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
- <math> A \subseteq cl(A) \! <math> (Isotonicity)
- <math> cl(cl(A)) = cl(A) \! <math> (Idempotence)
- <math> cl(A \cup B) = cl(A) \cup cl(B) \! <math> (Preservation of binary unions)
- <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 a 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.