Ugly duckling theorem

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 we refer to the bitopological space as $(X,\sigma ,\tau )$.

Bi-continuity

A map $f:X\to X^{\prime }$ from a bitopological space $(X,{\tau }_1,{\tau }_2)$ to another bitopological space $(X^{\prime },{{\tau }_1}^{\prime },{{\tau }_2}^{\prime })$ is called bi-continuous if $f$ is continuous both as a map from $(X,{\tau }_1)$ to $(X^{\prime },{{\tau }_1}^{\prime })$ and as map from $(X,{\tau }_2)$ to $(X^{\prime },{{\tau }_2}^{\prime })$.

Bitopological variants of topological properties

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

• A bitopological space $(X,{\tau }_1,{\tau }_2)$ is pairwise compact if each cover $\{U_i\mid i\in I\}$ of $X$ with $U_i\in {\tau }_1\cup {\tau }_2$, contains a finite subcover.

• A bitopological space $(X,{\tau }_1,{\tau }_2)$ is pairwise Hausdorff if for any two distinct points $x,y\in X$ there exist disjoint $U_1\in {\tau }_1$ and $U_2\in {\tau }_2$ with either $x\in U_1$ and $y\in U_2$ or $x\in U_2$ and $y\in U_1$.

• A bitopological space $(X,{\tau }_1,{\tau }_2)$ is pairwise zero-dimensional if opens in $(X,{\tau }_1)$ which are closed in $(X,{\tau }_2)$ form a basis for $(X,{\tau }_1)$, and opens in $(X,{\tau }_2)$ which are closed in $(X,{\tau }_1)$ form a basis for $(X,{\tau }_2)$.

• A bitopological space $(X,\sigma ,\tau )$ is called binormal if for every $F_{\sigma }$ $\sigma$-closed and $F_{\tau }$ $\tau$-closed sets there are $G_{\sigma }$ $\sigma$-open and $G_{\tau }$ $\tau$-open sets such that $F_{\sigma }\subseteq G_{\tau }$ $F_{\tau }\subseteq G_{\sigma }$, and $G_{\sigma }\cap G_{\tau }=\mathrm{\varnothing }.$

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.