In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which 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 introduced by Kazimierz Kuratowski, in a slightly different form that applied only to Hausdorff spaces. A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.
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Definition
A topological space <math>(X,\operatorname{cl})</math> is a set <math>X</math> with a function
- <math>\operatorname{cl}:\mathcal{P}(X) \to \mathcal{P}(X)</math>
called the closure operator where <math>\mathcal{P}(X)</math> is the power set of <math>X</math>. The closure operator has to satisfy the following properties
- <math> A \subseteq \operatorname{cl}(A) \! </math> (Extensivity)
- <math> \operatorname{cl}(\operatorname{cl}(A)) = \operatorname{cl}(A) \! </math> (Idempotence)
- <math> \operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B) \! </math> (Preservation of binary unions)
- <math> \operatorname{cl}(\varnothing) = \varnothing \! </math> (Preservation of nullary unions)
If the second axiom, that of idempotence, is relaxed, then the axioms define a preclosure.
Notes
Axioms (3) and (4) can be generalised (using a proof by mathematical induction) to the equivalent single statement:
- <math> \operatorname{cl}(A_{1} \cup \cdots \cup A_{n}) = \operatorname{cl}(A_{1}) \cup \cdots \cup \operatorname{cl}(A_{n}), n \geq 0 \! </math> (Preservation of finitary unions).
Recovering topological definitions
A function between two topological spaces
- <math>f:(X,\operatorname{cl}) \to (X',\operatorname{cl}')</math>
is called continuous if for all subsets <math>A</math> of <math>X</math>
- <math>f(\operatorname{cl}(A)) \subset \operatorname{cl}(f(A))</math>
A point <math>p</math> is called close to <math>A</math> in <math>(X,\operatorname{cl})</math> if <math>p\in \operatorname{cl}(A)</math> <math>A</math> is called closed in <math>(X,\operatorname{cl})</math> if <math>A=\operatorname{cl}(A)</math>. In other words the closed sets of <math>X</math> are the fixed points of the closure operator.
