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 $\scriptstyle f:X\to X'$ from a bitopological space $\scriptstyle (X,\tau _{1},\tau _{2})$ to another bitopological space $\scriptstyle (X',\tau _{1}',\tau _{2}')$ is called bi-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.

• 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 either $\scriptstyle x\in U_{1}$ and $\scriptstyle y\in U_{2}$ or $\scriptstyle x\in U_{2}$ and $\scriptstyle y\in U_{1}$.

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

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.