Kuratowski closure axioms

In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski,^[1] and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro,^[2] among others.

A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.^[3]

Definition

Kuratowski closure operators and weakenings

Let $X$ be an arbitrary set and $\wp (X)$ its power set. A Kuratowski closure operator is a unary operation $\mathbf {c} :\wp (X)\to \wp (X)$ with the following properties:

[K1] It preserves the empty set: $\mathbf {c} (\varnothing )=\varnothing$ ;
[K2] It is extensive: for all $A\subseteq X$ , $A\subseteq \mathbf {c} (A)$ ;
[K3] It is idempotent: for all $A\subseteq X$ , $\mathbf {c} (A)=\mathbf {c} (\mathbf {c} (A))$ ;
[K4] It preserves/distributes over binary unions: for all $A,B\subseteq X$ , $\mathbf {c} (A\cup B)=\mathbf {c} (A)\cup \mathbf {c} (B)$ .

A consequence of $\mathbf {c}$ preserving binary unions is the following condition:^[4]

[K4'] It is monotone: $A\subseteq B\Rightarrow \mathbf {c} (A)\subseteq \mathbf {c} (B)$ .

In fact if we rewrite the equality in [K4] as an inclusion, giving the weaker axiom [K4''] (subadditivity):

[K4''] It is subadditive: for all $A,B\subseteq X$ , $\mathbf {c} (A\cup B)\subseteq \mathbf {c} (A)\cup \mathbf {c} (B)$ ,

then it is easy to see that axioms [K4'] and [K4''] together are equivalent to [K4] (see the next-to-last paragraph of Proof 2 below).

Kuratowski (1966) includes a fifth (optional) axiom requiring that singleton sets should be stable under closure: for all $x\in X$ , $\mathbf {c} (\{x\})=\{x\}$ . He refers to topological spaces which satisfy all five axioms as T₁-spaces in contrast to the more general spaces which only satisfy the four listed axioms. Indeed, these spaces correspond exactly to the topological T₁-spaces via the usual correspondence (see below).^[5]

If requirement [K3] is omitted, then the axioms define a Čech closure operator.^[6] If [K1] is omitted instead, then an operator satisfying [K2], [K3] and [K4'] is said to be a Moore closure operator.^[7] A pair $(X,\mathbf {c} )$ is called Kuratowski, Čech or Moore closure space depending on the axioms satisfied by $\mathbf {c}$ .

Alternative axiomatizations

The four Kuratowski closure axioms can be replaced by a single condition, given by Pervin:^[8]

[P] For all $A,B\subseteq X$ , $A\cup \mathbf {c} (A)\cup \mathbf {c} (\mathbf {c} (B))=\mathbf {c} (A\cup B)\setminus \mathbf {c} (\varnothing )$ .

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

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

Alternatively, Monteiro (1945) had proposed a weaker axiom that only entails [K2]–[K4]:^[9]

[M] For all $A,B\subseteq X$ , ${\textstyle A\cup \mathbf {c} (A)\cup \mathbf {c} (\mathbf {c} (B))\subseteq \mathbf {c} (A\cup B)}$ .

Requirement [K1] is independent of [M] : indeed, if $X\neq \varnothing$ , the operator $\mathbf {c} ^{\star }:\wp (X)\to \wp (X)$ defined by the constant assignment $A\mapsto \mathbf {c} ^{\star }(A):=X$ satisfies [M] but does not preserve the empty set, since $\mathbf {c} ^{\star }(\varnothing )=X$ . 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]:^[2]

[BT] For all $A,B\subseteq X$ , ${\textstyle A\cup B\cup \mathbf {c} (\mathbf {c} (A))\cup \mathbf {c} (\mathbf {c} (B))=\mathbf {c} (A\cup B)}$ .

Analogous structures

Interior, exterior and boundary operators

A dual notion to Kuratowski closure operators is that of Kuratowski interior operator, which is a map $\mathbf {i} :\wp (X)\to \wp (X)$ satisfying the following similar requirements:^[3]

[I1] It preserves the total space: $\mathbf {i} (X)=X$ ;
[I2] It is intensive: for all $A\subseteq X$ , $\mathbf {i} (A)\subseteq A$ ;
[I3] It is idempotent: for all $A\subseteq X$ , $\mathbf {i} (\mathbf {i} (A))=\mathbf {i} (A)$ ;
[I4] It preserves binary intersections: for all $A,B\subseteq X$ , $\mathbf {i} (A\cap B)=\mathbf {i} (A)\cap \mathbf {i} (B)$ .

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 $\wp (X)$ , the map $\mathbf {n} :\wp (X)\to \wp (X)$ sending $A\mapsto \mathbf {n} (A):=X\setminus A$ . This map is an orthocomplementation on the power set lattice, meaning it satisfies De Morgan's laws: if ${\mathcal {I}}$ is an arbitrary set of indices and $\{A_{i}\}_{i\in {\mathcal {I}}}\subseteq \wp (X)$ ,

\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}).

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

Pervin (1964) further provides analogous axioms for Kuratowski exterior operators^[3] and Kuratowski boundary operators,^[10] which also induce Kuratowski closures via the relations $\mathbf {c} :=\mathbf {ne}$ and $\mathbf {c} (A):=A\cup \mathbf {b} (A)$ .

Abstract operators

Notice that axioms [K1]–[K4] may be adapted to define an abstract unary operation $\mathbf {c} :L\to L$ on a general bounded lattice $(L,\land ,\lor ,\mathbf {0} ,\mathbf {1} )$ , 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 $\mathbf {c} :S\to S$ on an arbitrary poset $S$ .

Connection to other axiomatizations of topology

Induction of topology from closure

A closure operator naturally induces a topology as follows. Let $X$ be an arbitrary set. We shall say that a subset $C\subseteq X$ is closed with respect to a Kuratowski closure operator $\mathbf {c} :\wp (X)\to \wp (X)$ if and only if it is a fixed point of said operator, or in other words it is stable under $\mathbf {c}$ , i.e. $\mathbf {c} (C)=C$ . 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 ${\mathfrak {S}}[\mathbf {c} ]$ of all closed sets satisfies the following:

[T1] It is a bounded sublattice of $\wp (X)$ , i.e. $X,\varnothing \in {\mathfrak {S}}[\mathbf {c} ]$ ;
[T2] It is complete under arbitrary intersections, i.e. if ${\mathcal {I}}$ is an arbitrary set of indices and $\{C_{i}\}_{i\in {\mathcal {I}}}\subseteq {\mathfrak {S}}[\mathbf {c} ]$ , then ${\textstyle \bigcap _{i\in {\mathcal {I}}}C_{i}\in {\mathfrak {S}}[\mathbf {c} ]}$ ;
[T3] It is complete under finite unions, i.e. if ${\mathcal {I}}$ is a finite set of indices and $\{C_{i}\}_{i\in {\mathcal {I}}}\subseteq {\mathfrak {S}}[\mathbf {c} ]$ , then ${\textstyle \bigcup _{i\in {\mathcal {I}}}C_{i}\in {\mathfrak {S}}[\mathbf {c} ]}$ .

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

Proof 1.
[T1] By extensivity [K2], $X\subseteq \mathbf {c} (X)$ and since closure maps the power set of $X$ into itself (that is, the image of any subset is a subset of $X$ ), $\mathbf {c} (X)\subseteq X$ we have $X=\mathbf {c} (X)$ . Thus $X\in {\mathfrak {S}}[\mathbf {c} ]$ . The preservation of the empty set [K1] readily implies $\varnothing \in {\mathfrak {S}}[\mathbf {c} ]$ . [T2] Next, let ${\mathcal {I}}$ be an arbitrary set of indices and let $C_{i}$ be closed for every $i\in {\mathcal {I}}$ . By extensivity [K2], ${\textstyle \bigcap _{i\in {\mathcal {I}}}C_{i}\subseteq \mathbf {c} \left(\bigcap _{i\in {\mathcal {I}}}C_{i}\right)}$ . Also, by isotonicity [K4'], if ${\textstyle \bigcap _{i\in {\mathcal {I}}}C_{i}\subseteq C_{i}}$ for all indices $i\in {\mathcal {I}}$ , then ${\textstyle \mathbf {c} \left(\bigcap _{i\in {\mathcal {I}}}C_{i}\right)\subseteq \mathbf {c} (C_{i})=C_{i}}$ for all $i\in {\mathcal {I}}$ , which implies ${\textstyle \mathbf {c} \left(\bigcap _{i\in {\mathcal {I}}}C_{i}\right)\subseteq \bigcap _{i\in {\mathcal {I}}}C_{i}}$ . Therefore, ${\textstyle \bigcap _{i\in {\mathcal {I}}}C_{i}=\mathbf {c} \left(\bigcap _{i\in {\mathcal {I}}}C_{i}\right)}$ , meaning ${\textstyle \bigcap _{i\in {\mathcal {I}}}C_{i}\in {\mathfrak {S}}[\mathbf {c} ]}$ . [T3] Finally, let ${\mathcal {I}}$ be a finite set of indices and let $C_{i}$ be closed for every $i\in {\mathcal {I}}$ . From the preservation of binary unions [K4], and using induction on the number of subsets of which we take the union, we have ${\textstyle \bigcup _{i\in {\mathcal {I}}}C_{i}=\mathbf {c} \left(\bigcup _{i\in {\mathcal {I}}}C_{i}\right)}$ . Thus, ${\textstyle \bigcup _{i\in {\mathcal {I}}}C_{i}\in {\mathfrak {S}}[\mathbf {c} ]}$ .

Proof 1.

[T1] By extensivity [K2], $X\subseteq \mathbf {c} (X)$ and since closure maps the power set of $X$ into itself (that is, the image of any subset is a subset of $X$ ), $\mathbf {c} (X)\subseteq X$ we have $X=\mathbf {c} (X)$ . Thus $X\in {\mathfrak {S}}[\mathbf {c} ]$ . The preservation of the empty set [K1] readily implies $\varnothing \in {\mathfrak {S}}[\mathbf {c} ]$ .

[T2] Next, let ${\mathcal {I}}$ be an arbitrary set of indices and let $C_{i}$ be closed for every $i\in {\mathcal {I}}$ . By extensivity [K2], ${\textstyle \bigcap _{i\in {\mathcal {I}}}C_{i}\subseteq \mathbf {c} \left(\bigcap _{i\in {\mathcal {I}}}C_{i}\right)}$ . Also, by isotonicity [K4'], if ${\textstyle \bigcap _{i\in {\mathcal {I}}}C_{i}\subseteq C_{i}}$ for all indices $i\in {\mathcal {I}}$ , then ${\textstyle \mathbf {c} \left(\bigcap _{i\in {\mathcal {I}}}C_{i}\right)\subseteq \mathbf {c} (C_{i})=C_{i}}$ for all $i\in {\mathcal {I}}$ , which implies ${\textstyle \mathbf {c} \left(\bigcap _{i\in {\mathcal {I}}}C_{i}\right)\subseteq \bigcap _{i\in {\mathcal {I}}}C_{i}}$ . Therefore, ${\textstyle \bigcap _{i\in {\mathcal {I}}}C_{i}=\mathbf {c} \left(\bigcap _{i\in {\mathcal {I}}}C_{i}\right)}$ , meaning ${\textstyle \bigcap _{i\in {\mathcal {I}}}C_{i}\in {\mathfrak {S}}[\mathbf {c} ]}$ .

[T3] Finally, let ${\mathcal {I}}$ be a finite set of indices and let $C_{i}$ be closed for every $i\in {\mathcal {I}}$ . From the preservation of binary unions [K4], and using induction on the number of subsets of which we take the union, we have ${\textstyle \bigcup _{i\in {\mathcal {I}}}C_{i}=\mathbf {c} \left(\bigcup _{i\in {\mathcal {I}}}C_{i}\right)}$ . Thus, ${\textstyle \bigcup _{i\in {\mathcal {I}}}C_{i}\in {\mathfrak {S}}[\mathbf {c} ]}$ .

Induction of closure from topology

Conversely, given a family $\kappa$ satisfying axioms [T1]–[T3], it is possible to construct a Kuratowski closure operator in the following way: if $A\in \wp (X)$ and $A^{\uparrow }=\{B\in \wp (X)\ |\ A\subseteq B\}$ is the inclusion upset of $A$ , then

\mathbf {c} _{\kappa }(A):=\bigcap _{B\in (\kappa \cap A^{\uparrow })}B

defines a Kuratowski closure operator $\mathbf {c} _{\kappa }$ on $\wp (X)$ .

Proof 2.
[K1] Since $\varnothing ^{\uparrow }=\wp (X)$ , $\mathbf {c} _{\kappa }(\varnothing )$ reduces to the intersection of all sets in the family $\kappa$ ; but $\varnothing \in \kappa$ by axiom [T1], so the intersection collapses to the null set and [K1] follows. [K2] By definition of $A^{\uparrow }$ , we have that $A\subseteq B$ for all $B\in \left(\kappa \cap A^{\uparrow }\right)$ , and thus $A$ must be contained in the intersection of all such sets. Hence follows extensivity [K2]. [K3] Notice that, for all $A\in \wp (X)$ , the family $\mathbf {c} _{\kappa }(A)^{\uparrow }\cap \kappa$ contains $\mathbf {c} _{\kappa }(A)$ itself as a minimal element w.r.t. inclusion. Hence ${\textstyle \mathbf {c} _{\kappa }^{2}(A)=\bigcap _{B\in \mathbf {c} _{\kappa }(A)^{\uparrow }\cap \kappa }B=\mathbf {c} _{\kappa }(A)}$ , which is idempotence [K3]. [K4'] Let $A\subseteq B\subseteq X$ : then $B^{\uparrow }\subseteq A^{\uparrow }$ , and thus $\kappa \cap B^{\uparrow }\subseteq \kappa \cap A^{\uparrow }$ . Since the latter family may contain more elements than the former, we find $\mathbf {c} _{\kappa }(A)\subseteq \mathbf {c} _{\kappa }(B)$ , which is isotonicity [K4']. Notice that isotonicity implies $\mathbf {c} _{\kappa }(A)\subseteq \mathbf {c} _{\kappa }(A\cup B)$ and $\mathbf {c} _{\kappa }(B)\subseteq \mathbf {c} _{\kappa }(A\cup B)$ , which together imply $\mathbf {c} _{\kappa }(A)\cup \mathbf {c} _{\kappa }(B)\subseteq \mathbf {c} _{\kappa }(A\cup B)$ . [K4] Finally, fix $A,B\in \wp (X)$ . Axiom [T2] implies $\mathbf {c} _{\kappa }(A),\mathbf {c} _{\kappa }(B)\in \kappa$ ; furthermore, axiom [T2] implies that $\mathbf {c} _{\kappa }(A)\cup \mathbf {c} _{\kappa }(B)\in \kappa$ . By extensivity [K2] one has $\mathbf {c} _{\kappa }(A)\in A^{\uparrow }$ and $\mathbf {c} _{\kappa }(B)\in B^{\uparrow }$ , so that $\mathbf {c} _{\kappa }(A)\cup \mathbf {c} _{\kappa }(B)\in \left(A^{\uparrow }\right)\cap \left(B^{\uparrow }\right)$ . But $\left(A^{\uparrow }\right)\cap \left(B^{\uparrow }\right)=(A\cup B)^{\uparrow }$ , so that all in all $\mathbf {c} _{\kappa }(A)\cup \mathbf {c} _{\kappa }(B)\in \kappa \cap (A\cup B)^{\uparrow }$ . Since then $\mathbf {c} _{\kappa }(A\cup B)$ is a minimal element of $\kappa \cap (A\cup B)^{\uparrow }$ w.r.t. inclusion, we find $\mathbf {c} _{\kappa }(A\cup B)\subseteq \mathbf {c} _{\kappa }(A)\cup \mathbf {c} _{\kappa }(B)$ . Point 4. ensures additivity [K4].

Proof 2.

[K1] Since $\varnothing ^{\uparrow }=\wp (X)$ , $\mathbf {c} _{\kappa }(\varnothing )$ reduces to the intersection of all sets in the family $\kappa$ ; but $\varnothing \in \kappa$ by axiom [T1], so the intersection collapses to the null set and [K1] follows.

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

[K3] Notice that, for all $A\in \wp (X)$ , the family $\mathbf {c} _{\kappa }(A)^{\uparrow }\cap \kappa$ contains $\mathbf {c} _{\kappa }(A)$ itself as a minimal element w.r.t. inclusion. Hence ${\textstyle \mathbf {c} _{\kappa }^{2}(A)=\bigcap _{B\in \mathbf {c} _{\kappa }(A)^{\uparrow }\cap \kappa }B=\mathbf {c} _{\kappa }(A)}$ , which is idempotence [K3].

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

[K4] Finally, fix $A,B\in \wp (X)$ . Axiom [T2] implies $\mathbf {c} _{\kappa }(A),\mathbf {c} _{\kappa }(B)\in \kappa$ ; furthermore, axiom [T2] implies that $\mathbf {c} _{\kappa }(A)\cup \mathbf {c} _{\kappa }(B)\in \kappa$ . By extensivity [K2] one has $\mathbf {c} _{\kappa }(A)\in A^{\uparrow }$ and $\mathbf {c} _{\kappa }(B)\in B^{\uparrow }$ , so that $\mathbf {c} _{\kappa }(A)\cup \mathbf {c} _{\kappa }(B)\in \left(A^{\uparrow }\right)\cap \left(B^{\uparrow }\right)$ . But $\left(A^{\uparrow }\right)\cap \left(B^{\uparrow }\right)=(A\cup B)^{\uparrow }$ , so that all in all $\mathbf {c} _{\kappa }(A)\cup \mathbf {c} _{\kappa }(B)\in \kappa \cap (A\cup B)^{\uparrow }$ . Since then $\mathbf {c} _{\kappa }(A\cup B)$ is a minimal element of $\kappa \cap (A\cup B)^{\uparrow }$ w.r.t. inclusion, we find $\mathbf {c} _{\kappa }(A\cup B)\subseteq \mathbf {c} _{\kappa }(A)\cup \mathbf {c} _{\kappa }(B)$ . Point 4. ensures additivity [K4].

Exact correspondence between the two structures

In fact, these two complementary constructions are inverse to one another: if $\mathrm {Cls} _{\text{K}}(X)$ is the collection of all Kuratowski closure operators on $X$ , and $\mathrm {Atp} (X)$ 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 ${\mathfrak {S}}:\mathrm {Cls} _{\text{K}}(X)\to \mathrm {Atp} (X)$ such that $\mathbf {c} \mapsto {\mathfrak {S}}[\mathbf {c} ]$ is a bijection, whose inverse is given by the assignment ${\mathfrak {C}}:\kappa \mapsto \mathbf {c} _{\kappa }$ .

Proof 3.
First we prove that ${\mathfrak {C}}\circ {\mathfrak {S}}={\mathfrak {1}}_{\mathrm {Cls} _{\text{K}}(X)}$ , the identity operator on $\mathrm {Cls} _{\text{K}}(X)$ . For a given Kuratowski closure $\mathbf {c} \in \mathrm {Cls} _{\text{K}}(X)$ , define $\mathbf {c} ':={\mathfrak {C}}[{\mathfrak {S}}[\mathbf {c} ]]$ ; then if $A\in \wp (X)$ its primed closure $\mathbf {c} '(A)$ is the intersection of all $\mathbf {c}$ -stable sets that contain $A$ . Its non-primed closure $\mathbf {c} (A)$ satisfies this description: by extensivity [K2] we have $A\subseteq \mathbf {c} (A)$ , and by idempotence [K3] we have $\mathbf {c} (\mathbf {c} (A))=\mathbf {c} (A)$ , and thus $\mathbf {c} (A)\in \left(A^{\uparrow }\cap {\mathfrak {S}}[\mathbf {c} ]\right)$ . Now, let $C\in \left(A^{\uparrow }\cap {\mathfrak {S}}[\mathbf {c} ]\right)$ such that $A\subseteq C\subseteq \mathbf {c} (A)$ : by isotonicity [K4'] we have $\mathbf {c} (A)\subseteq \mathbf {c} (C)$ , and since $\mathbf {c} (C)=C$ we conclude that $C=\mathbf {c} (A)$ . Hence $\mathbf {c} (A)$ is the minimal element of $A^{\uparrow }\cap {\mathfrak {S}}[\mathbf {c} ]$ w.r.t. inclusion, implying $\mathbf {c} '(A)=\mathbf {c} (A)$ . Now we prove that ${\mathfrak {S}}\circ {\mathfrak {C}}={\mathfrak {1}}_{\mathrm {Atp} (X)}$ . If $\kappa \in \mathrm {Atp} (X)$ and $\kappa ':={\mathfrak {S}}[{\mathfrak {C}}[\kappa ]]$ is the family of all sets that are stable under $\mathbf {c} _{\kappa }$ , the result follows if both $\kappa '\subseteq \kappa$ and $\kappa \subseteq \kappa '$ . Let $A\in \kappa '$ : hence $\mathbf {c} _{\kappa }(A)=A$ . Since $\mathbf {c} _{\kappa }(A)$ is the intersection of an arbitrary subfamily of $\kappa$ , and the latter is complete under arbitrary intersections by [T2], then $A=\mathbf {c} _{\kappa }(A)\in \kappa$ . Conversely, if $A\in \kappa$ , then $\mathbf {c} _{\kappa }(A)$ is the minimal superset of $A$ that is contained in $\kappa$ . But that is trivially $A$ itself, implying $A\in \kappa '$ .

Proof 3.

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

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

We observe that one may also extend the bijection ${\mathfrak {S}}$ to the collection $\mathrm {Cls} _{\check {C}}(X)$ of all Čech closure operators, which strictly contains $\mathrm {Cls} _{\text{K}}(X)$ ; this extension ${\overline {\mathfrak {S}}}$ is also surjective, which signifies that all Čech closure operators on $X$ also induce a topology on $X$ .^[11] However, this means that ${\overline {\mathfrak {S}}}$ is no longer a bijection.

Examples

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

Fix an arbitrary $S\subsetneq X$ , and let $\mathbf {c} _{S}:\wp (X)\to \wp (X)$ be such that $\mathbf {c} _{S}(A):=A\cup S$ for all $A\in \wp (X)$ . Then $\mathbf {c} _{S}$ defines a Kuratowski closure; the corresponding family of closed sets ${\mathfrak {S}}[\mathbf {c} _{S}]$ coincides with $S^{\uparrow }$ , the family of all subsets that contain $S$ . When $S=\varnothing$ , we once again retrieve the discrete topology $\wp (X)$ (i.e. $\mathbf {c} _{\varnothing }=\mathbf {c} _{\bot }$ , as can be seen from the definitions).
If $\lambda$ is an infinite cardinal number such that $\lambda \leq \operatorname {crd} (X)$ , then the operator $\mathbf {c} _{\lambda }:\wp (X)\to \wp (X)$ such that
$\mathbf {c} _{\lambda }(A)={\begin{cases}A&\operatorname {crd} (A)<\lambda ,\\X&\operatorname {crd} (A)\geq \lambda \end{cases}}$
satisfies all four Kuratowski axioms.^[12] If $\lambda =\aleph _{0}$ , this operator induces the cofinite topology on $X$ ; if $\lambda =\aleph _{1}$ , 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 $\langle \mathbf {c} :\wp (X)\to \mathrm {im} (\mathbf {c} );\iota :\mathrm {im} (\mathbf {c} )\hookrightarrow \wp (X)\rangle$ , provided one views $\wp (X)$ as a poset with respect to inclusion, and $\mathrm {im} (\mathbf {c} )$ as a subposet of $\wp (X)$ . Indeed, it can be easily verified that, for all $A\in \wp (X)$ and $C\in \mathrm {im} (\mathbf {c} )$ , $\mathbf {c} (A)\subseteq C$ if and only if $A\subseteq \iota (C)$ .
If $\{A_{i}\}_{i\in {\mathcal {I}}}$ is a subfamily of $\wp (X)$ , then
$\bigcup _{i\in {\mathcal {I}}}\mathbf {c} (A_{i})\subseteq \mathbf {c} \left(\bigcup _{i\in {\mathcal {I}}}A_{i}\right),\qquad \mathbf {c} \left(\bigcap _{i\in {\mathcal {I}}}A_{i}\right)\subseteq \bigcap _{i\in {\mathcal {I}}}\mathbf {c} (A_{i}).$
If $A,B\in \wp (X)$ , then $\mathbf {c} (A)\setminus \mathbf {c} (B)\subseteq \mathbf {c} (A\setminus B)$ .

Topological concepts in terms of closure

Refinements and subspaces

A pair of Kuratowski closures $\mathbf {c} _{1},\mathbf {c} _{2}:\wp (X)\to \wp (X)$ such that $\mathbf {c} _{2}(A)\subseteq \mathbf {c} _{1}(A)$ for all $A\in \wp (X)$ induce topologies $\tau _{1},\tau _{2}$ such that $\tau _{1}\subseteq \tau _{2}$ , and vice versa. In other words, $\mathbf {c} _{1}$ dominates $\mathbf {c} _{2}$ if and only if the topology induced by the latter is a refinement of the topology induced by the former, or equivalently ${\mathfrak {S}}[\mathbf {c} _{1}]\subseteq {\mathfrak {S}}[\mathbf {c} _{2}]$ .^[13] For example, $\mathbf {c} _{\top }$ clearly dominates $\mathbf {c} _{\bot }$ (the latter just being the identity on $\wp (X)$ ). Since the same conclusion can be reached substituting $\tau _{i}$ with the family $\kappa _{i}$ containing the complements of all its members, if $\mathrm {Cls} _{\text{K}}(X)$ is endowed with the partial order $\mathbf {c} \leq \mathbf {c} '\iff \mathbf {c} (A)\subseteq \mathbf {c} '(A)$ for all $A\in \wp (X)$ and $\mathrm {Atp} (X)$ is endowed with the refinement order, then we may conclude that ${\mathfrak {S}}$ 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: $\mathbf {c} _{A}(B)=A\cap \mathbf {c} _{X}(B)$ , for all $B\subseteq A$ .^[14]

Continuous maps, closed maps and homeomorphisms

A function $f:(X,\mathbf {c} )\to (Y,\mathbf {c} ')$ is continuous at a point $p$ iff $p\in \mathbf {c} (A)\Rightarrow f(p)\in \mathbf {c} '(f(A))$ , and it is continuous everywhere iff

f(\mathbf {c} (A))\subseteq \mathbf {c} '(f(A))

for all subsets

A\in \wp (X)

.^[15] The mapping

f

is a closed map iff the reverse inclusion holds,^[16] and it is a homeomorphism iff it is both continuous and closed, i.e. iff equality holds.^[17]

Separation axioms

Let $(X,\mathbf {c} )$ be a Kuratowski closure space. Then

$X$ is a T₀-space iff $x\neq y$ implies $\mathbf {c} (\{x\})\neq \mathbf {c} (\{y\})$ ;^[18]
$X$ is a T₁-space iff $\mathbf {c} (\{x\})=\{x\}$ for all $x\in X$ ;^[19]
$X$ is a T₂-space iff $x\neq y$ implies that there exists a set $A\in \wp (X)$ such that both $x\notin \mathbf {c} (A)$ and $y\notin \mathbf {c} (\mathbf {n} (A))$ , where $\mathbf {n}$ is the set complement operator.^[20]

Closeness and separation

A point $p$ is close to a subset $A$ if $p\in \mathbf {c} (A).$ This can be used to define a proximity relation on the points and subsets of a set.^[21]

Two sets $A,B\in \wp (X)$ are separated iff $(A\cap \mathbf {c} (B))\cup (B\cap \mathbf {c} (A))=\varnothing$ . The space $X$ is connected iff it cannot be written as the union of two separated subsets.^[22]

Notes

^ Kuratowski (1922).
^ ^a ^b Monteiro (1945), p. 160.
^ ^a ^b ^c Pervin (1964), p. 44.
^ Pervin (1964), p. 43, Exercise 6.
^ Kuratowski (1966), p. 38.
^ Arkhangel'skij & Fedorchuk (1990), p. 25.
^ "Moore closure". nLab. March 7, 2015. Retrieved August 19, 2019.
^ Pervin (1964), p. 42, Exercise 5.
^ Monteiro (1945), p. 158.
^ Pervin (1964), p. 46, Exercise 4.
^ Arkhangel'skij & Fedorchuk (1990), p. 26.
^ A proof for the case $\lambda =\aleph _{0}$ can be found at "Is the following a Kuratowski closure operator?!". Stack Exchange. November 21, 2015.
^ Pervin (1964), p. 43, Exercise 10.
^ Pervin (1964), p. 49, Theorem 3.4.3.
^ Pervin (1964), p. 60, Theorem 4.3.1.
^ Pervin (1964), p. 66, Exercise 3.
^ Pervin (1964), p. 67, Exercise 5.
^ Pervin (1964), p. 69, Theorem 5.1.1.
^ Pervin (1964), p. 70, Theorem 5.1.2.
^ A proof can be found at this link.
^ Pervin (1964), pp. 193–196.
^ Pervin (1964), p. 51.

References

Kuratowski, Kazimierz (1922) [1920], "Sur l'opération A de l'Analysis Situs" [On the operation A in Analysis Situs] (PDF), Fundamenta Mathematicae (in French), vol. 3, pp. 182–199.
Kuratowski, Kazimierz (1966) [1958], Topology, vol. I, translated by Jaworowski, J., Academic Press, ISBN 0-12-429201-1, LCCN 66029221.
- —— (2010). "On the Operation Ā Analysis Situs". ResearchGate. Translated by Mark Bowron.
Pervin, William J. (1964), Boas, Ralph P. Jr. (ed.), Foundations of General Topology, Academic Press, ISBN 9781483225159, LCCN 64-17796.
Arkhangel'skij, A.V.; Fedorchuk, V.V. (1990) [1988], Gamkrelidze, R.V.; Arkhangel'skij, A.V.; Pontryagin, L.S. (eds.), General Topology I, Encyclopaedia of Mathematical Sciences, vol. 17, translated by O'Shea, D.B., Berlin: Springer-Verlag, ISBN 978-3-642-64767-3, LCCN 89-26209.
Monteiro, António (1945), "Caractérisation de l'opération de fermeture par un seul axiome" [Characterization of the operation of closure by a single axiom], Portugaliae mathematica (in French), vol. 4, no. 4, pp. 158–160, MR 0012310, Zbl 0060.39406.