Английская Википедия:Bitopological space
Шаблон:Short description In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is <math>X</math> and the topologies are <math>\sigma</math> and <math>\tau</math> then the bitopological space is referred to as <math>(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>\scriptstyle f:X\to X'</math> from a bitopological space <math>\scriptstyle (X,\tau_1,\tau_2)</math> to another bitopological space <math>\scriptstyle (X',\tau_1',\tau_2')</math> is called continuous or sometimes pairwise continuous if <math>\scriptstyle f</math> is continuous both as a map from <math>\scriptstyle (X,\tau_1)</math> to <math>\scriptstyle (X',\tau_1')</math> and as map from <math>\scriptstyle (X,\tau_2)</math> to <math>\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>\scriptstyle (X,\tau_1,\tau_2)</math> is pairwise compact if each cover <math>\scriptstyle \{U_i\mid i\in I\}</math> of <math>\scriptstyle X</math> with <math>\scriptstyle U_i\in \tau_1\cup\tau_2</math>, contains a finite subcover. In this case, <math>\scriptstyle \{U_i\mid i\in I\}</math> must contain at least one member from <math>\tau_1</math> and at least one member from <math>\tau_2</math>
- A bitopological space <math>\scriptstyle (X,\tau_1,\tau_2)</math> is pairwise Hausdorff if for any two distinct points <math>\scriptstyle x,y\in X</math> there exist disjoint <math>\scriptstyle U_1\in \tau_1</math> and <math>\scriptstyle U_2\in\tau_2</math> with <math>\scriptstyle x\in U_1</math> and <math>\scriptstyle y\in U_2</math>.
- A bitopological space <math>\scriptstyle (X,\tau_1,\tau_2)</math> is pairwise zero-dimensional if opens in <math>\scriptstyle (X,\tau_1)</math> which are closed in <math>\scriptstyle (X,\tau_2)</math> form a basis for <math>\scriptstyle (X,\tau_1)</math>, and opens in <math>\scriptstyle (X,\tau_2)</math> which are closed in <math>\scriptstyle (X,\tau_1)</math> form a basis for <math>\scriptstyle (X,\tau_2)</math>.
- A bitopological space <math>\scriptstyle (X,\sigma,\tau)</math> is called binormal if for every <math>\scriptstyle F_\sigma</math> <math>\scriptstyle \sigma</math>-closed and <math>\scriptstyle F_\tau</math> <math>\scriptstyle \tau</math>-closed sets there are <math>\scriptstyle G_\sigma</math> <math>\scriptstyle \sigma</math>-open and <math>\scriptstyle G_\tau</math> <math>\scriptstyle \tau</math>-open sets such that <math>\scriptstyle F_\sigma\subseteq G_\tau</math> <math>\scriptstyle F_\tau\subseteq G_\sigma</math>, and <math>\scriptstyle G_\sigma\cap G_\tau= \empty.</math>
Notes
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.
