Kuratowski closure axioms
From Biocrawler, the free encyclopedia.
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 (X,cl) is a set X with a function
called the closure operator where 
is the power set of X.
The closure operator has to satisfy the following properties
-
(Isotonicity)
-
(Idempotence)
-
(Preservation of binary unions)
-
(Preservation of nullary unions)
Notes
Axioms (3) and (4) can be generalised (using a proof by mathematical induction) to the single statement:
(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
is a called continuous if for all subsets A of X
A point p is called close to A in (X,cl) if 
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.
