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,<ref>Template:Harvp.</ref> and the idea was further studied by mathematicians such as Wacław Sierpiński and 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 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:

A consequence of <math>\mathbf{c}</math> preserving binary unions is the following condition:<ref>Template:Harvp, Exercise 6.</ref> Template:Quote frame In fact if we rewrite the equality in [K4] as an inclusion, giving the weaker axiom [K4''] (subadditivity): Template:Quote frame 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).

Template:Harvp 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 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).<ref name=":0">Template:Harvp.</ref>

If requirement [K3] is omitted, then the axioms define a Čech closure operator.<ref>Template:Harvp.</ref> If [K1] is omitted instead, then an operator satisfying [K2], [K3] and [K4'] is said to be a Moore closure operator.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</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>Template:Harvp, Exercise 5.</ref> Template:Quote frame

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

1. 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].
2. 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].
3. 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].
4. Choose arbitrary <math>A,B \subseteq X</math>. Applying axioms [K1]–[K3], one derives [K4].

Alternatively, Template:Harvp had proposed a weaker axiom that only entails [K2]–[K4]:<ref>Template:Harvp.</ref> Template:Quote frame 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">Template:Harvp.</ref> Template:Quote frame

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">Template:Harvp.</ref>

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>.

Template:Harvp further provides analogous axioms for Kuratowski exterior operators<ref name=":2" /> and Kuratowski boundary operators,<ref>Template:Harvp, 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

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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 poset <math>S</math>.

Connection to other axiomatizations of topology

Induction of topology from closure

A closure operator naturally induces a 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: Template:Quote frame

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

Template:Collapse top [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 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>. Template: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 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>.

Template:Collapse top [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]. Template: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>.

Template:Collapse top 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>. Template: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>Template:Harvp.</ref> However, this means that <math>\overline{\mathfrak{S}}</math> is no longer a bijection.

Examples

• 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 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 {{#invoke:citation/CS1|citation |CitationClass=web }}</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>Template:Harvp, 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>Template:Harvp, Theorem 3.4.3.</ref>

Continuous maps, closed maps and homeomorphisms

A function <math>f:(X,\mathbf{c})\to (Y,\mathbf{c}')</math> is 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>Template:Harvp, Theorem 4.3.1.</ref> The mapping <math>f</math> is a closed map iff the reverse inclusion holds,<ref>Template:Harvp, Exercise 3.</ref> and it is a homeomorphism iff it is both continuous and closed, i.e. iff equality holds.<ref>Template:Harvp, Exercise 5.</ref>

Separation axioms

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

• <math>X</math> is a T₀-space iff <math>x \neq y</math> implies <math>\mathbf{c}(\{x\}) \neq \mathbf{c}(\{y\})</math>;<ref>Template:Harvp, Theorem 5.1.1.</ref>
• <math>X</math> is a T₁-space iff <math>\mathbf{c}(\{x\})=\{x\}</math> for all <math>x \in X</math>;<ref>Template:Harvp, Theorem 5.1.2.</ref>
• <math>X</math> is a T₂-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 link.</ref>

Closeness and separation

A point <math>p</math> is close to a subset <math>A</math> if <math>p\in\mathbf{c}(A).</math>This can be used to define a proximity relation on the points and subsets of a set.<ref>Template:Harvp.</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 iff it cannot be written as the union of two separated subsets.<ref>Template:Harvp.</ref>

See also

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References

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External links

• Alternative Characterizations of Topological Spaces