# Bitopological space

In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is $X$ and the topologies are $\sigma$ and $\tau$ then the bitopological space is referred to as $(X,\sigma ,\tau )$ . The notion was introduced by Kelly in the study of quasimetrics, i.e. distance functions that are not required to be symmetric.

## Continuity

A map $f:X\to X'$ from a bitopological space $(X,\tau _{1},\tau _{2})$ to another bitopological space $(X',\tau _{1}',\tau _{2}')$ is called continuous or sometimes pairwise continuous if $f$ is continuous both as a map from $(X,\tau _{1})$ to $(X',\tau _{1}')$ and as map from $(X,\tau _{2})$ to $(X',\tau _{2}')$ .

## Bitopological variants of topological properties

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

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