View source for Kuratowski closure axioms

In [[topology]] and related branches of [[mathematics]], the '''Kuratowski closure axioms''' are a set of [[axiom]]s that can be used to define a [[topological structure]] on a [[Set (mathematics)|set]]. They are equivalent to the more commonly used [[open set]] definition. They were first formalized by [[Kazimierz Kuratowski]],<ref>{{Harvp|Kuratowski|1922}}.</ref> and the idea was further studied by mathematicians such as [[Wacław Sierpiński]] and [[Antonio Monteiro (mathematician)|António Monteiro]],<ref name=":1" /> among others.

A similar set of axioms can be used to define a topological structure using only the dual notion of [[Interior (topology)#Interior operator|interior operator]].<ref name=":2" />

== Definition ==
===Kuratowski closure operators and weakenings===
Let <math>X</math> be an arbitrary set and <math>\wp(X)</math> its [[power set]]. A '''Kuratowski closure operator''' is a [[unary operation]] <math>\mathbf{c}:\wp(X) \to \wp(X)</math> with the following properties:

{{Quote frame|
quote='''[K1]''' It ''preserves the empty set'': <math> \mathbf{c}(\varnothing) = \varnothing </math>;

'''[K2]''' It is ''extensive'': for all <math>A \subseteq X</math>, <math> A \subseteq \mathbf{c}(A)</math>;

'''[K3]''' It is ''idempotent'': for all <math>A \subseteq X</math>, <math> \mathbf{c}(A) = \mathbf{c}(\mathbf{c}(A))</math>;

'''[K4]''' It {{em|preserves}}/{{em|[[Distributive property|distributes over]] binary unions}}: for all <math>A,B \subseteq X</math>, <math> \mathbf{c}(A \cup B) = \mathbf{c}(A) \cup \mathbf{c}(B)</math>.
}}

A consequence of <math>\mathbf{c}</math> preserving binary unions is the following condition:<ref>{{Harvp|Pervin|1964|p=43}}, Exercise 6.</ref>
{{Quote frame|quote='''[K4']''' It is ''[[monotonic function|monotone]]'': <math> A \subseteq B \Rightarrow \mathbf{c}(A) \subseteq \mathbf{c}(B) </math>.}}
In fact if we rewrite the equality in '''[K4]''' as an inclusion, giving the weaker axiom '''[K4<nowiki>''</nowiki>]''' (''subadditivity''):
{{Quote frame|quote='''[K4<nowiki>''</nowiki>]''' It is ''subadditive'': for all <math>A,B \subseteq X</math>, <math> \mathbf{c}(A \cup B) \subseteq \mathbf{c}(A) \cup \mathbf{c}(B)</math>,}}
then it is easy to see that axioms '''[K4']''' and '''[K4<nowiki>''</nowiki>]''' together are equivalent to '''[K4]''' (see the next-to-last paragraph of Proof 2 below).

{{Harvp|Kuratowski|1966}} includes a fifth (optional) axiom requiring that singleton sets should be stable under closure: for all <math>x \in X</math>, <math>\mathbf{c}(\{x\}) = \{x\}</math>. He refers to topological spaces which satisfy all five axioms as ''T<sub>1</sub>-spaces'' in contrast to the more general spaces which only satisfy the four listed axioms. Indeed, these spaces correspond exactly to the [[T1 space|topological T<sub>1</sub>-spaces]] via the usual correspondence (see below).<ref name=":0">{{Harvp|Kuratowski|1966|p=38}}.</ref>

If requirement '''[K3]''' is omitted, then the axioms define a '''Čech closure operator'''.<ref>{{Harvp|Arkhangel'skij|Fedorchuk|1990|p=25}}.</ref> If '''[K1]''' is omitted instead, then an operator satisfying '''[K2]''', '''[K3]''' and '''[K4']''' is said to be a '''Moore closure operator'''.<ref>{{Cite web|url=https://ncatlab.org/nlab/show/Moore+closure#InTermsOfClosureOperators|title=Moore closure|date=March 7, 2015|website=nLab|access-date=August 19, 2019}}</ref> A pair <math>(X, \mathbf{c})</math> is called '''Kuratowski''', '''Čech''' or '''Moore closure space''' depending on the axioms satisfied by <math>\mathbf{c}</math>.

=== Alternative axiomatizations ===
The four Kuratowski closure axioms can be replaced by a single condition, given by Pervin:<ref>{{Harvp|Pervin|1964|p=42}}, Exercise 5.</ref>
{{Quote frame|quote='''[P]''' For all <math>A,B \subseteq X</math>, <math>A \cup \mathbf{c}(A) \cup \mathbf{c}(\mathbf{c}(B)) = \mathbf{c}(A \cup B) \setminus \mathbf{c}(\varnothing)</math>.}}

Axioms '''[K1]'''–'''[K4]''' can be derived as a consequence of this requirement:

# Choose <math>A = B = \varnothing</math>. Then <math>\varnothing \cup \mathbf{c}(\varnothing) \cup \mathbf{c}(\mathbf{c}(\varnothing)) = \mathbf{c}(\varnothing) \setminus \mathbf{c}(\varnothing) = \varnothing</math>, or <math>\mathbf{c}(\varnothing) \cup \mathbf{c}(\mathbf{c}(\varnothing)) = \varnothing</math>. This immediately implies '''[K1]'''.
# Choose an arbitrary <math>A \subseteq X</math> and <math>B = \varnothing</math>. Then, applying axiom '''[K1]''', <math>A \cup \mathbf{c}(A) = \mathbf{c}(A)</math>, implying '''[K2]'''.
# Choose <math>A = \varnothing</math> and an arbitrary <math>B \subseteq X</math>. Then, applying axiom '''[K1]''', <math>\mathbf{c}(\mathbf{c}(B)) = \mathbf{c}(B)</math>, which is '''[K3]'''.
# Choose arbitrary <math>A,B \subseteq X</math>. Applying axioms '''[K1]'''–'''[K3]''', one derives '''[K4]'''.

Alternatively, {{Harvp|Monteiro|1945}} had proposed a weaker axiom that only entails '''[K2]'''–'''[K4]''':<ref>{{Harvp|Monteiro|1945|p=158}}.</ref>
{{Quote frame|quote='''[M]''' For all <math>A,B \subseteq X</math>, <math display="inline">A \cup \mathbf{c}(A)\cup \mathbf{c}(\mathbf{c}(B)) \subseteq \mathbf{c}(A \cup B)</math>.}}
Requirement '''[K1]''' is independent of '''[M]''' : indeed, if <math>X \neq \varnothing</math>, the operator <math>\mathbf{c}^\star : \wp(X) \to \wp(X)</math> defined by the constant assignment <math>A \mapsto \mathbf{c}^\star(A) := X</math> satisfies '''[M]''' but does not preserve the empty set, since <math>\mathbf{c}^\star(\varnothing) = X</math>. Notice that, by definition, any operator satisfying '''[M]''' is a Moore closure operator.

A more symmetric alternative to '''[M]''' was also proven by M. O. Botelho and M. H. Teixeira to imply axioms '''[K2]'''–'''[K4]''':<ref name=":1">{{Harvp|Monteiro|1945|p=160}}.</ref>
{{Quote frame|quote='''[BT]''' For all <math>A,B \subseteq X</math>, <math display="inline">A \cup B \cup \mathbf{c}(\mathbf{c}(A)) \cup \mathbf{c}(\mathbf{c}(B)) = \mathbf{c}(A \cup B)</math>.}}

== Analogous structures ==
=== Interior, exterior and boundary operators ===
A dual notion to Kuratowski closure operators is that of '''Kuratowski interior operator''', which is a map <math>\mathbf{i} : \wp(X) \to \wp(X)</math> satisfying the following similar requirements:<ref name=":2">{{Harvp|Pervin|1964|p=44}}.</ref>

{{Quote frame|quote='''[I1]''' It ''preserves the total space'': <math> \mathbf{i}(X) = X </math>;

'''[I2]''' It is ''intensive'': for all <math>A \subseteq X</math>, <math> \mathbf{i}(A) \subseteq A</math>;

'''[I3]''' It is ''idempotent'': for all <math>A \subseteq X</math>, <math> \mathbf{i}(\mathbf{i}(A)) = \mathbf{i}(A)</math>;

'''[I4]''' It ''preserves binary intersections'': for all <math>A,B \subseteq X</math>, <math> \mathbf{i}(A \cap B) = \mathbf{i}(A) \cap \mathbf{i}(B)</math>.}}

For these operators, one can reach conclusions that are completely analogous to what was inferred for Kuratowski closures. For example, all Kuratowski interior operators are ''isotonic'', i.e. they satisfy '''[K4']''', and because of intensivity '''[I2]''', it is possible to weaken the equality in '''[I3]''' to a simple inclusion.

The duality between Kuratowski closures and interiors is provided by the natural '''complement operator''' on <math>\wp(X)</math>, the map <math>\mathbf{n} : \wp(X) \to \wp(X)</math> sending <math>A \mapsto \mathbf{n}(A):= X \setminus A</math>. This map is an [[orthocomplementation]] on the power set lattice, meaning it satisfies [[De Morgan's laws]]: if <math>\mathcal{I}</math> is an arbitrary set of indices and <math>\{A_i\}_{i\in\mathcal I} \subseteq \wp(X)</math>,
<math display="block">
  \mathbf{n}\left(\bigcup_{i \in \mathcal I} A_i\right) = \bigcap_{i\in \mathcal I} \mathbf{n}(A_i), \qquad
  \mathbf{n}\left(\bigcap_{i \in \mathcal I} A_i\right) = \bigcup_{i\in \mathcal I} \mathbf{n}(A_i).
</math>

By employing these laws, together with the defining properties of <math>\mathbf{n}</math>, one can show that any Kuratowski interior induces a Kuratowski closure (and vice versa), via the defining relation <math>\mathbf {c} := \mathbf{nin}</math> (and <math>\mathbf {i} := \mathbf{ncn}</math>). Every result obtained concerning <math>\mathbf{c}</math> may be converted into a result concerning <math>\mathbf{i}</math> by employing these relations in conjunction with the properties of the orthocomplementation <math>\mathbf{n}</math>.

{{Harvp|Pervin|1964}} further provides analogous axioms for '''Kuratowski exterior operators'''<ref name=":2" /> and '''Kuratowski boundary operators''',<ref>{{Harvp|Pervin|1964|p=46}}, Exercise 4.</ref> which also induce Kuratowski closures via the relations <math>\mathbf{c} := \mathbf{ne}</math> and <math>\mathbf{c}(A):= A \cup \mathbf{b}(A)</math>.

=== Abstract operators ===
{{Main|Interior algebra}}
Notice that axioms '''[K1]'''–'''[K4]''' may be adapted to define an ''abstract'' unary operation <math>\mathbf c : L \to L</math> on a general bounded lattice <math>(L,\land,\lor,\mathbf 0, \mathbf 1)</math>, by formally substituting set-theoretic inclusion with the partial order associated to the lattice, set-theoretic union with the join operation, and set-theoretic intersections with the meet operation; similarly for axioms '''[I1]'''–'''[I4]'''. If the lattice is orthocomplemented, these two abstract operations induce one another in the usual way. Abstract closure or interior operators can be used to define a [[generalized topology]] on the lattice.

Since neither unions nor the empty set appear in the requirement for a Moore closure operator, the definition may be adapted to define an abstract unary operator <math>\mathbf{c} : S \to S</math> on an arbitrary [[Partially ordered set|poset]] <math>S</math>.

== Connection to other axiomatizations of topology ==

=== Induction of topology from closure ===
A closure operator naturally induces a [[Topological space|topology]] as follows. Let <math>X</math> be an arbitrary set. We shall say that a subset <math> C\subseteq X </math> is '''closed''' with respect to a Kuratowski closure operator <math>\mathbf{c} : \wp(X) \to \wp(X)</math> if and only if it is a ''fixed point'' of said operator, or in other words it is ''stable under'' <math>\mathbf{c}</math>, i.e. <math> \mathbf{c}(C) = C </math>. The claim is that the family of all subsets of the total space that are complements of closed sets satisfies the three usual requirements for a topology, or equivalently, the family <math>\mathfrak{S}[\mathbf{c}]</math> of all closed sets satisfies the following:
{{Quote frame|quote='''[T1]''' It is a ''bounded [[sublattice]]'' of <math>\wp(X)</math>, i.e. <math>X,\varnothing \in\mathfrak{S}[\mathbf{c}]</math>;

'''[T2]''' It is ''complete under arbitrary intersections'', i.e. if <math>\mathcal{I}</math> is an arbitrary set of indices and <math>\{C_i\}_{i\in\mathcal I} \subseteq \mathfrak{S}[\mathbf{c}]</math>, then <math display="inline">\bigcap_{i\in\mathcal I} C_i \in \mathfrak{S}[\mathbf{c}]</math>;

'''[T3]''' It is ''complete under finite unions'', i.e. if <math>\mathcal{I}</math> is a finite set of indices and <math>\{C_i\}_{i\in\mathcal I} \subseteq \mathfrak{S}[\mathbf{c}]</math>, then <math display="inline">\bigcup_{i\in\mathcal I} C_i \in \mathfrak{S}[\mathbf{c}]</math>.
|}}

Notice that, by idempotency '''[K3]''', one may succinctly write <math>\mathfrak{S}[\mathbf{c}] = \operatorname{im}(\mathbf{c})</math>.

{{collapse top|title=Proof 1.|expand=yes|left=true}}
'''[T1] '''By extensivity '''[K2]''', <math> X\subseteq\mathbf{c}(X) </math> and since closure maps the power set of <math>X</math> into itself (that is, the image of any subset is a subset of <math>X</math>), <math> \mathbf{c}(X)\subseteq X </math> we have <math> X = \mathbf{c}(X)</math>. Thus <math> X \in \mathfrak{S}[\mathbf{c}]</math>.  The preservation of the empty set '''[K1]''' readily implies <math> \varnothing \in\mathfrak{S}[\mathbf{c}] </math>.

'''[T2] '''Next, let <math> \mathcal{I} </math> be an arbitrary set of indices and let <math> C_i </math> be closed for every <math> i\in\mathcal{I}</math>.  By extensivity '''[K2]''', <math display="inline"> \bigcap_{i\in\mathcal{I}}C_i \subseteq \mathbf{c}\left(\bigcap_{i\in\mathcal{I}}C_i\right)</math>.  Also, by isotonicity '''[K4']''', if <math display="inline">\bigcap_{i\in\mathcal I} C_i \subseteq C_i</math>for all indices <math>i \in \mathcal I</math>, then <math display="inline"> \mathbf{c}\left(\bigcap_{i\in\mathcal{I}}C_i \right) \subseteq \mathbf{c}(C_i) = C_i</math> for all <math>i \in \mathcal I</math>, which implies <math display="inline">\mathbf{c}\left(\bigcap_{i\in\mathcal{I}}C_i \right) \subseteq \bigcap_{i\in\mathcal{I}}C_i</math>.  Therefore, <math display="inline"> \bigcap_{i\in\mathcal{I}}C_i = \mathbf{c}\left(\bigcap_{i\in\mathcal{I}}C_i\right) </math>, meaning <math display="inline">\bigcap_{i\in\mathcal{I}}C_i \in \mathfrak{S}[\mathbf{c}]</math>.

'''[T3] '''Finally, let <math> \mathcal{I} </math> be a finite set of indices and let <math> C_i </math> be closed for every <math> i\in\mathcal{I} </math>.  From the preservation of binary unions '''[K4]''', and using [[mathematical induction|induction]] on the number of subsets of which we take the union, we have <math display="inline"> \bigcup_{i\in\mathcal{I}}C_i = \mathbf{c}\left(\bigcup_{i\in\mathcal{I}}C_i \right) </math>. Thus, <math display="inline"> \bigcup_{i\in\mathcal{I}}C_i \in \mathfrak{S}[\mathbf{c}] </math>.
{{collapse bottom}}

=== Induction of closure from topology ===

Conversely, given a family <math>\kappa</math> satisfying axioms '''[T1]'''–'''[T3]''', it is possible to construct a Kuratowski closure operator in the following way: if <math>A \in \wp(X)</math> and <math>A^\uparrow = \{B \in \wp(X)\ |\ A \subseteq B \}</math> is the inclusion [[Upper set|upset]] of <math>A</math>, then
<math display="block">\mathbf{c}_\kappa(A) := \bigcap_{B \in (\kappa \cap A^\uparrow)} B</math>

defines a Kuratowski closure operator <math>\mathbf{c}_\kappa</math> on <math>\wp(X)</math>.

{{collapse top|title=Proof 2.|expand=yes|left=true}}
'''[K1] '''Since <math>\varnothing^\uparrow = \wp(X)</math>, <math>\mathbf{c}_\kappa(\varnothing)</math> reduces to the intersection of all sets in the family <math>\kappa</math>; but <math>\varnothing \in \kappa</math> by axiom '''[T1]''', so the intersection collapses to the null set and '''[K1]''' follows.

'''[K2]''' By definition of <math>A^\uparrow</math>, we have that <math>A \subseteq B</math> for all <math>B \in \left(\kappa \cap A^\uparrow\right)</math>, and thus <math>A</math> must be contained in the intersection of all such sets. Hence follows extensivity '''[K2]'''.

'''[K3]''' Notice that, for all <math>A \in \wp(X)</math>, the family <math>\mathbf{c}_\kappa(A)^\uparrow \cap \kappa</math> contains <math>\mathbf{c}_\kappa(A)</math> itself as a minimal element w.r.t. inclusion. Hence <math display="inline">\mathbf{c}_\kappa^2(A) = \bigcap_{B \in \mathbf{c}_\kappa(A)^\uparrow \cap \kappa}B = \mathbf{c}_\kappa(A)</math>, which is idempotence '''[K3]'''.

'''[K4']''' Let <math>A \subseteq B \subseteq X</math>: then <math>B^\uparrow \subseteq A^\uparrow</math>, and thus <math>\kappa \cap B^\uparrow \subseteq \kappa \cap A^\uparrow</math>. Since the latter family may contain more elements than the former, we find <math>\mathbf{c}_\kappa(A) \subseteq \mathbf{c}_\kappa(B)</math>, which is isotonicity '''[K4']'''. Notice that isotonicity implies <math>\mathbf{c}_\kappa(A) \subseteq \mathbf{c}_\kappa(A\cup B)</math> and <math>\mathbf{c}_\kappa(B) \subseteq \mathbf{c}_\kappa(A\cup B)</math>, which together imply <math>\mathbf{c}_\kappa(A) \cup \mathbf{c}_\kappa(B) \subseteq \mathbf{c}_\kappa(A\cup B)</math>.

'''[K4]''' Finally, fix <math>A,B \in \wp(X)</math>. Axiom '''[T2]''' implies <math>\mathbf{c}_\kappa(A), \mathbf{c}_\kappa(B) \in \kappa</math>; furthermore, axiom '''[T2]''' implies that <math>\mathbf{c}_\kappa(A) \cup \mathbf{c}_\kappa(B) \in \kappa</math>. By extensivity '''[K2]''' one has <math>\mathbf{c}_\kappa(A) \in A^\uparrow</math> and <math>\mathbf{c}_\kappa(B) \in B^\uparrow</math>, so that <math>\mathbf{c}_\kappa(A) \cup \mathbf{c}_\kappa(B) \in \left(A^\uparrow\right) \cap \left(B^\uparrow\right)</math>. But <math>\left(A^\uparrow\right) \cap \left(B^\uparrow\right) = (A \cup B)^\uparrow</math>, so that all in all <math>\mathbf{c}_\kappa(A) \cup \mathbf{c}_\kappa(B) \in \kappa\cap (A \cup B)^\uparrow</math>. Since then <math>\mathbf{c}_\kappa(A \cup B)</math> is a minimal element of <math>\kappa \cap (A \cup B)^\uparrow</math> w.r.t. inclusion, we find <math>\mathbf{c}_\kappa(A \cup B) \subseteq \mathbf{c}_\kappa(A) \cup \mathbf{c}_\kappa(B)</math>. Point 4. ensures additivity '''[K4]'''.
{{collapse bottom}}

=== Exact correspondence between the two structures ===
In fact, these two complementary constructions are inverse to one another: if <math>\mathrm{Cls}_\text{K}(X)</math> is the collection of all Kuratowski closure operators on <math>X</math>, and <math>\mathrm{Atp}(X)</math> is the collection of all families consisting of complements of all sets in a topology, i.e. the collection of all families satisfying '''[T1]'''–'''[T3]''', then <math>\mathfrak{S} : \mathrm{Cls}_\text{K}(X) \to \mathrm{Atp}(X)</math> such that <math>\mathbf{c} \mapsto \mathfrak{S}[\mathbf{c}]</math> is a bijection, whose inverse is given by the assignment <math>\mathfrak{C}: \kappa \mapsto \mathbf{c}_\kappa</math>.

{{collapse top|title=Proof 3.|expand=yes|left=true}}
First we prove that <math>\mathfrak{C} \circ \mathfrak{S} = \mathfrak{1}_{\mathrm{Cls}_\text{K}(X)}</math>, the identity operator on <math>\mathrm{Cls}_\text{K}(X)</math>. For a given Kuratowski closure <math>\mathbf{c} \in \mathrm{Cls}_\text{K}(X)</math>, define <math>\mathbf{c}' := \mathfrak{C}[\mathfrak{S}[\mathbf{c}]]</math>; then if <math>A \in \wp(X)</math> its primed closure <math>\mathbf{c}'(A)</math> is the intersection of all <math>\mathbf{c}</math>-stable sets that contain <math>A</math>. Its non-primed closure <math>\mathbf{c}(A)</math> satisfies this description: by extensivity '''[K2]''' we have <math>A \subseteq \mathbf{c}(A)</math>, and by idempotence '''[K3]''' we have <math>\mathbf{c}(\mathbf{c}(A)) = \mathbf{c}(A)</math>, and thus <math>\mathbf{c}(A) \in \left(A^\uparrow \cap \mathfrak{S}[\mathbf{c}]\right)</math>. Now, let <math>C \in \left(A^\uparrow \cap \mathfrak{S}[\mathbf{c}]\right)</math> such that <math>A \subseteq C \subseteq \mathbf{c}(A)</math>: by isotonicity '''[K4']''' we have <math>\mathbf{c}(A) \subseteq \mathbf{c}(C)</math>, and since <math>\mathbf{c}(C) = C</math> we conclude that <math>C = \mathbf{c}(A)</math>. Hence <math>\mathbf{c}(A)</math> is the minimal element of <math>A^\uparrow \cap \mathfrak{S}[\mathbf{c}]</math> w.r.t. inclusion, implying <math>\mathbf{c}'(A) = \mathbf{c}(A)</math>.

Now we prove that <math>\mathfrak{S} \circ \mathfrak{C} = \mathfrak{1}_{\mathrm{Atp}(X)}</math>. If <math>\kappa \in \mathrm{Atp}(X)</math> and <math>\kappa':= \mathfrak{S}[\mathfrak{C}[\kappa]]</math> is the family of all sets that are stable under <math>\mathbf{c}_\kappa</math>, the result follows if both <math>\kappa' \subseteq \kappa</math> and <math>\kappa \subseteq \kappa'</math>. Let <math>A \in \kappa'</math>: hence <math>\mathbf{c}_\kappa(A) = A</math>. Since <math>\mathbf{c}_\kappa(A)</math> is the intersection of an arbitrary subfamily of <math>\kappa</math>, and the latter is complete under arbitrary intersections by '''[T2]''', then <math>A = \mathbf{c}_\kappa(A) \in \kappa</math>. Conversely, if <math>A \in \kappa</math>, then <math>\mathbf{c}_\kappa(A)</math> is the minimal superset of <math>A</math> that is contained in <math>\kappa</math>. But that is trivially <math>A</math> itself, implying <math>A \in \kappa'</math>.
{{collapse bottom}}

We observe that one may also extend the bijection <math>\mathfrak{S}</math> to the collection <math>\mathrm{Cls}_{\check C}(X)</math> of all Čech closure operators, which strictly contains <math>\mathrm{Cls}_\text{K}(X)</math>; this extension <math>\overline{\mathfrak{S}}</math> is also surjective, which signifies that all Čech closure operators on <math>X</math> also induce a topology on <math>X</math>.<ref>{{Harvp|Arkhangel'skij|Fedorchuk|1990|p=26}}.</ref> However, this means that <math>\overline{\mathfrak{S}}</math> is no longer a bijection.

== Examples ==
{{Expand section|date=August 2019}}

* As discussed above, given a topological space <math>X</math> we may define the closure of any subset <math>A \subseteq X</math> to be the set <math>\mathbf{c}(A)=\bigcap\{C\text{ a closed subset of }X| A\subseteq C\}</math>, i.e. the intersection of all closed sets of <math>X</math> which contain <math>A</math>. The set <math>\mathbf{c}(A)</math> is the smallest closed set of <math>X</math> containing <math>A</math>, and the operator <math>\mathbf{c}:\wp(X) \to \wp(X)</math> is a Kuratowski closure operator.
* If <math>X</math> is any set, the operators <math>\mathbf{c}_\top, \mathbf{c}_\bot : \wp(X) \to \wp(X)</math> such that <math display="block">\mathbf{c}_\top(A) = \begin{cases}
\varnothing & A = \varnothing, \\
X & A \neq \varnothing,
\end{cases} \qquad \mathbf{c}_\bot(A) = A\quad \forall A \in \wp(X),</math>are Kuratowski closures. The first induces the [[Trivial topology|indiscrete topology]] <math>\{\varnothing,X\}</math>, while the second induces the [[discrete topology]] <math>\wp(X)</math>.

*Fix an arbitrary <math>S \subsetneq X</math>, and let <math>\mathbf{c}_S: \wp(X) \to \wp(X)</math> be such that <math>\mathbf{c}_S(A) := A \cup S</math> for all <math>A \in \wp(X)</math>. Then <math>\mathbf{c}_S</math> defines a Kuratowski closure; the corresponding family of closed sets <math>\mathfrak{S}[\mathbf{c}_S]</math> coincides with <math>S^\uparrow</math>, the family of all subsets that contain <math>S</math>. When <math>S = \varnothing</math>, we once again retrieve the discrete topology <math>\wp(X)</math> (i.e. <math>\mathbf{c}_{\varnothing}=\mathbf{c}_\bot</math>, as can be seen from the definitions).
* If <math>\lambda</math> is an infinite cardinal number such that <math>\lambda \leq \operatorname{crd}(X)</math>, then the operator <math>\mathbf{c}_\lambda : \wp(X) \to \wp(X)</math> such that<math display="block">\mathbf{c}_\lambda(A) = \begin{cases}
A & \operatorname{crd}(A) < \lambda, \\
X & \operatorname{crd}(A) \geq \lambda
\end{cases}</math>satisfies all four Kuratowski axioms.<ref>A proof for the case <math>\lambda = \aleph_0</math> can be found at {{cite web |title=Is the following a Kuratowski closure operator?! |work=Stack Exchange |date=November 21, 2015 |url=https://math.stackexchange.com/q/1539449 }}</ref> If <math>\lambda = \aleph_0</math>, this operator induces the [[cofinite topology]] on <math>X</math>; if <math>\lambda = \aleph_1</math>, it induces the [[cocountable topology]].

== Properties ==

* Since any Kuratowski closure is isotonic, and so is obviously any inclusion mapping, one has the (isotonic) [[Galois connection]] <math>\langle \mathbf{c}: \wp(X) \to \mathrm{im}(\mathbf{c});\iota : \mathrm{im}(\mathbf{c}) \hookrightarrow \wp(X) \rangle</math>, provided one views <math>\wp(X)</math>as a poset with respect to inclusion, and <math>\mathrm{im}(\mathbf{c})</math> as a subposet of <math>\wp(X)</math>. Indeed, it can be easily verified that, for all <math>A \in \wp(X)</math> and <math>C \in \mathrm{im}(\mathbf{c})</math>, <math>\mathbf{c}(A) \subseteq C</math> if and only if <math>A \subseteq \iota(C)</math>.
* If <math>\{A_i\}_{i\in\mathcal I}</math> is a subfamily of <math>\wp(X)</math>, then <math display="block">\bigcup_{i\in\mathcal I} \mathbf{c}(A_i) \subseteq \mathbf{c}\left(\bigcup_{i\in\mathcal I} A_i\right), \qquad \mathbf{c}\left(\bigcap_{i\in\mathcal I} A_i\right) \subseteq \bigcap_{i\in\mathcal I} \mathbf{c}(A_i). </math>
* If <math>A,B \in \wp(X)</math>, then <math>\mathbf{c}(A) \setminus \mathbf{c}(B) \subseteq \mathbf{c}(A\setminus B)</math>.

== Topological concepts in terms of closure ==

=== Refinements and subspaces ===
A pair of Kuratowski closures <math>\mathbf{c}_1, \mathbf{c}_2 : \wp(X) \to \wp(X)</math> such that <math>\mathbf{c}_2(A) \subseteq \mathbf{c}_1(A)</math> for all <math>A \in \wp(X)</math> induce topologies <math>\tau_1,\tau_2</math> such that <math>\tau_1 \subseteq \tau_2</math>, and vice versa. In other words, <math>\mathbf{c}_1</math> dominates <math>\mathbf{c}_2</math> if and only if the topology induced by the latter is a refinement of the topology induced by the former, or equivalently <math>\mathfrak{S}[\mathbf{c}_1] \subseteq \mathfrak{S}[\mathbf{c}_2]</math>.<ref>{{Harvp|Pervin|1964|p=43}}, Exercise 10.</ref> For example,  <math>\mathbf{c}_\top</math> clearly dominates <math>\mathbf{c}_\bot</math>(the latter just being the identity on <math>\wp(X)</math>). Since the same conclusion can be reached substituting <math>\tau_i</math> with the family <math>\kappa_i</math> containing the complements of all its members, if <math>\mathrm{Cls}_\text{K}(X)</math> is endowed with the partial order <math>\mathbf{c} \leq \mathbf{c}' \iff \mathbf{c}(A) \subseteq \mathbf{c}'(A)</math> for all <math>A \in \wp(X)</math> and <math>\mathrm{Atp}(X)</math> is endowed with the refinement order, then we may conclude that <math>\mathfrak{S}</math> is an antitonic mapping between posets.

In any induced topology (relative to the subset ''A'') the closed sets induce a new closure operator that is just the original closure operator restricted to ''A'': <math> \mathbf{c}_A(B) = A \cap \mathbf{c}_X(B) </math>, for all <math>B \subseteq A</math>.<ref>{{Harvp|Pervin|1964|p=49}}, Theorem 3.4.3.</ref>

=== Continuous maps, closed maps and homeomorphisms ===
A function <math>f:(X,\mathbf{c})\to (Y,\mathbf{c}')</math> is [[Continuity (topology)|continuous]] at a point <math>p</math> iff <math>p\in\mathbf{c}(A) \Rightarrow f(p)\in\mathbf{c}'(f(A))</math>, and it is continuous everywhere iff <math display="block">f(\mathbf{c}(A)) \subseteq \mathbf{c}'(f(A))</math> for all subsets <math>A \in \wp(X)</math>.<ref>{{Harvp|Pervin|1964|p=60}}, Theorem 4.3.1.</ref> The mapping <math>f</math> is a closed map iff the reverse inclusion holds,<ref>{{Harvp|Pervin|1964|p=66}}, Exercise 3.</ref> and it is a [[homeomorphism]] iff it is both continuous and closed, i.e. iff equality holds.<ref>{{Harvp|Pervin|1964|p=67}}, Exercise 5.</ref>

=== Separation axioms ===
Let <math>(X, \mathbf{c})</math> be a Kuratowski closure space. Then

* <math>X</math> is a [[T0 space|T<sub>0</sub>-space]] iff <math>x \neq y</math> implies <math>\mathbf{c}(\{x\}) \neq \mathbf{c}(\{y\})</math>;<ref>{{Harvp|Pervin|1964|p=69}}, Theorem 5.1.1.</ref>
* <math>X</math> is a [[T1 space|T<sub>1</sub>-space]] iff <math>\mathbf{c}(\{x\})=\{x\}</math> for all <math>x \in X</math>;<ref>{{Harvp|Pervin|1964|p=70}}, Theorem 5.1.2.</ref>
* <math>X</math> is a [[Hausdorff space|T<sub>2</sub>-space]] iff <math>x \neq y</math> implies that there exists a set <math>A \in \wp(X)</math> such that both <math>x \notin \mathbf{c}(A)</math> and <math>y \notin \mathbf{c}(\mathbf{n}(A))</math>, where <math>\mathbf{n}</math> is the set complement operator.<ref>A proof can be found at this [https://math.stackexchange.com/a/3144373/332594 link].</ref>

=== Closeness and separation ===
A point <math>p</math> is [[Closeness (topology)|close]] to a subset <math>A</math> if <math>p\in\mathbf{c}(A).</math>This can be used to define a [[Proximity space|proximity]] relation on the points and subsets of a set.<ref>{{Harvp|Pervin|1964|pp=193–196}}.</ref>

Two sets <math>A,B \in \wp(X)</math> are separated iff <math>(A \cap \mathbf{c}(B)) \cup (B \cap \mathbf{c}(A)) = \varnothing</math>. The space <math>X</math> is [[Connected space|connected]] iff it cannot be written as the union of two separated subsets.<ref>{{Harvp|Pervin|1964|p=51}}.</ref>

== See also ==

* {{annotated link|Characterizations of the category of topological spaces}}
* {{annotated link|Čech closure operator}}
* {{annotated link|Closure operator}}
* {{annotated link|Interior algebra|Closure algebra}}
* {{annotated link|Preclosure operator}}
* {{annotated link|Pretopological space}}
* {{annotated link|Topological space}}

==Notes==
{{reflist|3}}

==References==
*{{Citation|last=Kuratowski|first=Kazimierz|title=Sur l'opération A de l'Analysis Situs|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3121.pdf|work=Fundamenta Mathematicae|volume=3|pages=182–199|publication-date=1922|orig-date=1920|trans-title=On the operation A in Analysis Situs|language=fr|author-link=Kazimierz Kuratowski}}.
*{{citation|first=Kazimierz|last=Kuratowski|title=Topology|volume=I|year=1966|publisher=Academic Press|orig-date=1958|isbn=0-12-429201-1|lccn=66029221|author-link=Kazimierz Kuratowski|translator-last=Jaworowski|translator-first=J.}}.
**{{cite web |author=Kazimierz Kuratowski |author-mask=2 |translator=Mark Bowron |date=2010 |title=On the Operation Ā Analysis Situs |website=ResearchGate |url=https://www.researchgate.net/publication/318445643}}
* {{citation|first=William J.|last=Pervin|title=Foundations of General Topology|year=1964|editor-last=Boas|editor-first=Ralph P. Jr.|publisher=Academic Press|isbn=9781483225159|lccn=64-17796}}.
*{{Citation|last1=Arkhangel'skij|first1=A.V.|title=General Topology I|volume=17|publication-date=1990|editor-last=Gamkrelidze|editor-first=R.V.|orig-date=1988|series=Encyclopaedia of Mathematical Sciences|place=Berlin|publisher=Springer-Verlag|isbn=978-3-642-64767-3|lccn=89-26209|last2=Fedorchuk|first2=V.V.|editor2-last=Arkhangel'skij|editor2-first=A.V.|editor3-last=Pontryagin|editor3-first=L.S.|translator-last=O'Shea|translator-first=D.B.}}.
*{{Citation|last=Monteiro|first=António|title=Caractérisation de l'opération de fermeture par un seul axiome|work=Portugaliae mathematica|volume=4|issue=4|pages=158–160|date=1945|url=http://purl.pt/2135|trans-title=Characterization of the operation of closure by a single axiom|language=fr|zbl=0060.39406|author-link=Antonio Monteiro (mathematician)|mr=0012310}}.

==External links==
* [http://math.uga.edu/~pete/TopSection3.pdf  Alternative Characterizations of Topological Spaces]

[[Category:Closure operators]]
[[Category:Mathematical axioms]]