Bitopological space

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

Bitopological variants of topological properties

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

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