Bitopological space explained

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

\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.

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.