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.

A topological space $(X,cl)$ is a set $X$ with a function

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

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

cl(A_{1} \cup \cdots \cup A_{n}) = cl(A_{1}) \cup \cdots \cup cl(A_{n}), n \geq 0 \!

(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.

A function between two topological spaces

f:(X,cl) \to (X',cl')

is called continuous if for all subsets

A

X

f(cl(A)) \subset cl'(f(A))

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

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

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