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

Question

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.

Answer 1

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

1. Introductory Comments and 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

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Answer 2

THIS CONTINUES MY ANSWER THAT BEGINS HERE

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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

Answer 3

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.

Answer 4

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.