A consequence of preserving binary unions is the following condition:
[K4'] It is isotonic: .
In fact if we rewrite the equality in [K4] as an inclusion, giving the weaker axiom [K4''] (subadditivity):
[K4''] It is subadditive: for all , ,
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 , . 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).
If requirement [K3] is omitted, then the axioms define a Čech closure operator. If [K1] is omitted instead, then an operator satisfying [K2], [K3] and [K4'] is said to be a Moore closure operator. A pair is called Kuratowski, Čech or Moore closure space depending on the axioms satisfied by .
The four Kuratowski closure axioms can be replaced by a single condition, given by Pervin:
[P] For all , .
Axioms [K1]—[K4] can be derived as a consequence of this requirement:
Choose . Then , or . This immediately implies [K1].
Choose an arbitrary and . Then, applying axiom [K1], , implying [K2].
Choose and an arbitrary . Then, applying axiom [K1], , which is [K3].
Choose arbitrary . 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]:
[M] For all , .
Requirement [K1] is independent of [M] : indeed, if , the operator defined by the constant assignment satisfies [M] but does not preserve the empty set, since . 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]:
A dual notion to Kuratowski closure operators is that of Kuratowski interior operator, which is a map satisfying the following similar requirements:
[I1] It preserves the total space: ;
[I2] It is intensive: for all , ;
[I3] It is idempotent: for all , ;
[I4] It preserves binary intersections: for all , .
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 , the map sending . This map is an orthocomplementation on the power set lattice, meaning it satisfies De Morgan's laws: if is an arbitrary set of indices and ,
By employing these laws, together with the defining properties of , one can show that any Kuratowski interior induces a Kuratowski closure (and vice versa), via the defining relation (and ). Every result obtained concerning may be converted into a result concerning by employing these relations in conjunction with the properties of the orthocomplementation .
Pervin (1964) further provides analogous axioms for Kuratowski exterior operators and Kuratowski boundary operators, which also induce Kuratowski closures via the relations and .
Notice that axioms [K1]—[K4] may be adapted to define an abstract unary operation on a general bounded lattice , 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 on an arbitrary poset.
Connection to other axiomatizations of topology
A closure operator naturally induces a topology as follows. Let be an arbitrary set. We shall say that a subset is closed with respect to a Kuratowski closure operator if and only if it is a fixed point of said operator, or in other words it is stable under, i.e. . 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 of all closed sets satisfies the following:
[T2] It is complete under arbitrary intersections, i.e. if is an arbitrary set of indices and , then ;
[T3] It is complete under finite unions, i.e. if is a finite set of indices and , then .
Notice that, by idempotency [K3], one may succinctly write .
[T1] By extensivity [K2], and since closure maps the power set of into itself (that is, the image of any subset is a subset of ), we have . Thus . The preservation of the empty set [K1] readily implies .
[T2] Next, let be an arbitrary set of indices and let be closed for every . By extensivity [K2], . Also, by isotonicity [K4'], if for all indices , then for all , which implies . Therefore, , meaning .
[T3] Finally, let be a finite set of indices and let be closed for every . From the preservation of binary unions [K4], and using induction on the number of subsets of which we take the union, we have . Thus, .
Conversely, given a family satisfying axioms [T1]—[T3], it is possible to construct a Kuratowski closure operator in the following way: if and is the inclusion upset of , then
defines a Kuratowski closure operator on .
[K1] Since , reduces to the intersection of all sets in the family ; but by axiom [T1], so the intersection collapses to the null set and [K1] follows.
[K2] By definition of , we have that for all , and thus must be contained in the intersection of all such sets. Hence follows extensivity [K2].
[K3] Notice that, for all , the family contains itself as a minimal element w.r.t. inclusion. Hence , which is idempotence [K3].
[K4’] Let : then , and thus . Since the latter family may contain more elements than the former, we find , which is isotonicity [K4']. Notice that isotonicity implies and , which together imply .
[K4] Finally, fix . Axiom [T2] implies ; furthermore, axiom [T2] implies that . By extensivity [K2] one has and , so that . But , so that all in all . Since then is a minimal element of w.r.t. inclusion, we find . Point 4. ensures additivity [K4].
Exact correspondence between the two structures
In fact, these two complementary constructions are inverse to one another: if is the collection of all Kuratowski closure operators on , and 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 such that is a bijection, whose inverse is given by the assignment .
First we prove that , the identity operator on . For a given Kuratowski closure , define ; then if its primed closure is the intersection of all -stable sets that contain . Its non-primed closure satisfies this description: by extensivity [K2] we have , and by idempotence [K3] we have , and thus . Now, let such that : by isotonicity [K4'] we have , and since we conclude that . Hence is the minimal element of w.r.t. inclusion, implying .
Now we prove that . If and is the family of all sets that are stable under , the result follows if both and . Let : hence . Since is the intersection of an arbitrary subfamily of , and the latter is complete under arbitrary intersections by [T2], then . Conversely, if , then is the minimal superset of that is contained in . But that is trivially itself, implying .
We observe that one may also extend the bijection to the collection of all Čech closure operators, which strictly contains ; this extension is also surjective, which signifies that all Čech closure operators on also induce a topology on . However, this means that is no longer a bijection.
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As discussed above, given a topological space we may define the closure of any subset to be the set , i.e. the intersection of all closed sets of which contain . The set is the smallest closed set of containing , and the operator is a Kuratowski closure operator.
Fix an arbitrary , and let be such that for all . Then defines a Kuratowski closure; the corresponding family of closed sets coincides with , the family of all subsets that contain . When , we once again retrieve the discrete topology (i.e. , as can be seen from the definitions).
If is a cardinal number such that , then the operator such that
Since any Kuratowski closure is isotonic, and so is obviously any inclusion mapping, one has the (isotonic) Galois connection, provided one views as a poset with respect to inclusion, and as a subposet of . Indeed, it can be easily verified that, for all and , if and only if .
A pair of Kuratowski closures such that for all induce topologies such that , and vice versa. In other words, dominates if and only if the topology induced by the latter is a refinement of the topology induced by the former, or equivalently . For example, clearly dominates (the latter just being the identity on ). Since the same conclusion can be reached substituting with the family containing the complements of all its members, if is endowed with the partial order for all and is endowed with the refinement order, then we may conclude that 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: , for all .
Continuous maps, closed maps and homeomorphisms
A function is continuous at a point iff , and it is continuous everywhere iff
for all subsets . The mapping is a closed map iff the reverse inclusion holds, and it is a homeomorphism iff it is both continuous and closed, i.e. iff equality holds.