Bitopological space

In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is X {\displaystyle X} and the topologies are σ {\displaystyle \sigma } and τ {\displaystyle \tau } then the bitopological space is referred to as ( X , σ , τ ) {\displaystyle (X,\sigma ,\tau )} . 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 f : X → X ′ {\displaystyle f:X\to X'} from a bitopological space ( X , τ 1 , τ 2 ) {\displaystyle (X,\tau _{1},\tau _{2})} to another bitopological space ( X ′ , τ 1 ′ , τ 2 ′ ) {\displaystyle (X',\tau _{1}',\tau _{2}')} is called continuous or sometimes pairwise continuous if f {\displaystyle f} is continuous both as a map from ( X , τ 1 ) {\displaystyle (X,\tau _{1})} to ( X ′ , τ 1 ′ ) {\displaystyle (X',\tau _{1}')} and as map from ( X , τ 2 ) {\displaystyle (X,\tau _{2})} to ( X ′ , τ 2 ′ ) {\displaystyle (X',\tau _{2}')} .

Bitopological variants of topological properties

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

• A bitopological space ( X , τ 1 , τ 2 ) {\displaystyle (X,\tau _{1},\tau _{2})} is pairwise compact if each cover { U i ∣ i ∈ I } {\displaystyle \{U_{i}\mid i\in I\}} of X {\displaystyle X} with U i ∈ τ 1 ∪ τ 2 {\displaystyle U_{i}\in \tau _{1}\cup \tau _{2}} , contains a finite subcover. In this case, { U i ∣ i ∈ I } {\displaystyle \{U_{i}\mid i\in I\}} must contain at least one member from τ 1 {\displaystyle \tau _{1}} and at least one member from τ 2 {\displaystyle \tau _{2}}
• A bitopological space ( X , τ 1 , τ 2 ) {\displaystyle (X,\tau _{1},\tau _{2})} is pairwise Hausdorff if for any two distinct points x , y ∈ X {\displaystyle x,y\in X} there exist disjoint U 1 ∈ τ 1 {\displaystyle U_{1}\in \tau _{1}} and U 2 ∈ τ 2 {\displaystyle U_{2}\in \tau _{2}} with x ∈ U 1 {\displaystyle x\in U_{1}} and y ∈ U 2 {\displaystyle y\in U_{2}} .
• A bitopological space ( X , τ 1 , τ 2 ) {\displaystyle (X,\tau _{1},\tau _{2})} is pairwise zero-dimensional if opens in ( X , τ 1 ) {\displaystyle (X,\tau _{1})} that are closed in ( X , τ 2 ) {\displaystyle (X,\tau _{2})} form a basis for ( X , τ 1 ) {\displaystyle (X,\tau _{1})} , and opens in ( X , τ 2 ) {\displaystyle (X,\tau _{2})} that are closed in ( X , τ 1 ) {\displaystyle (X,\tau _{1})} form a basis for ( X , τ 2 ) {\displaystyle (X,\tau _{2})} .
• A bitopological space ( X , σ , τ ) {\displaystyle (X,\sigma ,\tau )} is called binormal if for every F σ {\displaystyle F_{\sigma }} σ {\displaystyle \sigma } -closed and F τ {\displaystyle F_{\tau }} τ {\displaystyle \tau } -closed sets there are G σ {\displaystyle G_{\sigma }} σ {\displaystyle \sigma } -open and G τ {\displaystyle G_{\tau }} τ {\displaystyle \tau } -open sets such that F σ ⊆ G τ {\displaystyle F_{\sigma }\subseteq G_{\tau }} F τ ⊆ G σ {\displaystyle F_{\tau }\subseteq G_{\sigma }} , and G σ ∩ G τ = ∅ . {\displaystyle G_{\sigma }\cap G_{\tau }=\emptyset .}

See also

• Polytopological space

References

• Kelly, J. C. (1963). Bitopological spaces. Proc. London Math. Soc., 13(3) 71–89.
• Reilly, I. L. (1972). On bitopological separation properties. Nanta Math., (2) 14–25.
• Reilly, I. L. (1973). Zero dimensional bitopological spaces. Indag. Math., (35) 127–131.
• Salbany, S. (1974). Bitopological spaces, compactifications and completions. Department of Mathematics, University of Cape Town, Cape Town.
• Kopperman, R. (1995). Asymmetry and duality in topology. Topology Appl., 66(1) 1--39.
• Fletcher. P, Hoyle H.B. III, and Patty C.W. (1969). The comparison of topologies. Duke Math. J.,36(2) 325–331.
• Dochviri, I., Noiri T. (2015). On some properties of stable bitopological spaces. Topol. Proc., 45 111–119.