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 is a set with a function
called the closure operator where is the power set of .
The closure operator has to satisfy the following properties for all
- (Preservation of binary unions)
- (Preservation of nullary unions)
If the second axiom, that of idempotence, is relaxed, then the axioms define a preclosure operator.
Connection to other axiomatizations of topology
Induction of a topology
A point is called close to in if
By defining a closure operator on a topology as defined usually (a set containing all open sets) is naturally induced as follows. A set is called open if and only if and we construct . The pair then obeys the Open Sets Definition of a topological space:
Empty and complete sets are open:
By Extensitivity and since we know that , therefore . From Preservation of nullary unions follows similarly .
Any union of open sets is open:
Let be a set of Indices and we unite every where is open for every . By De Morgan's laws is
And by Preservation of binary unions:
Hence And together with Extensivity follows Therefore, A is open.
The intersection of any finite number of open sets is open.
Let be a finite set of Indices with open .
From the Preservation of nullary unions follows by induction:
- is open.
Recovering topological definitions
A function between two topological spaces
is called continuous if for all subsets of