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

Bitopological variants of topological properties

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

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

See also

• Polytopological space

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.