In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is X and the topologies are σ and τ then the bitopological space is referred to as ( X , σ , τ ) .
A map f : X → X ′ from a bitopological space ( X , τ 1 , τ 2 ) to another bitopological space ( X ′ , τ 1 ′ , τ 2 ′ ) is called continuous or sometimes pairwise continuous if f is continuous both as a map from ( X , τ 1 ) to ( X ′ , τ 1 ′ ) and as map from ( X , τ 2 ) to ( X ′ , τ 2 ′ ) .
Corresponding to well-known properties of topological spaces, there are versions for bitopological spaces.
A bitopological space ( X , τ 1 , τ 2 ) is pairwise compact if each cover { U i ∣ i ∈ I } of X with U i ∈ τ 1 ∪ τ 2 , contains a finite subcover. In this case, { U i ∣ i ∈ I } must contain at least one member from τ 1 and at least one member from τ 2 A bitopological space ( X , τ 1 , τ 2 ) is pairwise Hausdorff if for any two distinct points x , y ∈ X there exist disjoint U 1 ∈ τ 1 and U 2 ∈ τ 2 with x ∈ U 1 and y ∈ U 2 .A bitopological space ( X , τ 1 , τ 2 ) is pairwise zero-dimensional if opens in ( X , τ 1 ) which are closed in ( X , τ 2 ) form a basis for ( X , τ 1 ) , and opens in ( X , τ 2 ) which are closed in ( X , τ 1 ) form a basis for ( X , τ 2 ) .A bitopological space ( X , σ , τ ) is called binormal if for every F σ σ -closed and F τ τ -closed sets there are G σ σ -open and G τ τ -open sets such that F σ ⊆ G τ F τ ⊆ G σ , and G σ ∩ G τ = ∅ . 