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 $\scriptstyle G_{\sigma }\cap G_{\tau }=\emptyset .$

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.