Bitopological space

A map $\backslash scriptstyle\; f:X\backslash to\; X\text{'}$ from a bitopological space $\backslash scriptstyle\; (X,\backslash tau\_1,\backslash tau\_2)$ to another bitopological space $\backslash scriptstyle\; (X\text{'},\backslash tau\_1\text{'},\backslash tau\_2\text{'})$ is called continuous or sometimes pairwise continuous if $\backslash scriptstyle\; f$ is continuous both as a map from $\backslash scriptstyle\; (X,\backslash tau\_1)$ to $\backslash scriptstyle\; (X\text{'},\backslash tau\_1\text{'})$ and as map from $\backslash scriptstyle\; (X,\backslash tau\_2)$ to $\backslash scriptstyle\; (X\text{'},\backslash tau\_2\text{'})$.

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

• A bitopological space $\backslash scriptstyle\; (X,\backslash tau\_1,\backslash tau\_2)$ is pairwise compact if each cover $\backslash scriptstyle\; \backslash \{U\_i\backslash mid\; i\backslash in\; I\backslash \}$ of $\backslash scriptstyle\; X$ with $\backslash scriptstyle\; U\_i\backslash in\; \backslash tau\_1\backslash cup\backslash tau\_2$, contains a finite subcover. In this case, $\backslash scriptstyle\; \backslash \{U\_i\backslash mid\; i\backslash in\; I\backslash \}$ must contain at least one member from $\backslash tau\_1$ and at least one member from $\backslash tau\_2$
• A bitopological space $\backslash scriptstyle\; (X,\backslash tau\_1,\backslash tau\_2)$ is pairwise Hausdorff if for any two distinct points $\backslash scriptstyle\; x,y\backslash in\; X$ there exist disjoint $\backslash scriptstyle\; U\_1\backslash in\; \backslash tau\_1$ and $\backslash scriptstyle\; U\_2\backslash in\backslash tau\_2$ with $\backslash scriptstyle\; x\backslash in\; U\_1$ and $\backslash scriptstyle\; y\backslash in\; U\_2$.
• A bitopological space $\backslash scriptstyle\; (X,\backslash tau\_1,\backslash tau\_2)$ is pairwise zero-dimensional if opens in $\backslash scriptstyle\; (X,\backslash tau\_1)$ that are closed in $\backslash scriptstyle\; (X,\backslash tau\_2)$ form a basis for $\backslash scriptstyle\; (X,\backslash tau\_1)$, and opens in $\backslash scriptstyle\; (X,\backslash tau\_2)$ that are closed in $\backslash scriptstyle\; (X,\backslash tau\_1)$ form a basis for $\backslash scriptstyle\; (X,\backslash tau\_2)$.
• A bitopological space $\backslash scriptstyle\; (X,\backslash sigma,\backslash tau)$ is called binormal if for every $\backslash scriptstyle\; F\_\backslash sigma$ $\backslash scriptstyle\; \backslash sigma$-closed and $\backslash scriptstyle\; F\_\backslash tau$ $\backslash scriptstyle\; \backslash tau$-closed sets there are $\backslash scriptstyle\; G\_\backslash sigma$ $\backslash scriptstyle\; \backslash sigma$-open and $\backslash scriptstyle\; G\_\backslash tau$ $\backslash scriptstyle\; \backslash tau$-open sets such that $\backslash scriptstyle\; F\_\backslash sigma\backslash subseteq\; G\_\backslash tau$ $\backslash scriptstyle\; F\_\backslash tau\backslash subseteq\; G\_\backslash sigma$, and $\backslash scriptstyle\; G\_\backslash sigma\backslash cap\; G\_\backslash tau=\; \backslash empty.$

{{Reflist}}

• 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.

Category:Topology

Category:Topological spaces