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 T_{1}-spaces in contrast to the more general spaces which only satisfy the four listed axioms. Indeed, these spaces correspond exactly to the topological T_{1}-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]}
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) harvp error: no target: CITEREFMonteiro1945 (help) 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]}
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 $\{A_{i}\}_{i\in {\mathcal {I))}\subseteq \wp (X)$,
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 $\mathbf {c} (A):=A\cup \mathbf {b} (A)$.
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, ${\textstyle \bigcup _{i\in {\mathcal {I))}C_{i}\in {\mathfrak {S))[\mathbf {c} ]}$.
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
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].
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 $A\in \kappa '$.
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
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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
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
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
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 T_{0}-space iff $x\neq y$ implies $\mathbf {c} (\{x\})\neq \mathbf {c} (\{y\})$;^{[18]}
$X$ is a T_{1}-space iff $\mathbf {c} (\{x\})=\{x\))$ for all $x\in X$;^{[19]}
$X$ is a T_{2}-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]}