Ugly duckling theorem: Difference between revisions

Revision as of 18:01, 21 March 2013

In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is $X$ and the topologies are $σ$ and $τ$ then we refer to the bitopological space as $(X, σ, τ)$ .

Bi-continuity

A map $f : X \to X^{'}$ from a bitopological space $(X, τ_{1}, τ_{2})$ to another bitopological space $(X^{'}, {τ_{1}}^{'}, {τ_{2}}^{'})$ is called bi-continuous if $f$ is continuous both as a map from $(X, τ_{1})$ to $(X^{'}, {τ_{1}}^{'})$ and as map from $(X, τ_{2})$ to $(X^{'}, {τ_{2}}^{'})$ .

Bitopological variants of topological properties

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

A bitopological space $(X, τ_{1}, τ_{2})$ is pairwise compact if each cover ${U_{i} ∣ i \in I}$ of $X$ with $U_{i} \in τ_{1} \cup τ_{2}$ , contains a finite subcover.

A bitopological space $(X, τ_{1}, τ_{2})$ is pairwise Hausdorff if for any two distinct points $x, y \in X$ there exist disjoint $U_{1} \in τ_{1}$ and $U_{2} \in τ_{2}$ with either $x \in U_{1}$ and $y \in U_{2}$ or $x \in U_{2}$ and $y \in U_{1}$ .

A bitopological space $(X, τ_{1}, τ_{2})$ is pairwise zero-dimensional if opens in $(X, τ_{1})$ which are closed in $(X, τ_{2})$ form a basis for $(X, τ_{1})$ , and opens in $(X, τ_{2})$ which are closed in $(X, τ_{1})$ form a basis for $(X, τ_{2})$ .

A bitopological space $(X, σ, τ)$ is called binormal if for every $F_{σ}$ $σ$ -closed and $F_{τ}$ $τ$ -closed sets there are $G_{σ}$ $σ$ -open and $G_{τ}$ $τ$ -open sets such that $F_{σ} \subseteq G_{τ}$ $F_{τ} \subseteq G_{σ}$ , and $G_{σ} \cap G_{τ} = \emptyset .$

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.

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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>.
+==Bi-continuity==
+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>.
+==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.
+* 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>.
+* 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>
+==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.
+[[Category:Topology]]

Ugly duckling theorem: Difference between revisions

Revision as of 18:01, 21 March 2013

Bi-continuity

Bitopological variants of topological properties

References

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