# Kuratowski closure axioms

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.

## Definition

A topological space [itex](X,cl)[itex] is a set [itex]X[itex] with a function

[itex]cl:\mathcal{P}(X) \to \mathcal{P}(X)[itex]

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

The closure operator has to satisfy the following properties

1. [itex] A \subseteq cl(A) \! [itex] (Isotonicity)
2. [itex] cl(cl(A)) = cl(A) \! [itex] (Idempotence)
3. [itex] cl(A \cup B) = cl(A) \cup cl(B) \! [itex] (Preservation of binary unions)
4. [itex] cl(\varnothing) = \varnothing \! [itex] (Preservation of nullary unions)

## Notes

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

[itex] c(A_{1} \cup \cdots \cup A_{n}) = c(A_{1}) \cup \cdots \cup c(A_{n}) \! [itex] (Preservation of finitary unions).

An operator that only satisfies axioms (1) and (2) is called a Moore closure. Moore closure operators are often studied in lattice theory.

## Recovering topological definitions

A function between two topological spaces

[itex]f:(X,cl) \to (X',cl')[itex]

is a called continuous if for all subsets [itex]A[itex] of [itex]X[itex]

[itex]f(cl(A)) \subset cl'(f(A))[itex]

A point [itex]p[itex] is called close to [itex]A[itex] in [itex](X,cl)[itex] if [itex]p\in cl(A)[itex]

[itex]A[itex] is called closed in [itex](X,cl)[itex] if [itex]A=cl(A)[itex]. In other words the closed sets of [itex]X[itex] are the fixed points of the closure operator.

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