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| In [[mathematics]], a '''bitopological space''' is a [[set (mathematics)|set]] endowed with ''two'' [[Topological space|topologies]]. Typically, if the set is <math>X</math> and the topologies are <math>\sigma</math> and <math>\tau</math> then we refer to the bitopological space as <math>(X,\sigma,\tau)</math>.
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| ==Bi-continuity==
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| A [[map (mathematics)|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 '''bi-continuous''' if <math>\scriptstyle f</math> is [[Continuous function (topology)|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>.
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| ==Bitopological variants of topological properties==
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| Corresponding to well-known properties of topological spaces, there are versions for bitopological spaces.
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| * 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.
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| * 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 either <math>\scriptstyle x\in U_1</math> and <math>\scriptstyle y\in U_2</math> or <math>\scriptstyle x\in U_2</math> and <math>\scriptstyle y\in U_1</math>.
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| * 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>.
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| * 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>
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| ==References==
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| * Kelly, J. C. (1963). Bitopological spaces. ''Proc. London Math. Soc.'', 13(3) 71—89.
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| * Reilly, I. L. (1972). On bitopological separation properties. ''Nanta Math.'', (2) 14—25.
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| * Reilly, I. L. (1973). Zero dimensional bitopological spaces. ''Indag. Math.'', (35) 127—131.
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| * Salbany, S. (1974). ''Bitopological spaces, compactifications and completions''. Department of Mathematics, University of Cape Town, Cape Town.
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| * Kopperman, R. (1995). Asymmetry and duality in topology. ''Topology Appl.'', 66(1) 1--39.
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| [[Category:Topology]]
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The writer is known as Araceli Gulledge. Years in the past we moved to Kansas. Managing individuals is how I make money and it's something I really enjoy. One of his favorite hobbies is taking part in crochet but he hasn't made a dime with it.
Feel free to visit my webpage - Vinodjoseph.com