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 J. C. Kelly in the study of quasimetrics, i.e. distance functions that are not required to be symmetric.

Continuity
A map $$\scriptstyle f:X\to X'$$ from a bitopological space $$\scriptstyle (X,\tau_1,\tau_2)$$ to another bitopological space $$\scriptstyle (X',\tau_1',\tau_2')$$ is called continuous or sometimes pairwise continuous if $$\scriptstyle f$$ is continuous both as a map from $$\scriptstyle (X,\tau_1)$$ to $$\scriptstyle (X',\tau_1')$$ and as map from $$\scriptstyle (X,\tau_2)$$ to $$\scriptstyle (X',\tau_2')$$.

Bitopological variants of topological properties
Corresponding to well-known properties of topological spaces, there are versions for bitopological spaces.
 * A bitopological space $$\scriptstyle (X,\tau_1,\tau_2)$$ is pairwise compact if each cover $$\scriptstyle \{U_i\mid i\in I\}$$ of $$\scriptstyle X$$ with $$\scriptstyle U_i\in \tau_1\cup\tau_2$$, contains a finite subcover. In this case, $$\scriptstyle \{U_i\mid i\in I\}$$ must contain at least one member from $$\tau_1$$ and at least one member from $$\tau_2$$
 * A bitopological space $$\scriptstyle (X,\tau_1,\tau_2)$$ is pairwise Hausdorff if for any two distinct points $$\scriptstyle x,y\in X$$ there exist disjoint $$\scriptstyle U_1\in \tau_1$$ and $$\scriptstyle U_2\in\tau_2$$ with $$\scriptstyle x\in U_1$$ and $$\scriptstyle y\in U_2$$.
 * A bitopological space $$\scriptstyle (X,\tau_1,\tau_2)$$ is pairwise zero-dimensional if opens in $$\scriptstyle (X,\tau_1)$$ which are closed in $$\scriptstyle (X,\tau_2)$$ form a basis for $$\scriptstyle (X,\tau_1)$$, and opens in $$\scriptstyle (X,\tau_2)$$ which are closed in $$\scriptstyle (X,\tau_1)$$ form a basis for $$\scriptstyle (X,\tau_2)$$.
 * A bitopological space $$\scriptstyle (X,\sigma,\tau)$$ is called binormal if for every $$\scriptstyle F_\sigma$$ $$\scriptstyle \sigma$$-closed and $$\scriptstyle F_\tau$$ $$\scriptstyle \tau$$-closed sets there are $$\scriptstyle G_\sigma$$ $$\scriptstyle \sigma$$-open and $$\scriptstyle G_\tau$$ $$\scriptstyle \tau$$-open sets such that $$\scriptstyle F_\sigma\subseteq G_\tau$$ $$\scriptstyle F_\tau\subseteq G_\sigma$$, and $$\scriptstyle  G_\sigma\cap G_\tau= \empty.$$