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 \$ of $\scriptstyle X$ with $\scriptstyle U_i\in \tau_1\cup\tau_2$, contains a finite subcover. In this case, $\scriptstyle \$ 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 $\scriptstyle G_\sigma\cap G_\tau= \empty.$

Notes

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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.

Topology
Topological spaces