Kuratowski closure axioms
From Wikipedia, 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.
[edit] Definition
A topological space 
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
(Extensivity)
(Idempotence)
(Preservation of binary unions)
(Preservation of nullary unions)
[edit] Notes
Axioms (3) and (4) can be generalised (using a proof by mathematical induction) to the equivalent single statement:
(Preservation of finitary unions).
[edit] Recovering topological definitions
A function between two topological spaces
is called continuous if for all subsets A of X
A point p is called close to A in if 
A is called closed in if 
. In other words the closed sets of X are the fixed points of the closure operator.
