$$ to have the value $$\emptyset$$ at $$A = \emptyset:$$

$$T^{\text {base}}(A) \; \triangleq \; \left\{\begin{array}{c} \emptyset & \text{if} & A = \emptyset \\ T(A) & \text{if} & A \neq \emptyset \end{array} \right.$$

Note that $$T^{\text {base}}$$ is the greatest base closure operator less than or equal to $$T.$$

The prototypical example of a base closure operator is the derived set operator of a topological space:

Example (derived set operator): Let $$(X,\tau)$$ be a topological space and define $$T$$ by

$$T(A) \; = \; \{x \in X: \; A \cap (U - \{x\}) \neq \emptyset \;\; \text{for each} \; \tau \text{-open set} \; U \; \text{containing} \; x\}.$$

Then $$T$$ is a base closure operator on $$X.$$

By replacing “$$A \cap (U - \{x\}) \neq \emptyset$$” with “$$A \cap U$$ is not small” for various notions of “not small” (i.e. by using various stronger notions of being a limit point of a set), we obtain many other examples of base closure operators. For example, in the appropriate spaces we could require $$A \cap U$$ to not be finite, not be countable, not be a first Baire category set, not have Lebesgue measure zero, etc. In fact, the motivation in Lukeš/Malý/Zajíček [42] for considering the abstract notion of a base closure operator was to generalize all the various notions in potential theory for a set to be thin/not-thin at a point. Indeed, if $$T$$ is a base closure operator on $$X$$ and $$x \in X,$$ then $$\{A \in {\mathcal P}(X): \; x \notin T(A) \}$$ is an ideal of sets that we could call the $$T$$-thin sets at $$x.$$ Moreover, if $$X$$ is a set and if for each $$x \in X$$ we specify a nonempty ideal $${\mathcal I}_x$$ of subsets of $$X,$$ then there exists a base closure operator on $$X$$ such that, for each $$x \in X,$$ we have $${\mathcal I}_x$$ equal to the collection of $$T$$-thin sets at $$x.$$ Regarding the last two sentences, see Lukeš/Malý/Zajíček [42] (1.A.2 on p. 10 and 1.A.17 on p. 16).

Definition ($$T$$-topology): Let $$T$$ be a base closure operator on $$X$$ and let $$C \in {\mathcal P}(X).$$ We say that $$C$$ is a $$T$$-closed set if $$T(C) \subseteq C.$$

It is not difficult to show, for a fixed base closure operator $$T,$$ that the collection of $$T$$-closed sets satisfies the axioms to be the closed sets for some topology $${\tau}^T$$ on $$X.$$ Of course, the corresponding topological closure operator, which I will denote by $${\tau}^T{\text {-}}{\text {Cl}},$$ might not be equal to $$T,$$ but we always have $$\;T \leq {\tau}^T{\text {-}}{\text {Cl}}.$$ [[ Proof of this last inequality: Choose $$A \in {\mathcal P}(X)$$ and let $$C$$ be any $$T$$-closed set containing $$A.$$ Since $$A \subseteq C,$$ we have $$T(A) \subseteq T(C);$$ and since $$C$$ is $$T$$-closed, we have $$T(C) \subseteq C.$$ Therefore, $$T(A)$$ is a subset of any such set $$C,$$ and hence $$T(A)$$ is a subset of the intersection of all such sets $$C.$$ Thus, for each $$A \in {\mathcal P}(X)$$ we have shown that $$T(A) \subseteq {\tau}^T{\text {-}}{\text {Cl}}(A).$$ ]]

If $$(X,\tau)$$ is a topological space, then of course the topological closure operator $${\tau}{\text {-}}{\text {Cl}}$$ for $$(X,\tau)$$ is a base closure operator on $$X.$$ Moreover, as is easy to check and what one would expect, the topological closure operator corresponding to $${\tau}{\text {-}}{\text {Cl}}$$ (regarding $${\tau}{\text {-}}{\text {Cl}}$$ as a base closure operator) is equal to $${\tau}{\text {-}}{\text {Cl}}.$$ That is, we have $${\tau}^{({\tau}{\text {-}}{\text {Cl}})}{\text {-}}{\text {Cl}} = {\tau}{\text {-}}{\text {Cl}}.$$

The topologies that arise in this way in potential theory $$-$$ the topologies $${\tau}^T$$ where $$T$$ is a base closure operator, defined using a limit point notion in which a set is NOT $$T$$-thin at a point $$x$$ $$-$$ are called fine topologies (Wikipedia page and google search) because, in Euclidean spaces, these topologies are finer (i.e. have more open sets) than the usual Euclidean topology. [Note that a more stringent condition to be a limit point causes fewer points to be added to the set to get its topological closure, hence the topological closure is smaller, hence more closed sets are needed to obtain (via intersection) this smaller topological closure, hence there are more open sets. (There is probably a less circuitous way of seeing this.)] An alternative way that fine topologies like this arise in potential theory is that it is useful to consider topologies on $${\mathbb R}^n$$ (or more general spaces) such that, for various collections of functions from $${\mathbb R}^n$$ to $$\mathbb R$$ that are larger than the collection of ordinary continuous functions, all functions in the larger collection will be continuous. This is accomplished by using finer topologies on $${\mathbb R}^n$$ and the usual topology on $${\mathbb R}.$$

# 6. Čech Closure Operators $$\;$$ [satisfy (K1), (K2), (K3)]

Definition: We say that $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$$ is a Čech closure operator on $$X$$ if $$T$$ is a base closure operator on $$X$$ such that for each $$A \in {\mathcal P}(X)$$ we have $$A \subseteq T(A).$$

Note that a Čech closure operator differs from a topological closure operator in that (K4), which says that $$T = T^{2},$$ is weakened to $$T \leq T^{2}.$$ Thus, if a Čech closure operator $$T$$ satisfies $$T^2 \leq T,$$ then $$T$$ is a topological closure operator.

The corresponding generalized topological spaces are also called pretopological spaces. I’m using Čech’s name because Eduard Čech (1893-1960) studied these generalized closure functions extensively in his massive 893 page book [9] (1966 English translation). Čech also published a survey study of these notions in 1937. For some historical details about this part of Čech’s work, see Koutsk [40] (especially pp. 108-109). For instance, Čech was motivated by the fact that the idempotency property of closure was not satisfied by many spaces of functions for various notions of convergence. “He supposed that this transition to the more general concept would create no difficulties for the participants of the seminar and therefore without any hesitation he went on with his lectures. However, there were some difficulties. The concept of closures of sets supposed by axiom (4) [= idempotency] had been meanwhile so deeply rooted in the considerations of some members that misunderstandings occurred too frequently.” (Koutsk [40], near bottom of p. 108)

An example of a base closure operator that is not a Čech closure operator is the ordinary derived set operator $$D:{\mathcal P}(\mathbb {R}) \rightarrow {\mathcal P}(\mathbb {R}),$$ where for example $$D(\mathbb Z) = \emptyset,$$ and hence $$A \subseteq D(A)$$ is false for $$A = {\mathbb Z}.$$

Example (zero distance from a set): Let $$X$$ be a set and let $$d:X \times X \rightarrow [0,\infty)$$ satisfy $$d(x,x) = 0$$ and $$d(x,y) = d(y,x)$$ for all $$x,\,y \in X.$$ Thus, $$(X,d)$$ is what we get if we begin with a metric space and drop the triangle inequality and drop the assumption that $$d(x,y) = 0$$ implies $$x = y.$$ Čech [9] calls this a semi-pseudometric space. Then $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$$ is a Čech closure operator, where $$T$$ is defined by

$$T(A) \; = \; \left\{x \in X: \; \inf_{a \in A} d(x,a) \leq 0 \right\}.$$

The reason for using $$\leq 0$$ rather than $$= 0$$ is to accommodate the convention that $$\inf \emptyset = -\infty.$$ Čech [9] (p. 302) observes that if $${\lambda}^{*}$$ and $${\lambda}_{*}$$ are the Lebesgue outer and inner measures on $$[0,1],$$ then $$d^{*}(A,B) = {\lambda}^{*}(A\,\triangle\,B)$$ defines a pseudometric on $${\mathcal P}([0,1])$$ and $$d_{*}(A,B) = {\lambda}_{*}(A\,\triangle\,B)$$ defines a semi-pseudometric on $${\mathcal P}([0,1])$$ that is not a pseudometric on $${\mathcal P}([0,1]).$$ $$(A\,\triangle\,B$$ is the symmetric difference of $$A$$ and $$B.)$$

Čech Modification of a Base Closure Operator: Given a base closure operator $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X),$$ we can obtain a Čech closure operator $$T^{\text {Čech}}$$ on $$X$$ by letting $$T^{\text {Čech}}(A) \triangleq A \cup T(A)$$ for each $$A \in {\mathcal P}(X).$$

It is not difficult to show that $$T^{\text {Čech}}$$ is the smallest Čech closure operator greater than or equal to $$T.$$

Since every Čech closure operator is a base closure operator, the same notion of a $$T$$-closed set continues to apply when $$T$$ is a Čech closure operator and, for a fixed Čech closure operator $$T,$$ the collection of $$T$$-closed sets satisfies the axioms to be the closed sets for some topology $${\tau}^T$$ on $$X.$$ Thus, repeating what I’ve already said in the more general case of base closure operators, for any Čech closure operator $$T$$ we have $$\;T \leq {\tau}^T{\text {-}}{\text {Cl}},$$ where $${\tau}^T{\text {-}}{\text {Cl}}$$ is the topological closure operator for the topological space $$(X,{\tau}^T).$$

Now suppose we begin with a base closure operator $$T.$$ From $$T$$ we obtain the Čech modification $$T^{\text {Čech}},$$ and from $$T^{\text {Čech}}$$ we obtain the collection of $$T^{\text {Čech}}$$-closed sets, and from the collection of $$T^{\text {Čech}}$$-closed sets we obtain the topological closure operator $${\tau}^{T^{\text {Čech}}}{\text {-}}{\text {Cl}}$$ for the topological space $$(X,{\tau}^{T^{\text {Čech}}}).$$ Since the collection of $$T$$-closed sets is identical to the collection of $$T^{\text {Čech}}$$-closed sets (proof in a moment), it follows that $$(X,{\tau}^T) = (X,{\tau}^{T^{\text {Čech}}}).$$ [[ Proof: $$(\subseteq)$$ Let $$A \subseteq X$$ be $$T$$-closed. Then $$T(A) \subseteq A.$$ Hence, $$T^{\text {Čech}}(A) \triangleq A \cup T(A) \subseteq A,$$ which shows that $$A$$ is $$T^{\text {Čech}}$$-closed. $$(\supseteq)$$ Now let $$A \subseteq X$$ be $$T^{\text {Čech}}$$-closed. Then $$T^{\text {Čech}}(A) \subseteq A,$$ and hence $$A \cup T(A) \subseteq A,$$ which implies $$T(A) \subseteq A,$$ so $$A$$ is $$T$$-closed. ]]

Again, suppose $$T$$ is a base closure operator. To summarize the results above, we have

$$T \; \leq \; T^{\text {Čech}} \; \leq \; {\tau}^T{\text {-}}{\text {Cl}}$$

where $$T^{\text {Čech}}$$ is the Čech modification of $$T$$ and $${\tau}^T{\text {-}}{\text {Cl}}$$ is the identical topological closure operator associated with both $$T$$ and $$T^{\text {Čech}}.$$ Lukeš/Malý/Zajíček [42] use the term strong base operator (defined on middle of p. 8 and used frequently thereafter) for a base closure operator $$T$$ such that $$T^{\text {Čech}} = {\tau}^T{\text {-}}{\text {Cl}}$$ (the main interest being, of course, when $$T \neq T^{\text {Čech}}),$$ which they show is equivalent (Theorem 1.2 at the bottom of p. 7) to the property $$T^2 \leq T.$$ [[ To prevent possible confusion, if $$T$$ were a Čech closure operator, then as we mentioned at the beginning of this section, the inequality $$T^2 \leq T$$ would imply that $$T$$ is a topological closure operator, and hence we would have $$T = {\tau}^T{\text {-}}{\text {Cl}}.$$ However, the $$T$$ that we're considering here is only assumed to be a base closure operator, and it happens to be the case that we can have both $$T^2 \leq T$$ and $$T \neq {\tau}^T{\text {-}}{\text {Cl}}.$$ ]] Moreover, if $$T$$ is a base closure operator, then $$T^{\text {*base}}:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$$ defined by $$T^{\text {*base}}(A) = {\tau}^T{\text {-}}{\text {Cl}}(T(A))$$ is the smallest strong base closure operator greater than $$T$$ (see top half of p. 9), and thus we can consider $$T^{\text {*base}}$$ to be the strong base modification of the base closure operator $$T.$$ Finally, at least for here (see [42] for still more such results), 1.A.8 at the bottom of p. 12 states that for a base closure operator $$T,$$ the operators $$T$$ and $${\tau}^T{\text {-}}{\text {Cl}}$$ commute $$-$$ that is, we have $$T \circ ({\tau}^T{\text {-}}{\text {Cl}}) = ({\tau}^T{\text {-}}{\text {Cl}}) \circ T.$$

Various results about Čech closure operators can be found in many of the references below, but probably the most helpful along with the specific citations above are Čech [9] (especially pp. 237-241, 250-254, 272-275, 555-564, 832-833, 861-863), Kannan [35] (pp. 42-43), Roth [55], Roth/Carlson [56], and Thron [65]. Dolecki/Greco [17] study, among other things, equivalence classes of Čech closure operators that generate the same topological closure operator.

# 7. Topological Closure Operators $$\;$$ [satisfy (K1), (K2), (K3), (K4)]

Definition: We say that $$T:{\mathcal P}(X) \rightarrow {\mathcal P}(X)$$ is a topological closure operator on $$X$$ if $$T$$ is a Čech closure operator on $$X$$ such that for each $$A \in {\mathcal P}(X)$$ we have $$T(T(A)) = T(A).$$

Recall that if $$T$$ is a Čech closure operator, then $$T \leq T^{2}.$$ Since the composition of two Čech closure operators is a Čech closure operator, it follows that if $$T$$ is a Čech closure operator, then we have

$$T \; \leq \; T^2 \; \leq \; T^3 \; \leq \; T^4 \; \leq \; \cdots,$$

where $$\;T^2 = T \circ T,\;$$ $$\;T^3 = T \circ T \circ T,\;$$ etc. are the finite composition iterates of $$T.$$ This is exactly the same situation we had earlier when we were iterating a monotone closure operator. In this case, each Čech closure operator can be transfinitely iterated to obtain a topological closure operator.

Example (Baire function hierarchy): Let $$X = {\mathbb R}^{\mathbb R}$$ be the set of functions from $$\mathbb R$$ to $${\mathbb R}.$$ Given $${\mathcal A} \in {\mathcal P}(X)\;$$ (thus, $$\mathcal A$$ is a set of functions from $$\mathbb R$$ to $${\mathbb R}),\;$$ let $$T(\mathcal A) = \{f \in X: \; f \; {\text {is a pointwise limit of a sequence of functions in}} \; \mathcal A \}.$$ Then $$T$$ is a Čech closure operator on $$X.$$ Moreover, if $$\mathcal C$$ is the set of continuous functions, then $$T(\mathcal C)$$ is the set of Baire $$1$$ functions, $$T^2(\mathcal C)$$ is the set of Baire $$2$$ functions, etc. In this case $$T$$ has to be iterated $${\omega}_1$$ many times before we get a topology (the topology of pointwise convergence), and $${\tau}^T{\text {-}}{\text {Cl}}(\mathcal C)$$ $$-$$ the corresponding topological closure operator applied to the set of continuous functions $$-$$ is equal to the set of Borel measurable functions. See this Mathematics Stack Exchange answer by Andrés E. Caicedo for more details.

Topological Modification of a Čech Closure Operator: Let $$T$$ be a Čech closure operator on $$X.$$ For each ordinal $$\alpha \geq 1$$ we define $$T^{\alpha}$$ by transfinite induction:

$$\begin{array}{l} T^1 & = & T \\ T^{\alpha + 1} & = & T \circ T^{\alpha} \\ T^{\lambda} & = & \sup \,\{T^{\gamma}: \; \gamma < \lambda \}, \;\; {\text{if}} \; \lambda \; {\text {is a nonzero limit ordinal}} \end{array}$$

The following results hold:

(a) $$\;\alpha \leq \beta \;$$ implies $$\;T^{\alpha} \leq T^{\beta}$$

(b) $$\;$$For each ordinal $$\alpha \geq 1,$$ we have $$T^{\alpha}$$ is a Čech closure operator on $$X.$$

(c) $$\;$$For each ordinal $$\alpha \geq 1,$$ we have $$\;{\tau}^{T^{\alpha}}{\text {-}}{\text {Cl}} = {\tau}^T{\text {-}}{\text {Cl}}.\;$$ That is, all the iterates of $$T$$ have the same corresponding topological closure operator.

(d) $$\;$$There exists an ordinal $$\lambda \leq |X|^{+}$$ such that $$T^{\alpha} = T^{\lambda}$$ for all $$\alpha \geq \lambda.$$

(e) $$\;$$Let $$|T|$$ be the least of the ordinals $$\lambda$$ in (d) above. If $$\;1 \leq \alpha < \beta \leq |T|,\;$$ then $$\;T^{\alpha} < T^{\beta}.$$

(f) $$\;$$If $$\alpha \geq |T|,\;$$ then $$\;T^{\alpha} = {\tau}^T{\text {-}}{\text {Cl}}.$$

(g) $$\;$$Let $$(X,\tau)$$ be a topological space and let $${\tau}{\text {-}}{\text {Cl}}$$ be the topological closure operator for $$(X,\tau).$$ Then $${\tau}{\text {-}}{\text {Cl}} = \sup {\mathcal C},$$ where $$\mathcal C$$ is the collection of all Čech closure operators $$T$$ such that $${\tau}^T{\text {-}}{\text {Cl}} = {\tau}{\text {-}}{\text {Cl}}.$$

Regarding (g), note that each of the iterates $$T^{\alpha}$$ belongs to $${\mathcal C},$$ but since there might be other Čech closure operators, besides these, that have $${\tau}{\text {-}}{\text {Cl}}$$ for their corresponding topological closure operator, (g) is not automatic from (c) alone. In fact, this possibility can actually arise, and examples can be found in Dolecki/Greco [17].

Various aspects of the above results can be found in Čech [9] (pp. 274-275, 836), Hammer [23], Hausdorff [30], Kannan [35] (pp. 42-43), Kent/Richardson [37], Lukeš/Malý/Zajíček [42] (Exercise 1.A.10 on p. 13), and Novák [51].

SELECTED REFERENCES

# 8. Selected References

The references below are mostly restricted to those that specifically involve, entirely or partly, various generalized closure operator notions. Thus, the literature on generalized topological notions dealing with the vast zoo of various semi-open set notions, various types of separation axioms in generalized spaces, various generalized continuity notions, category-theoretic connections, etc. are not included unless I thought it had specific relevance to something discussed here.

[1] Peter Henry George Aczel, An introduction to inductive definitions, pp. 739-782 in Jon Kenneth Barwise (editor), Handbook of Mathematical Logic, North-Holland, 1977, xi + 1165 pages.

[2] Jon Kenneth Barwise, Admissible Sets and Structures. An Approach to Definability Theory, Perspectives in Mathematical Logic #13, Springer-Verlag, 1975, xiii + 394 pages.

[3] Jon Kenneth Barwise and Lawrence Stuart Moss, Vicious Circles. On the Mathematics of Non-Wellfounded Phenomena, CSLI Lecture Notes #60, 1996, x + 390 pages.

[4] Mary Katherine Bennett, Convexity closure operators, Algebra Universalis 10 #3 (1980), 345-354.

[5] Garrett Birkhoff, Lattice Theory, corrected reprint of 3rd edition, American Mathematical Society Colloquium Publications #25, American Mathematical Society, 1979, vi + 418 pages.

[6] Jürgen Bliedtner and Peter Albert Loeb, The optimal differentiation basis and liftings of $$L^{\infty}$$, Transactions of the American Mathematical Society 352 #10 (October 2000), 4693-4710.

[7] Donald Jerome Brown and Roman Suszko, Abstract Logics, Dissertationes Mathematicae 102 (1973), 9-41.

[8] Stanley Neal Burris, Theory of Pre-Closures, Ph.D. dissertation (under Allen Seymour Davis), University of Oklahoma, 1968, iv + 47 pages.

This is a study of closure operators that satisfy (semi-K1) and (K3).

[9] Eduard Čech, Topological Spaces, revised and translated by Zdeněk Frolík and Miroslav Katětov, John Wiley and Sons, 1966, 893 pages.

Čech published a survey of his Čech closure operator work in 1937: Topologické prostory, Časopis pro Pěstování Matematiky a Fysiky 66 #4 (1937), D225-D264.

[10] Ákos Császár, Generalized open sets, Acta Mathematica Hungarica 75 #1-2 (April 1997), 65-87.

In this paper and the next paper, Császár studies various topological ideas in the context of a generalized topological space defined by a monotone closure operator, although for many of the results he (explicitly) assumes additional hypotheses, as needed.

[11] Ákos Császár, On the $$\gamma$$-interior and* $$\gamma$$-closure of a set*, Acta Mathematica Hungarica 80 #1-2 (July 1998), 89-93.

[12] Sterling Gene Crossley, Semi-Topological Properties and Related Topics, Ph.D. dissertation (under Shelby Keith Hildebrand), Texas Technological College [after 1968: Texas Tech University], 1968, iii + 76 pages.

[13] Dirk van Dalen, Hans Cornelis Doets, and Henricus Cornelius Maria de Swart, Sets: Naïve, Axiomatic and Applied, International Series in Pure and Applied Mathematics #106, Pergamon Press, 1978, xviii + 339 pages.

[14] Mahlon Marsh Day, Convergence, closure and neighborhoods, Duke Mathematical Journal 11 #1 (March 1944), 181-199.

[15] Vladimir Devidé, On monotone mapings [sic] of the power-set, Portugaliae Mathematica 21 #2 (1962), 111-112.

[16] Keith James Devlin, The Joy of Sets. Fundamentals of Contemporary Set Theory, 2nd edition, Undergraduate Texts in Mathematics, Springer-Verlag, 1993, x + 192 pages.

[17] Szymon Dolecki and Gabriele H. Greco, Topologically maximal pretopologies, Studia Mathematica 77 #3 (1984), 265-281.

[18] Herbert Bruce Enderton, Elements of Set Theory, Academic Press, 1977, xiv + 279 pages.

[19] Preston Clarence Hammer, General topology, symmetry, and convexity, Transactions of the Wisconsin Academy of Sciences, Arts and Letters 44 (1955), 221-255.

[20] Preston Clarence Hammer, Kuratowski’s closure theorem, Nieuw Archief voor Wiskunde (3) 8 (1960), 74-80.

[21] Preston Clarence Hammer, Extended topology: reduction of limit functions, Nieuw Archief voor Wiskunde (3) 9 #1 (1961), 16-24.

[22] Preston Clarence Hammer, Semispaces and the topology of convexity, pp. 305-316 in Victor LaRue Klee (editor), Convexity, Proceedings of Symposia in Pure Mathematics 7, American Mathematical Society, 1963.

[23] Preston Clarence Hammer, Extended topology: set-valued set-functions, Nieuw Archief voor Wiskunde (3) 10 #? (1962), 55-77.

[24] Preston Clarence Hammer, Extended topology: additive and subadditive subfunctions of a function, Rendiconti del Circolo Matematico di Palermo (2) 11 #3 (1962), 262-270.

[25] Preston Clarence Hammer, Extended topology: perfect sets, Portugaliae Mathematica 23 #1 (1964), 27-34.

[26] Preston Clarence Hammer, Extended topology: the continuity concept, Mathematics Magazine 36 #2 (March 1963), 101-105.

[27] Preston Clarence Hammer, Extended topology: Continuity I, Portugaliae Mathematica 23 #2 (1964), 77-93.

[28] Preston Clarence Hammer, Extended topology: connected sets and Wallace separations, Portugaliae Mathematica 22 #4 (1963), 167-187.

[29] Preston Clarence Hammer, Extended topology: structure of isotonic functions, Journal für die reine und angewandte Mathematik 213 #3-4 (1964), 174-186.

[30] Felix Hausdorff, Gestufte räume, Fundamenta Mathematicae 25 (1935), 486-502.

[31] Arie Hinkis, Proofs of the Cantor-Bernstein Theorem. A Mathematical Excursion, Science Networks / Historical Studies #45, Birkhäuser, 2013, xxiv + 429 pages.

[32] Peter Greayer Hinman, Recursion-Theoretic Hierarchies, Perspectives in Mathematical Logic #7, Springer-Verlag, 1978, xii + 480 pages.

[33] Karel Hrbacek and Thomas J. Jech, Introduction to Set Theory, 3rd edition, Pure and Applied Mathematics #220, Marcel Dekker, 1999, xii + 291 pages.

[34] Robert Edward Jamison, A General Theory of Convexity, Ph.D. dissertation (under Victor LaRue Klee), University of Washington, 1974, v + 120 pages.

[35] Varadachariar Kannan, Ordinal Invariants in Topology, Memoirs of the American Mathematical Society 32 #245 (July 1981), vi + 164 pages.

[36] T. Kavitha, Some Problems on Čech Closure Spaces, Ph.D. dissertation (under P. T. Ramachandran), University of Calicut (Kerala, India), August 2017, viii + 139 pages.

Generalized Čech closure operators $$-$$ closure operators that satisfy (semi-K1), (K2), (K3) $$-$$ are studied in Sections 5.4-5.4 on pp. 120-131.

[37] Darrell Conley Kent and Gary Douglas Richardson, The decomposition series of a convergence space, Czechoslovak Mathematical Journal 23 #3 (1973), 437-446.

[38] Bronisaw Knaster, Un théorème sur les fonctions d’ensembles [A theorem on functions of sets], Annales de la Société Polonaise de Mathématique [= Rocznik Polskiego Towarzystwa Matematycznego] 6 (1927), 133-134.

This is not a published paper, but rather an abstract from the section titled Comptes-rendus des séances de la Société Polonaise de Mathématique Section de Varsovie [Reports of the sessions of the Warsaw Section of the Polish Mathematics Society] (session taking place 9 December 1927). The results were obtained jointly with Tarski and presented by Knaster.

[39] William B. Koenen, The Kuratowski closure problem in the topology of convexity, American Mathematical Monthly 73 #7 (August-September 1966), 704-708.

[40] Karel Koutský, Čech's topological seminar in Brno, 1936-1939, Časopis pro Pěstování Matematiky 90 #1 (1965), 104-117.

[41] Yinbin Lei and Jun Zhang, Generalizing topological set operators, Electronic Notes in Theoretical Computer Science 345 (28 August 2019), 63-76.

[42] Jaroslav Lukeš, Jan Malý, and Luděk Zajíček, Fine Topology Methods in Real Analysis and Potential Theory, Lecture Notes in Mathematics #1189, Springer-Verlag, 1986, x + 472 pages.

[43] Jacek Malinowski, On generalizations of consequence operation, Bulletin of the Section of Logic (University of Łódź) 31 #2 (2002), 135-143.

[44] Zlatko P. Mamuzić, Introduction to General Topology, translated by William Joseph Pervin and James Leo Sieber and Robert C. Moore, P. Noordhoff (Netherlands), 1963, 159 pages.

[45] Norman Marshall Martin and Stephen Randall Pollard, Closure Spaces and Logic, Mathematics and Its Applications #369, Kluwer Academic Publishers, 1996, xviii + 230 pages.

The closure spaces of this book satisfy (semi-K1), (K3), and $$T^2 \leq T.$$ That is, from the topological closure operator axioms we drop half of (K1), (K2), and half of (K4).

[46] Elliott Mendelson, Schaums Outline of Theory and Problems of Boolean Algebra and Switching Circuits, Schaums Outline Series, McGraw-Hill Book Company, 1970, viii + 213 pages.

[47] Eliakim Hastings Moore, Introduction to a Form of General Analysis, pp. vii-viii + 11-50 in American Mathematical Society Colloquium Publications #2, Yale University Press, 1910, x + 222 pages.

Moore’s extensionally attainable property (defined on middle of p. 59) is often cited as an early, and perhaps the earliest, formulation of a general closure operator notion. Supposedly it is equivalent to a closure operator $$T$$ on a set that satisfies (semi-K1), (K3), and $$T^2 \leq T$$ (e.g. pp. 16-17 of Martin/Pollard [45]), but I have not attempted to verify this or consider whether such claims might be somewhat ahistorical. For a historical study of Moore’s General Analysis program, see Reinhard Siegmund-Schultze, Eliakim Hastings Moore’s “General Analysis”, Archive for History of Exact Sciences 52 #1 (January 1998), 51-89.

[48] Yiannis Nicholas Moschovakis, Elementary Induction on Abstract Structures, Studies in Logic and the Foundations of Mathematics #77, North-Holland, 1974, x + 218 pages.

[49] Yiannis Nicholas Moschovakis, Descriptive Set Theory, Studies in Logic and the Foundations of Mathematics #100, North-Holland, 1980, xii + 637 pages.

[50] Yiannis Nicholas Moschovakis, Notes on Set Theory, 2nd edition, Undergraduate Texts in Mathematics, Springer-Verlag, 2006, xii + 276 pages.

[51] Josef Novák, On some problems concerning multivalued convergences, Czechoslovak Mathematical Journal 14 #4 (1964), 548-561.

[52] Horst Wolfram Pohlers, Proof Theory, Lecture Notes in Mathematics #1407, Springer-Verlag, 1989, viii + 213 pages.

[53] Marlon Cecil Rayburn, On the lattice of $$\sigma$$-algebras, Canadian Journal of Mathematics 21 (1969), 755-761.

[54] Sheldon Theodore Rio, On the Hammer Topological System, Ph.D. dissertation (under Henry Arnold Bradford), Oregon State College [after 1961: Oregon State University], June 1959, iv + 53 pages.

[55] David Niel Roth, Čech Closure Spaces, Master of Arts thesis (under John Warnock Carlson), Emporia State University (Emporia, Kansas), July 1979, iv + 25 pages.

[56] David Niel Roth and John Warnock Carlson, Čech closure spaces, Kyungpook Mathematical Journal 20 #1 (June 1980), 11-30.

[57] Marcin Jan Schroeder, Modification of pre-closure spaces as closure/interior operations on the lattice of pre-closure operators, RIMS Kôkyûroku #1712 (Algebras, Languages, Algorithms in Algebraic Systems and Computations), Research Institute for Mathematical Sciences (Kyoto University, Japan), September 2010, 148-155.

[58] Gideon Ernst Schwarz, A note on transfinite iteration, Journal of Symbolic Logic 21 #3 (September 1956), 265-266.

See the half-page review by Dana Stewart Scott in Journal of Symbolic Logic 22 #3 (September 1957), p. 303.

[59] Wacław Franciszek Sierpiński, Algèbra des Ensembles [Algebra of Sets], Monograe Matematyczne #23, Nakadem Polskiego Towarzystwa Matematycznego (Warszawa-Wrocaw), 1951, ii + 205 pages.

[60] Wacław Franciszek Sierpiński, General Topology, translated by Cypra Cecilia Krieger, 2nd edition, Mathematical Expositions #7, University of Toronto Press, 1952, xii + 290 pages.

[61] Bärbel M. R. Stadler and Peter Florian Stadler, Generalized topological spaces in evolutionary theory and combinatorial chemistry, Journal of Chemical Information and Computer Sciences 42 #3 (May 2002), 577-585.

[62] Bärbel M. R. Stadler and Peter Florian Stadler, Basic properties of closure spaces, unpublished manuscript, 29 January 2002 (web release date), 20 pages.

[63] Alfred Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics 5 #2 (June 1955), 285-309.

The main results were announced in A fixpoint theorem for lattices and its applications (abstract #496, received 27 June 1949), Bulletin of the American Mathematical Society 55 #11 (November 1949), 1051-1052.

[64] Wolfgang Joseph Thron, Topological Structures, Holt, Rinehart and Winston, 1966, x + 240 pages.

[65] Wolfgang Joseph Thron, What results are valid on closure spaces, Topology Proceedings 6 (1981), 135-158.

[66] Marcel Lodewijk Johanna van de Vel, Theory of Convex Structures, North-Holland Mathematical Library #50, Elsevier Science Publishers B.V., 1993, xv + 540 pages.

[67] Stephen W. Willard, General Topology, Addison-Wesley, 1970, xii + 369 pages.

[68] Ryszard Wójcicki, Theory of Logical Calculi. Basic Theory of Consequence Operations, Synthese Library: Studies in Epistemology, Logic, Methodology, and Philosophy of Science #199, Kluwer Academic Publishers, 1988, xviii + 473 pages.

[69] Rafał Ryszard Zduńczyk, Simple systems and closure operators, Bulletin de la Société des Science et des Lettres de Łódź, Recherches sur les Déformations, 66 # 3 (2016), 65-72.

Couldn't read the Russian image text but I'll take a stab at it:

If there are no restrictions on $$M\mapsto\overline M$$ at all then it's just a unary set operation: a function from the power set of $$R$$ to itself.

• Sure thing. I'll accept the answer, but I want to wait if the structure has any specific name. In the source book, it is used to axiomatize all alternative definitions. – Leonid Dworzanski Sep 24 '19 at 2:45
• I've found a paper where "closure operator" is stated as in the question: 'A closure operator on X is a function c: ℘(X)→℘(X)' Thron, W. J. "What results are valid on closure spaces." Topology Proc. Vol. 6. 1981. – Leonid Dworzanski Sep 24 '19 at 3:49
• In another paper, it is explicitly stated that such function is arbitrary. – Leonid Dworzanski Sep 24 '19 at 4:05
• @LeonidDworzanski Great! I had a hunch that complementation and closure have basically nothing in common... – Bjørn Kjos-Hanssen Sep 24 '19 at 4:14


text:
# Calculus

to evaluate (integral) 3x^2 cos (2x^3-4) dx, it is necessary to let

A. u=3x^2
B. u=6x
C. u=2x^3-4
D. u=6x^2

1. 👍 0
2. 👎 0
3. 👁 102
1. C

then du = 2(3x^2) dx

1. 👍 0
2. 👎 0

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format: plain


text:
# How to get sentence embedding using BERT?

How to get sentence embedding using BERT?

from transformers import BertTokenizer
tokenizer=BertTokenizer.from_pretrained('bert-base-uncased')
sentence='I really enjoyed this movie a lot.'
#1.Tokenize the sequence:
tokens=tokenizer.tokenize(sentence)
print(tokens)
print(type(tokens))

# 2. Add [CLS] and [SEP] tokens:

tokens = ['[CLS]'] + tokens + ['[SEP]']
print(" Tokens are \n {} ".format(tokens))

# 3. Padding the input:

T=15
attn_mask=[ 1 if token != '[PAD]' else 0 for token in padded_tokens  ]

# 4. Maintain a list of segment tokens:

seg_ids=[0 for _ in range(len(padded_tokens))]
print("Segment Tokens are \n {}".format(seg_ids))

# 5. Obtaining indices of the tokens in BERT’s vocabulary:

sent_ids=tokenizer.convert_tokens_to_ids(padded_tokens)
print("senetence idexes \n {} ".format(sent_ids))
token_ids = torch.tensor(sent_ids).unsqueeze(0)
seg_ids   = torch.tensor(seg_ids).unsqueeze(0)

# Feed them to BERT

hidden_reps, cls_head = bert_model(token_ids, attention_mask = attn_mask,token_type_ids = seg_ids)
print(type(hidden_reps))
print(hidden_reps.shape ) #hidden states of each token in inout sequence
print(cls_head.shape ) #hidden states of each [cls]

output:
hidden_reps size
torch.Size([1, 15, 768])

torch.Size([1, 768])

Which vector represents the sentence embedding here? Is it hidden_reps or cls_head ?

Is there any other way to get sentence embedding from BERT in order to perform similarity check with other sentences?

There is actually an academic paper for doing so. It is called S-BERT or Sentence-BERT.
They also have a github repo which is easy to work with.

Which vector represents the sentence embedding here? Is it hidden_reps or cls_head?

If we look in the forward() method of the BERT model, we see the following lines explaining the return types:

outputs = (sequence_output, pooled_output,) + encoder_outputs[1:]  # add hidden_states and attentions if they are here
return outputs  # sequence_output, pooled_output, (hidden_states), (attentions)

So the first element of the tuple is the "sentence output" - each token in the input is embedded in this tensor. In your example, you have 1 input sequence, which was 15 tokens long, and each token was embedding into a 768-dimensional space.

The second element of the tuple is the "pooled output". You'll notice that the "sequence" dimension has been squashed, so this represents a pooled embedding of the input sequence.

So they both represent the sentence embedding. You can think of hidden_reps as a "verbose" representation, where each token has been embedded. You can think of cls_head as a condensed representation, where the entire sequence has been pooled.

Is there any other way to get sentence embedding from BERT in order to perform similarity check with other sentences?

Using the transformers library is the easiest way I know of to get sentence embeddings from BERT.

There are, however, many ways to measure similarity between embedded sentences. The simplest approach would be to measure the Euclidean distance between the pooled embeddings (cls_head) for each sentence.

• zachdj thanks for the information . So should i use hidden_reps or cls_head to get sentence vector ? clas_head has only 1 vector with 768 dimension but hidden_reps has 15 vectors with 768 dimension . How should i convert these 15 vectors into single vector ? should i add or do mean or any other way to get the 15 token vectors represented into a single vector . – Aj_MLstater Nov 5 '19 at 12:53
• There are many ways to pool the 15 token embeddings into a single vector. You could take use mean pooling or max pooling. You could also avoid pooling altogether and use all 15 embeddings. – zachdj Nov 5 '19 at 17:29
• For your question about whether to use hidden_reps or cls_head, it just depends on what you're trying to do. They both represent the sentence. One represents each token, and one has already been pooled. – zachdj Nov 5 '19 at 17:30
• @zachdji thanks for the information .Can you share the syntax for mean pool and max pool i tired torch.mean(hidden_reps[0],1) but when i tried to find cosin similarity for 2 different sentences it gave me high score .So not sure whether im doing the right way to get the sentence embedding . – Aj_MLstater Nov 6 '19 at 11:15
• s1=what is your age? tensor([-0.0106, -0.0101, -0.0144, -0.0115, -0.0115, -0.0116, -0.0173, -0.0071, -0.0083, -0.0070], grad_fn=<MeanBackward1>) s2='Today is monday' tensor([-0.0092, -0.0094, -0.0113, -0.0106, -0.0166, -0.0071, -0.0073, -0.0074, -0.0080, -0.0076], grad_fn=<MeanBackward1>) cos = torch.nn.CosineSimilarity(dim=0) score was 0.93 .But ideally it should be very less as 2 sentences are not similry not sure why berth is giving high score . – Aj_MLstater Nov 6 '19 at 11:15

There is very cool tool called bert-as-service which does the job for you. It maps a sentence to a fixed length word embeddings based on the pre trained model you use. It also allows a lot of parameter tweaking which is covered extensively in the documentation.

In your example, the hidden state corresponding to the first token ([CLS]) in hidden_reps can be used as a sentence embedding.

By contrast, the pooled output (mistakenly referred to as hidden states of each [cls] in your code) proved a bad proxy for a sentence embedding in my experiments.

bert-as-service provides a very easy way to generate embeddings for sentences.

It is explained very well in the bert-as-service repository:

Installations:

pip install bert-serving-server  # server
pip install bert-serving-client  # client, independent of bert-serving-server

Download one of the pre-trained models available at here.

Start the service:

bert-serving-start -model_dir /your_model_directory/ -num_worker=4

Generate the vectors for the list of sentences:

from bert_serving.client import BertClient
bc = BertClient()
vectors=bc.encode(your_list_of_sentences)


text:
# Kinematics | Problem 1.4

Irodov I.E. - Problems in General Physics (1981)

## PHYSICAL FUNDAMENTALS OF MECHANICS | Kinematics

### Problem 1.4

A point moves rectilinearly in one direction. Fig.1.1 shows the distance $$s$$ traversed by the point as a function of the time $$t$$.
(c) the time moment $$t_0$$ at which the instantaneous velocity is equal to the mean velocity averaged over the first $$t_0$$ seconds.


text:
# MatLab updating a recursive variable.

See the code

x(1) = 0;
k = 1;
while x(k) <=6

x(k) = x(k) +2

k = k +1;

end

So I want the code to output

x(1) = 2

x(2) = 4

x(3) = 6

Basically I want to write out a list of x(i)s, that gets updated during each iteration. But I am getting an "out of bound" error and I realize that there is a conflict with the way MatLab sees "()" as a vector.

Anyway to get around this?

EDIT1

What if I want to do

x = 0;
y = 0;
k = 1;
while x <=6 && y<=13

x = x +2;
y = y + 3 +x;

k = k +1;

end

I want this to output

x = 2

y = 5

x = 4

y = 8

But it's not doing that, and I don't want to do this in two separate scripts. I want to handle everything at once.

-
Print Step #$k$ and then $x = ?$. –  John Habert Feb 7 '14 at 7:02
@JohnHabert, wait what? –  Hawk Feb 7 '14 at 7:06
Skip trying to label your variable recursively. Just print the step number and the answer for that step. You are already counting what step you are at with $k$. –  John Habert Feb 7 '14 at 7:07
You mean just have $x(k) + 2$ instead of $x(k) = x(k) +2$? I still get errors –  Hawk Feb 7 '14 at 7:10
I'm saying just do $x = x+2$; and for output generate something that looks like: Step #1 : x = 2, Step #2 : x = 4, etc. –  John Habert Feb 7 '14 at 7:12

There's an error in your code, in the line

x(k)=x(k) + 2

because x(k) is not yet defined. See the correct code:

x(1) = 0;
k = 1;
while x(k) <=6

x(k+1) = x(k) +2

k = k +1;

end
-
What if I want to update two variables at once? –  Hawk Feb 7 '14 at 8:13
Provide an example. –  5xum Feb 7 '14 at 8:13
See edit post please –  Hawk Feb 7 '14 at 8:15


text:
# zbMATH — the first resource for mathematics

On the structure and chromaticity of graphs in which any two colour classes induce a tree. (English) Zbl 0893.05004
Some families of chromatically unique graphs are constructed through the partition of the vertex set of a graph $$G$$ into independent sets $$A_1,A_2, \dots,A_r$$ such that the subgraph of $$G$$ induced by $$A_i \cup A_j$$ is a tree for all $$i$$ and $$j$$ with $$1\leq i<j \leq r$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C05 Trees
##### Keywords:
chromatically unique graphs; partition; tree
Full Text:
##### References:
 [1] Biggs, N., Algebraic graph theory, (1974), Cambridge Univ. Press Cambridge · Zbl 0284.05101 [2] Chao, C.Y.; Li, N.-Z.; Xu, S.-J., On q-trees, J. graph theory, 10, 129-136, (1986) · Zbl 0591.05021 [3] Chao, C.Y.; Whitehead, E.G., Chromatically unique graphs, Discrete math., 27, 171-177, (1979) · Zbl 0411.05035 [4] Chia, G.L., The chromaticity of wheels with a missing spoke, Discrete math., 82, 209-212, (1990) · Zbl 0712.05025 [5] Chia, G.L., On the join of graphs and chromatic uniqueness, J. graph theory, 19, 251-261, (1995) · Zbl 0819.05027 [6] Dirac, G.A., On rigid circuit graphs, Abh. math. sem. univ. Hamburg, 25, 71-76, (1967) · Zbl 0098.14703 [7] Erdős, P., On the number of complete subgraphs contained in certain graphs, Publ. math. inst. hung. acad. sci., 7, 459-464, (1962) [8] Fisher, D.C., Lower bounds on the number of triangles in a graph, J. graph theory, 13, 505-512, (1989) · Zbl 0673.05046 [9] Han, B.T., Chromaticity of qk-trees, Acta math. appl. sinica, 11, 457-467, (1988), (Chinese, English summary) · Zbl 0666.05032 [10] Koh, K.M.; Teo, K.L., The search for chromatically unique graphs, Graphs combin., 6, 259-285, (1990) · Zbl 0727.05023 [11] Koh, K.M.; Teo, K.L., The search for chromatically unique graphs II, Discrete math., 172, 59-78, (1997) · Zbl 0879.05031 [12] Lovàsz, L.; Simonovits, M., On complete subgraphs of a graph II, Stud. pure math., 459-496, (1983) [13] Read, R.C.; Tutte, W.T., Chromatic polynomials, (), 15-42 · Zbl 0667.05022 [14] Vaderlind, P., Chromatic uniqueness of k-trees, () [15] Whitney, H., A logic expansion in mathematics, Bull. amer. math. soc., 38, 572-579, (1932) · JFM 58.0605.08 [16] Xu, S.-J.; Li, N.-Z., The chromaticity of wheels, Discrete math., 51, 207-212, (1984) · Zbl 0547.05032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
Home > Relative Error > Relative Error Definition And Formula

# Relative Error Definition And Formula

## Contents

Back to Top Suppose the measurement has some errors compared to true values.Relative error decides how incorrect a quantity is from a number considered to be true. For example, when an absolute error in a temperature measurement given in Celsius is 1° and the true value is 2°C, the relative error is 0.5 and the percent error is There are two features of relative error that should be kept in mind. Thank you,,for signing up! http://supercgis.com/relative-error/relative-mean-error-definition.html

For example, if a measurement made with a metric ruler is 5.6 cm and the ruler has a precision of 0.1 cm, then the tolerance interval in this measurement is 5.6 Definition: Relative error is a measure of the uncertainty of measurement compared to the size of the measurement.Also Known As: relative uncertaintyExamples: Three weights are measured at 5.05 g, 5.00 g, Back to Top To calculate the relative error use the following way:Observe the true value (x) and approximate measured value (xo). Graph of f ( x ) = e x {\displaystyle f(x)=e^{x}} (blue) with its linear approximation P 1 ( x ) = 1 + x {\displaystyle P_{1}(x)=1+x} (red) at a = original site

## Relative Error Calculator

Any measurements within this range are "tolerated" or perceived as correct. Then find the absolute deviation using formulaAbsolute deviation $\Delta$ x = True value - measured value = x - xoThen substitute the absolute deviation value $\Delta$ x in relative error formula The approximation error is the gap between the curves, and it increases for x values further from 0. What is Dimensional Formula of Hubble constant ?

How to Calculate the Relative Error? Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view Follow Us Relative Errors Explained With a Definition, Equation, And Examples Relative error is estimated by comparison, unlike an Say we measure any given quantity for n number of times and a1, a2 , a3 …..an are the individual values then Arithmetic mean am = [a1+a2+a3+ …..an]/n am= [Σi=1i=n ai]/n Absolute Error Definition An approximation error can occur because the measurement of the data is not precise due to the instruments. (e.g., the accurate reading of a piece of paper is 4.5cm but since

Contents 1 Formal Definition 1.1 Generalizations 2 Examples 3 Uses of relative error 4 Instruments 5 See also 6 References 7 External links Formal Definition One commonly distinguishes between the relative Absolute Error Formula Question2: If the approximate value of $\pi$ is 3.14. The actual length of this field is 500 feet. The precision of a measuring instrument is determined by the smallest unit to which it can measure.

## Relative Error Chemistry

It gives us the exact number with the units of the quantity that are deviated from the true one. Should the accepted or true measurement NOT be known, the relative error is found using the measured value, which is considered to be a measure of precision. Relative Error Calculator Find the absolute error, relative error and percent of error of the approximation 3.14 to the value , using the TI-83+/84+ entry of pi as the actual value. Relative Error Formula good explanation Muthukumar January 24, 2016 at 11:13 pm Thank you very much.This is usefull for my viva ayushka July 26, 2016 at 12:14 pm very good and easy explanation!

b.) The relative error in the length of the field is c.) The percentage error in the length of the field is 3. his comment is here Generalizations These definitions can be extended to the case when v {\displaystyle v} and v approx {\displaystyle v_{\text{approx}}} are n-dimensional vectors, by replacing the absolute value with an n-norm.[1] Examples As ISBN0-8018-5413-X. ^ Helfrick, Albert D. (2005) Modern Electronic Instrumentation and Measurement Techniques. Anzila July 27, 2016 at 4:17 am simple & perfect explanation [email protected] August 21, 2016 at 5:52 am thanks Post a comment Name Email Website Cancel reply Categories Maths Algebra Physics Relative Error Physics

Relative ErrorProblems Back to Top Below are given some relative error examples you can go through it: Solved Examples Question1: John measures the size of metal ball as 3.97 cm but Avoid the error called "parallax" -- always take readings by looking straight down (or ahead) at the measuring device. Step-by-step Solutions» Walk through homework problems step-by-step from beginning to end. http://supercgis.com/relative-error/relative-error-definition-example.html Please try the request again.

For this same case, when the temperature is given in Kelvin, the same 1° absolute error with the same true value of 275.15 K gives a relative error of 3.63×10−3 and Can Relative Error Be Negative Many of these experiments have standard data values. For example, if you know a length is 3.535 m + 0.004 m, then 0.004 m is an absolute error.


text:
# What is the derivative of f(x)=log_2(cos(x)) ?

Mar 18, 2018

$- \tan \frac{x}{\ln} \left(2\right)$

#### Explanation:

$f \left(x\right) = {\log}_{2} \left(\cos \left(x\right)\right) = \ln \frac{\cos \left(x\right)}{\ln} \left(2\right)$

$\frac{1}{\ln} \left(2\right)$ is just a constant and can be ignored.

$\left(\ln \left(u\right)\right) ' = \frac{u '}{u}$

$u = \cos \left(x\right) , u ' = - \sin \left(x\right)$

$f ' \left(x\right) = \frac{1}{\ln} \left(2\right) \cdot \frac{- \sin \left(x\right)}{\cos} \left(x\right) = - \tan \frac{x}{\ln} \left(2\right)$


text:
# Stack Snippets Sandbox - Try It Out Here!

This post is dedicated to testing out the newly announced Stack Snippets feature. Feel free to answer with your own Stack Snippets, and have some fun!

• Plus points to whoever can find a way to abuse it? :) – Mysticial Aug 25 '14 at 21:10
• @Mysticial one way to find out... ;) – Haney Aug 25 '14 at 21:15
• I'll have to try this out when I get home- Work won't let me run the snippet in the answer below because it's from a "Newly Registered Website" :( – Kendra Aug 25 '14 at 21:42
• @Kendra sounds like a workplace firewall that blocks based on content. stacksnippets.net is a newly registered domain. – Haney Aug 25 '14 at 21:52
• @Haney That's what it was exactly. I'm sure it'll chill after a while. :) – Kendra Aug 25 '14 at 23:43
• Well, there goes PPCG. I honestly don't know if it's a good or bad thing there. Easily testable, yet obfuscated snippets :D – Geobits Aug 26 '14 at 3:07
• NoScript may be getting in the way. Only a few are running for me and I got one warning about cross-site-scripting. OS X 10.8.5; FF 31.0; NoScript 2.6.8.36. – dmckee --- ex-moderator kitten Aug 26 '14 at 3:10
• Seems like my workplace firewall is blocking most of these, which in turn then crashes my current tab, good idea, but if it is blocked at work / public networks then it's going to be a hindrance more than anything :( – Joe Aug 26 '14 at 8:49
• I wonder how long it will be before someone tries to write a snippet that when run applies an upvote to the containing answer (or question)... – JonK Aug 26 '14 at 13:26
• @JayBlanchard The snippets run fine for me in Firefox 31. – RevanProdigalKnight Aug 26 '14 at 13:36
• @JayBlanchard That's good, because I was going to resort to the old "Have you tried turning it off and on again?" and I really didn't want to have to do that. – RevanProdigalKnight Aug 26 '14 at 19:36
• @bfrohs you get points. I start the negotiations at whateverNumberOfPointsYouWant * -1 and let's settle in the middle? – Haney Aug 26 '14 at 21:04
• I would like -9001 points please ;) – 0b10011 Aug 26 '14 at 21:05
• Perfect, I'll counter with 9001 and we'll meet in the middle (0). – Haney Aug 26 '14 at 21:09
• Are the stack snippets broken? Because the server stacksnippets.net seems to have 500 server error issues. And it has been like this for a little while. Any updates on this? – Alexander Johansen Jan 29 '15 at 13:20

## 93 Answers

This could crash your browser. Have a nice day :-)

a=[];while(true){a.push(a)};

• Didn't realize this would crash the browser until I expanded it. I wish stackoverflow would protext against this. – Eric Oct 2 '15 at 8:43

Let

• $(\Omega,\mathcal A)$ be a measurable space and $\mathbb F=(\mathcal F_n)_{n\in\mathbb N_0}$ be a filtration on $(\Omega,\mathcal A)$
• $E$ be an at most countable set equipped with the discrete topology
• $X=(X_n)_{n\in\mathbb N_0}$ be a discrete Markov chain on $(\Omega,\mathcal A)$ with respect to $\mathbb F$ with values in $(E,\mathcal E)$

Let $x\in E$ be recurrent, $$\tau_x^k:=\inf\left{n>\tau_x^{k-1}:X_n=x\right}\;\;\;\text{with }\tau_x^0:=0\;,$$ $t_0:=x$ and $$t_k:=\tau_x^k-\tau_x^{k-1}\;\;\;$$ for $k\in\mathbb N$. How can we prove, that $(t_k)_{k\in\mathbb N_0}$ is independent and identically distributed?

Let $k\in\mathbb N_0$. Then, $\tau:=\tau_x^{k-1}$ is a $\mathbb F$-stopping time. Let $$\tilde X:=\left(X_{\tau+n}\right)_{n\in\mathbb N_0}\;.$$ The strong Markov property yields for all $n\in\mathbb N_0$ $$\begin{split} \operatorname P_x\left[t_k=n\right]&=\operatorname P_x\left[\tilde X_1\ne x,\ldots,\tilde X_{n-1}\ne x\text{ and }\tilde X_n=x\right]\ &=\operatorname P_{X_\tau}\left[X_1\ne x,\ldots,X_{n-1}\ne x\text{ and }X_n=x\right]\ &=\operatorname P_x\left[X_1\ne x,\ldots,X_{n-1}\ne x\text{ and }X_n=x\right] \end{split}\tag 1$$ $\operatorname P_x$-almost surely, since $\operatorname P_x\left[\tau<\infty\right]=1$ by the recurrence of $x$. Since the right-hand side of $(1)$ doesn't depend on $k$, the $t_k$ are indeed identically distributed.

How can we show, that they are independent, too?

Thankyou for the downvotes; I confirmed Fullscreen rendering when the answer is greyed out was wrong. They've now disabled the running of code snippets when the post gets downvoted.

Just testing...

• Hmm. I'm not sure we should worry too much about fullscreen rendering for a negative-points question, but the transparent overlay isn't exactly functional. – AmeliaBR Sep 9 '14 at 1:07
• @AmeliaBR It's not the rendering per se, but the fact that most of the rest of the display is still interactive while you're in fullscreen mode! I edited this comment while fullscreen was active, for example. – Mark Hurd Sep 9 '14 at 1:19
• Hadn't even thought to try that. But I think you'll have to leave a bug report on the current feedback thread to get something to happen -- unless someone is trying out the sandbox themselves,they aren't likely to discover your -13 answer when there are two pages of posts! – AmeliaBR Sep 9 '14 at 3:16
• @AmeliaBR Good point, but this was based upon a downvoted feedback post from the original feedback Q. – Mark Hurd Sep 9 '14 at 3:20
• @AmeliaBR Done. – Mark Hurd Sep 9 '14 at 3:24
• Now you deserve to get the rep back. +1 :) – Martin Braun Sep 10 '14 at 18:46
• @modiX, you don't lose rep on this meta site, so downvotes don't hurt you. – gunr2171 Sep 10 '14 at 19:12
• @modiX Note I made this CW to start with, to avoid any of that :-) But that's also partly why I followed AmeliaBR's suggestion to add a new answer here. – Mark Hurd Sep 11 '14 at 5:51
• @gunr2171 I should mention I did think this was Meta Stack Exchange too. – Mark Hurd Sep 11 '14 at 5:54