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 $(X,\sigma ,\tau )$ .

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

Notes

References

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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.

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