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.



General topological space with closure operation as in Russian translation of Hausdorff's 1914 and 1927 Mengenlehre
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](https://math.stackexchange.com/q/2254363/13130). 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](https://www.google.com/search?q="generalized+topology"+"closure+operator") and [extended topology](https://www.google.com/search?q="extended+topology"+"closure+operator") and [pretopology](https://www.google.com/search?q=%22pretopology%22+%22closure+operator%22).


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](https://www.encyclopediaofmath.org/index.php/Potential_theory,_abstract).
> 
> 
> 


**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**](https://mathoverflow.net/a/343402/15780)



> 
> 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](https://math.stackexchange.com/q/853836/13130) and [this](https://math.stackexchange.com/q/2549424/13130)), 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](https://math.stackexchange.com/q/336370/13130)). 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](https://www.google.com/search?q=%22monotone+inductive+definitions%22) 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 $\kappa$](https://math.stackexchange.com/q/1405858/13130), 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](https://en.wikipedia.org/wiki/Sequential_space#Sequential_closure) 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)](https://mumble.net/~jar/hammer/biography.html), 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*](https://www.google.com/search?q=Hammer+%22extended+topology%22). 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](https://genealogy.math.ndsu.nodak.edu/id.php?id=8559) (1965) and [George Clifford Gastl](https://genealogy.math.ndsu.nodak.edu/id.php?id=8604) (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](https://genealogy.math.ndsu.nodak.edu/id.php?id=13512), 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](https://math.stackexchange.com/q/269284/13130), 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](https://math.stackexchange.com/q/1279969/13130), 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*](https://en.wikipedia.org/wiki/Knaster%E2%80%93Tarski_theorem), [Mathworld: *Tarski's Fixed Point Theorem*](http://mathworld.wolfram.com/TarskisFixedPointTheorem.html), 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](https://math.stackexchange.com/a/54334/13130).


[**SELECTED REFERENCES**](https://mathoverflow.net/a/343402/15780)


[**ANSWER CONTINUES HERE**](https://mathoverflow.net/a/343401/15780)


