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In [[topology]] and related branches of [[mathematics]], the '''Kuratowski closure axioms''' are a set of [[axiom]]s which can be used to define a [[topological structure]] on a [[Set (mathematics)|set]]. They are equivalent to the more commonly used [[open set]] definition. They were first introduced by [[Kazimierz Kuratowski]], in a slightly different form that applied only to [[Hausdorff space]]s.

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

== Definition ==

Let <math>X</math> be a set and <math>\mathcal{P}(X)</math> its [[power set]]. 
A '''Kuratowski Closure Operator''' is an assignment <math>\operatorname{cl}:\mathcal{P}(X) \to \mathcal{P}(X)</math> with the following properties:
# <math> \operatorname{cl}(\varnothing) = \varnothing </math> (Preservation of Nullary Union)
# <math> A \subseteq \operatorname{cl}(A) </math> (Extensivity)
# <math> \operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B) </math> (Preservation of Binary Union)
# <math> \operatorname{cl}(\operatorname{cl}(A)) = \operatorname{cl}(A) \! </math> ([[Idempotent function|Idempotence]])

If the last axiom, Idempotence, is omitted, then the axioms define a [[preclosure operator|Preclosure Operator]]. 
A consequence from the third axiom is: <math> A \subseteq B \Rightarrow \operatorname{cl}(A) \subseteq \operatorname{cl}(B) </math> (Preservation of Inclusion)

== Connection to other Axiomatizations of Topology ==

=== Induction of Topology ===
'''Construction''' 
A closure operator naturally induces a topology as follows: 
A subset <math> C\subseteq X </math> is called closed if and only if <math> \operatorname{cl}(C) = C </math>.

'''Empty Set and Entire Space are closed:''' 
By Extensitivity <math> X\subseteq\operatorname{cl}(X) </math> and since Closure maps into itself <math> \operatorname{cl}(X)\subseteq X </math> we have <math> X = \operatorname{cl}(X)</math>. Thus <math> X </math> is closed. 
By Preservation of Nullary Unions follows <math> \operatorname{cl}(\varnothing) = \varnothing </math>. Thus <math> \varnothing </math> is closed

'''Arbitrary intersections of closed sets is closed:''' 
Let <math> \mathcal{I} </math> be an arbitrary set of indices and <math> C_i </math> closed for every <math> i\in\mathcal{I}</math>. 
Then by Extensitivity: <math> \bigcap_{i\in\mathcal{I}}C_i \subseteq \operatorname{cl}(\bigcap_{i\in\mathcal{I}}C_i) </math> 
Also by Preservation of Inclusions: <math> \bigcap_{i\in\mathcal{I}}C_i \subseteq C_i \forall i\in\mathcal{I} \Rightarrow \operatorname{cl}(\bigcap_{i\in\mathcal{I}}C_i) \subseteq \operatorname{cl}(C_i) = C_i \forall i\in\mathcal{I} \Rightarrow \operatorname{cl}(\bigcap_{i\in\mathcal{I}}C_i) \subseteq \bigcap_{i\in\mathcal{I}}C_i </math> 
And therefore <math> \bigcap_{i\in\mathcal{I}}C_i = \operatorname{cl}(\bigcap_{i\in\mathcal{I}}C_i) </math>. Thus <math> \bigcap_{i\in\mathcal{I}}C_i </math> is closed.

'''Finite unions of closed sets is closed:''' 
Let <math> \mathcal{I} </math> be a finite set of indices and <math> C_i </math> closed for every <math> i\in\mathcal{I} </math>. 
From the Preservation of binary unions and by [[mathematical induction|induction]] we have <math> \bigcup_{i\in\mathcal{I}}C_i = \operatorname{cl}(\bigcup_{i\in\mathcal{I}}C_i) </math>. Thus <math> \bigcup_{i\in\mathcal{I}}C_i </math> is closed.

=== Induction of Closure ===
The induced topology reinduces a closure which agrees with the original closure: <math> \bar{A}=\operatorname{cl}(A) </math> 
For a proof see [http://math.uga.edu/~pete/TopSection3.pdf Alternative Characterizations of Topological Spaces].

=== Recovering Notions from Topology ===

'''Closeness''' 
A point <math>p</math> is [[Closeness (topology)|close]] to a subset <math>A</math> iff <math>p\in\operatorname{cl}(A)</math>.

'''Continuity''' 
A function <math>f:X\to Y</math> is [[Continuity (topology)|continuous]] at a point <math>p</math> iff <math>p\in\operatorname{cl}(A) \Rightarrow f(p)\in\operatorname{cl}(f(A))</math>.

==See also==
* [[Preclosure operator]]

==External links==
* [http://math.uga.edu/~pete/TopSection3.pdf Alternative Characterizations of Topological Spaces]

[[Category:Closure operators]]
[[Category:Mathematical axioms]]

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