Kuratowski closure axioms

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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.

1 Definition
2 Notes
3 Recovering topological definitions
4 See also

[edit] Definition

A topological space $(X,\operatorname{cl})$ is a set $X$ with a function

$\operatorname{cl}:\mathcal{P}(X) \to \mathcal{P}(X)$

called the closure operator where $\mathcal{P}(X)$ is the power set of $X$ .

The closure operator has to satisfy the following properties

$A \subseteq \operatorname{cl}(A) \!$ (Extensivity)
$\operatorname{cl}(\operatorname{cl}(A)) = \operatorname{cl}(A) \!$ (Idempotence)
$\operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B) \!$ (Preservation of binary unions)
$\operatorname{cl}(\varnothing) = \varnothing \!$ (Preservation of nullary unions)

If the second axiom, that of idempotence, is relaxed, then the axioms define a preclosure operator.

[edit] Notes

Axioms (3) and (4) can be generalised (using a proof by mathematical induction) to the equivalent single statement:

$\operatorname{cl}(A_{1} \cup \cdots \cup A_{n}) = \operatorname{cl}(A_{1}) \cup \cdots \cup \operatorname{cl}(A_{n}), n \geq 0 \!$ (Preservation of finitary unions).

[edit] Recovering topological definitions

A function between two topological spaces

$f:(X,\operatorname{cl}) \to (X',\operatorname{cl}')$

is called continuous if for all subsets $A$ of $X$

$f(\operatorname{cl}(A)) \subset \operatorname{cl}(f(A))$

A point $p$ is called close to $A$ in $(X,\operatorname{cl})$ if $p\in \operatorname{cl}(A)$

$A$ is called closed in $(X,\operatorname{cl})$ if $A=\operatorname{cl}(A)$ . In other words the closed sets of $X$ are the fixed points of the closure operator.