Bitopological space

In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is and the topologies are and then the bitopological space is referred to as. The notion was introduced by J. C. Kelly in the study of quasimetrics, i.e. distance functions that are not required to be symmetric.

Continuity

A map from a bitopological space to another bitopological space is called continuous or sometimes pairwise continuous if is continuous both as a map from to and as map from to.

Bitopological variants of topological properties

Corresponding to well-known properties of topological spaces, there are versions for bitopological spaces.

A bitopological space is pairwise compact if each cover of with, contains a finite subcover. In this case, must contain at least one member from and at least one member from
A bitopological space is pairwise Hausdorff if for any two distinct points there exist disjoint and with and.
A bitopological space is pairwise zero-dimensional if opens in that are closed in form a basis for, and opens in that are closed in form a basis for.
A bitopological space is called binormal if for every -closed and -closed sets there are -open and -open sets such that, and