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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 $(X, c l)$ 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 single statement:

$c(A_{1} \cup \cdots \cup A_{n}) = c(A_{1}) \cup \cdots \cup c(A_{n}) \!$ (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 a called continuous if for all subsets $A$ of $X$

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

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

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

12-03-2008 10:22:39
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