In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a bitopological space is a set endowed with two topologies

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

. Typically, if the set is and the topologies are and then we refer to the bitopological space as .

Bi-continuity

A map

Map (mathematics)

In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...

  from a bitopological space to another bitopological space is called bi-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.

• A bitopological space is pairwise Hausdorff if for any two distinct points there exist disjoint and with either and or and .

• A bitopological space is pairwise zero-dimensional if opens in which are closed in form a basis for , and opens in which 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