Bitopological space

Short description: Set endowed with two topologies

In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is [math]\displaystyle{ X }[/math] and the topologies are [math]\displaystyle{ \sigma }[/math] and [math]\displaystyle{ \tau }[/math] then the bitopological space is referred to as [math]\displaystyle{ (X,\sigma,\tau) }[/math]. 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 [math]\displaystyle{ \scriptstyle f:X\to X' }[/math] from a bitopological space [math]\displaystyle{ \scriptstyle (X,\tau_1,\tau_2) }[/math] to another bitopological space [math]\displaystyle{ \scriptstyle (X',\tau_1',\tau_2') }[/math] is called continuous or sometimes pairwise continuous if [math]\displaystyle{ \scriptstyle f }[/math] is continuous both as a map from [math]\displaystyle{ \scriptstyle (X,\tau_1) }[/math] to [math]\displaystyle{ \scriptstyle (X',\tau_1') }[/math] and as map from [math]\displaystyle{ \scriptstyle (X,\tau_2) }[/math] to [math]\displaystyle{ \scriptstyle (X',\tau_2') }[/math].

Bitopological variants of topological properties

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

• A bitopological space [math]\displaystyle{ \scriptstyle (X,\tau_1,\tau_2) }[/math] is pairwise compact if each cover [math]\displaystyle{ \scriptstyle \{U_i\mid i\in I\} }[/math] of [math]\displaystyle{ \scriptstyle X }[/math] with [math]\displaystyle{ \scriptstyle U_i\in \tau_1\cup\tau_2 }[/math], contains a finite subcover. In this case, [math]\displaystyle{ \scriptstyle \{U_i\mid i\in I\} }[/math] must contain at least one member from [math]\displaystyle{ \tau_1 }[/math] and at least one member from [math]\displaystyle{ \tau_2 }[/math]
• A bitopological space [math]\displaystyle{ \scriptstyle (X,\tau_1,\tau_2) }[/math] is pairwise Hausdorff if for any two distinct points [math]\displaystyle{ \scriptstyle x,y\in X }[/math] there exist disjoint [math]\displaystyle{ \scriptstyle U_1\in \tau_1 }[/math] and [math]\displaystyle{ \scriptstyle U_2\in\tau_2 }[/math] with [math]\displaystyle{ \scriptstyle x\in U_1 }[/math] and [math]\displaystyle{ \scriptstyle y\in U_2 }[/math].
• A bitopological space [math]\displaystyle{ \scriptstyle (X,\tau_1,\tau_2) }[/math] is pairwise zero-dimensional if opens in [math]\displaystyle{ \scriptstyle (X,\tau_1) }[/math] which are closed in [math]\displaystyle{ \scriptstyle (X,\tau_2) }[/math] form a basis for [math]\displaystyle{ \scriptstyle (X,\tau_1) }[/math], and opens in [math]\displaystyle{ \scriptstyle (X,\tau_2) }[/math] which are closed in [math]\displaystyle{ \scriptstyle (X,\tau_1) }[/math] form a basis for [math]\displaystyle{ \scriptstyle (X,\tau_2) }[/math].
• A bitopological space [math]\displaystyle{ \scriptstyle (X,\sigma,\tau) }[/math] is called binormal if for every [math]\displaystyle{ \scriptstyle F_\sigma }[/math] [math]\displaystyle{ \scriptstyle \sigma }[/math]-closed and [math]\displaystyle{ \scriptstyle F_\tau }[/math] [math]\displaystyle{ \scriptstyle \tau }[/math]-closed sets there are [math]\displaystyle{ \scriptstyle G_\sigma }[/math] [math]\displaystyle{ \scriptstyle \sigma }[/math]-open and [math]\displaystyle{ \scriptstyle G_\tau }[/math] [math]\displaystyle{ \scriptstyle \tau }[/math]-open sets such that [math]\displaystyle{ \scriptstyle F_\sigma\subseteq G_\tau }[/math] [math]\displaystyle{ \scriptstyle F_\tau\subseteq G_\sigma }[/math], and [math]\displaystyle{ \scriptstyle G_\sigma\cap G_\tau= \empty. }[/math]

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.