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 au } then the bitopological space is referred to as ( X , σ , τ ) {displaystyle (X,sigma , au )} . The notion was introduced by Kelly in the study of quasimetrics, i.e. distance functions that are not required to be symmetric. 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 au } then the bitopological space is referred to as ( X , σ , τ ) {displaystyle (X,sigma , au )} . The notion was introduced by Kelly in the study of quasimetrics, i.e. distance functions that are not required to be symmetric. A map f : X → X ′ {displaystyle scriptstyle f:X o X'} from a bitopological space ( X , τ 1 , τ 2 ) {displaystyle scriptstyle (X, au _{1}, au _{2})} to another bitopological space ( X ′ , τ 1 ′ , τ 2 ′ ) {displaystyle scriptstyle (X', au _{1}', au _{2}')} is called continuous or sometimes pairwise continuous if f {displaystyle scriptstyle f} is continuous both as a map from ( X , τ 1 ) {displaystyle scriptstyle (X, au _{1})} to ( X ′ , τ 1 ′ ) {displaystyle scriptstyle (X', au _{1}')} and as map from ( X , τ 2 ) {displaystyle scriptstyle (X, au _{2})} to ( X ′ , τ 2 ′ ) {displaystyle scriptstyle (X', au _{2}')} .