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.

Definition

A topological space <math>(X,cl)</math> is a set <math>X</math> with a function

<math>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

1. <math> A \subseteq cl(A) \! </math> (Extensivity)
2. <math> cl(cl(A)) = cl(A) \! </math> (Idempotence)
3. <math> cl(A \cup B) = cl(A) \cup cl(B) \! </math> (Preservation of binary unions)
4. <math> cl(\varnothing) = \varnothing \! </math> (Preservation of nullary unions)

Notes

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

<math> c(A_{1} \cup \cdots \cup A_{n}) = c(A_{1}) \cup \cdots \cup c(A_{n}) \! </math> (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

<math>f:(X,cl) \to (X',cl')</math>

is called continuous if for all subsets <math>A</math> of <math>X</math>

<math>f(cl(A)) \subset cl'(f(A))</math>

A point <math>p</math> is called close to <math>A</math> in <math>(X,cl)</math> if <math>p\in cl(A)</math>

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