Open main menu

DefinitionEdit

Kuratowski closure operators and weakeningsEdit

Let   be an arbitrary set and   its power set. A Kuratowski closure operator is a unary operation   with the following properties:

[K1] It preserves the empty set:  ;

[K2] It is extensive: for all  ,  ;

[K3] It is idempotent: for all  ,  ;

[K4] It preserves binary unions: for all  ,  .

Because of extensivity [K2], it is possible to weaken the equality in [K3] to a simple inclusion. Furthermore, a consequence of   preserving binary unions is the stronger condition:[4]

[K4'] It is isotonic:  .

It is straightforward to show that weakening the equality in [K4] to a simple inclusion (subadditivity) and assuming [K4'] yields back 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).[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   is called Kuratowski, Čech or Moore closure space depending on the axioms satisfied by  .

Alternative axiomatizationsEdit

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

[P] For all  ,  .

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

  1. Choose  . Then  , or  . This immediately implies [K1].
  2. Choose an arbitrary   and  . Then, applying axiom [K1],  , implying [K2].
  3. Choose   and an arbitrary  . Then, applying axiom [K1],  , which is [K3].
  4. Choose arbitrary  . Applying axioms [K1][K3], one derives [K4].

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

[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]:[2]

[BT] For all  ,  .

Analogous structuresEdit

Interior, exterior and boundary operatorsEdit

A dual notion to Kuratowski closure operators is that of Kuratowski interior operator, which is a map   satisfying the following similar requirements:[3]

[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[3] and Kuratowski boundary operators[10], which also induce Kuratowski closures via the relations   and  .

Abstract operatorsEdit

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 topologyEdit

Induction of topology from closureEdit

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:

[T1] It is a bounded sublattice of  , i.e.  ;

[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 succintly write  .

Proof 1.

[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,  .

Induction of closure from topologyEdit

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  .
Proof 2.

[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 structuresEdit

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  .

Proof 3.

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  .[11] However, this means that   is no longer a bijection.

ExamplesEdit

  • If   is any set, the operators   such that
     
    are Kuratowski closures. The first induces the indiscrete topology  , while the second induces the discrete topology  .
  • If   is a cardinal number such that  , then the operator   such that
     
    satisfies all four Kuratowski axioms.[12] In the case where  , if  , this operator induces the cofinite topology on  ; if  , it induces the cocountable topology.
  • 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  , one retrieves the discrete topology  .

PropertiesEdit

  • 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  .
  • If   is a subfamily of  , then
     
  • If  , then  .

Topological concepts in terms of closureEdit

Refinements and subspacesEdit

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  .[13] 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  .[14]

Continuous maps, closed maps and homeomorphismsEdit

A function   is continuous at a point   iff  , and it is continuous everywhere iff

 
for all subsets  .[15] The mapping   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 axiomsEdit

Let   be a Kuratowski closure space. Then

  •   is a T0-space iff   implies  ;[18]
  •   is a T1-space iff   for all  ;[19]
  •   is a T2-space iff   implies that there exists a set   such that both   and  , where   is the set complement operator.[20]

Closeness and separationEdit

A point   is close to a subset   if  This can be used to define a proximity relation on the points and subsets of a set.[21]

Two sets   are separated iff  . The space   is connected iff no two subsets are separated.[22]

See alsoEdit

NotesEdit

  1. ^ Kuratowski (1922).
  2. ^ a b Monteiro (1945), p. 160.
  3. ^ a b c Pervin (1964), p. 44.
  4. ^ Pervin (1964), p. 43, Exercise 6.
  5. ^ Kuratowski (1966), p. 38.
  6. ^ Arkhangel'skij & Fedorchuk (1990), p. 25.
  7. ^ "Moore closure". nLab. March 7, 2015. Retrieved August 19, 2019.
  8. ^ Pervin (1964), p. 42, Exercise 5.
  9. ^ Monteiro (1945), p. 158.
  10. ^ Pervin (1964), p. 46, Exercise 4.
  11. ^ Arkhangel'skij & Fedorchuk (1990), p. 26.
  12. ^ A proof for the case   can be found at "Is the following a Kuratowski closure operator?!". Stack Exchange. November 21, 2015.
  13. ^ Pervin (1964), p. 43, Exercise 10.
  14. ^ Pervin (1964), p. 49, Theorem 3.4.3.
  15. ^ Pervin (1964), p. 60, Theorem 4.3.1.
  16. ^ Pervin (1964), p. 66, Exercise 3.
  17. ^ Pervin (1964), p. 67, Exercise 5.
  18. ^ Pervin (1964), p. 69, Theorem 5.1.1.
  19. ^ Pervin (1964), p. 70, Theorem 5.1.2.
  20. ^ A proof can be found at this link.
  21. ^ Pervin (1964), pp. 193–196.
  22. ^ Pervin (1964), p. 51.

ReferencesEdit

External linksEdit