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\section*{bitopological space}

\hypertarget{bitopological_space}{}\section*{{Bitopological space}}\label{bitopological_space}

\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{separation_axioms_for_topologies}{Separation axioms for topologies}\dotfill \pageref*{separation_axioms_for_topologies} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{related_articles}{Related Articles}\dotfill \pageref*{related_articles} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Recall that a [[topological space]] is a [[set]] $X$ equipped with a [[topological structure]] $\mathcal{T}$. Well, a \textbf{bitopological space} is simply a set equipped with \emph{two} topological structures $(X, \mathcal{T}, \mathcal{T}^*)$. Unlike with [[bialgebras]], no compatibility condition is required between these structures.

A \textbf{bicontinous map} is a [[function]] between bitopological spaces that is [[continuous map|continuous]] with respect to each topological structure.

Bitopological spaces and bicontinuous maps form a [[category]] $BiTop$.

\hypertarget{separation_axioms_for_topologies}{}\subsection*{{Separation axioms for topologies}}\label{separation_axioms_for_topologies}

Let $Cl$ denote the [[closure operator]] with respect to $\mathcal{T}$ and let $Cl^*$ denote the closure operator with respect to $\mathcal{T}^*$.

$\backslash$begin\{proposition\}$\backslash$label\{implications\} Let $(X, \mathcal{T}, \mathcal{T}^*)$ be a bitopological space. Consider the following properties of this space:

\begin{enumerate}%
\item for each point $x$ there is a $\mathcal{T}$-[[neighborhood base]] consisting of $\mathcal{T}^*$-closed sets;


\item for all $x\in X$ and all $\mathcal{T}$-opens $U$ containing $x$ there is a $\mathcal{T}^*$-closed $\mathcal{T}$-neighborhood $V$ of $x$ such that $V \subset U$;


\item $Cl^*(O) \subset Cl(O)$ for each $\mathcal{T}^*$-open $O$;


\item for all $x\in X$ and all $\mathcal{T}$-neighborhoods $U$ of $x$ the closure $Cl^*(U)$ is a $\mathcal{T}^*$-neighborhood;


\item for each point $x$ and each $\mathcal{T}$-closed $\mathcal{T}$-neighborhood $V$ of $x$ in $X$ there exists a $\mathcal{T}^*$-closed $\mathcal{T}$-neighborhood $U$ of $x$ in $X$ such that $U$ is contained in $V$.



\end{enumerate}
There are the following implications among these properties

$\backslash$begin\{centre\} $\backslash$begin\{tikzcd\} \& (2) \& (3) $\backslash$ (5) \& (1) \& (4) $\backslash$arrowRightarrow, from=2-1, to=1-2, ``$\backslash$text\{if $\mathcal{T}$ regular\}'' $\backslash$arrowLeftrightarrow, from=1-2, to=2-2 $\backslash$arrowLeftrightarrow, from=1-3, to=2-3 $\backslash$arrowRightarrow, from=2-2, to=2-1 $\backslash$arrowRightarrow, from=1-3, to=1-2, ``$\backslash$text\{if $\mathcal{T}$ regular\}'' $\backslash$arrowRightarrow, from=2-2, to=2-3 $\backslash$end\{tikzcd\} $\backslash$end\{centre\}

Especially, all properties are equivalent if $\mathcal{T}$ is [[regular space|regular]]. $\backslash$end\{proposition\}

$\backslash$begin\{proof\} \textbf{\hyperlink{regular1}{(1)} $\iff$ \hyperlink{regular2}{(2)}:} Given a neighborhood base for a point $x$ as guaranteed by the first property. When you spell out the properties of this neighborhood base, you end up with the second property. For the reverse direction start with an arbitrary $\mathcal{T}$-neighborhood base of a point $x$ consisting of open. Apply the second property to every element of this neighborhood base to the desired neighborhood base.

\textbf{\hyperlink{coupled1}{(3)} $\iff$ \hyperlink{coupled2}{(4)}:} Suppose property \hyperlink{coupled1}{(3)}, and let $U$ be a $\mathcal{T}$-neighborhood of an arbitrary point $x$. Then the [[complement]] $\widetilde{Cl^*(U)}$ is in $\mathcal{T}^*$, so that $Cl^*(\widetilde{Cl^*U}) \subset Cl(\widetilde{Cl^* U})$ by the first property. Hence $\widetilde{Cl(\widetilde{Cl^*U})} \subset \widetilde{Cl^*(\widetilde{Cl^* U})}$ for the complements. Since $U$ is a $\mathcal{T}$-neighborhood of $x$, $x$ does not belong to $Cl(\widetilde{Cl^*U})$. Moreover, $\widetilde{Cl^*(\widetilde{Cl^*U})}$ is $\mathcal{T}^*$-open and a subset of $Cl^*(U)$. Hence $Cl^*(U)$ is a $\mathcal{T}^*$-neighborhood of $x$.

For the converse suppose property \hyperlink{coupled2}{(4)}. Let $O$ be a nonempty $\mathcal{T}^*$-open set and $x$ an element of $Cl^*(O)$. Then if $U$ is any $\mathcal{T}$-neighborhood of $x$, some point $y \in O$ belongs to $Cl^*(U)$ due to the second property. Hence, as $O$ is a $\mathcal{T}^*$-neighborhood of $y$, some point of $U$ belongs to $O$. Thus $x \in Cl(O)$, and therefore $Cl^*(G) \subset Cl(O)$.

\textbf{\hyperlink{regular1}{(1)} $\implies$ \hyperlink{coupled2}{(4)}:} Given $x\in X$ and a $\mathcal{T}$-neighborhood $U$ by property \hyperlink{regular1}{(1)} there is a $\mathcal{T}^*$-open $O \subset U$ containing $x$. Hence $O \subset Cl^*(U)$, and $Cl^*(U)$ is a $\mathcal{T}^*$-neighborhood.

\textbf{\hyperlink{coupled1}{(3)} and $\mathcal{T}$ regular $\implies$ \hyperlink{regular2}{(2)}:} Let $x\in X$ and $U$ be a $\mathcal{T}$-open containing $x$. By regularity of $\mathcal{T}$ we can find disjoint $\mathcal{T}$-opens $V' \ni x$ and $U' \supset \tilde{U}$ ($\tilde{U}$ denotes the [[complement]]). Set $V \coloneqq Cl^*(V')$. This set is obviously a $\mathcal{T}^*$-closed $\mathcal{T}$-neighborhood of $x$. Due to property \hyperlink{coupled1}{(3)} $V \subset Cl(V')$. Since also $Cl(V') \subset \widetilde{U'}$, we have $V \subset U$. This is to say that $V$ is the $\mathcal{T}$-neighborhood we sought.

\textbf{\hyperlink{cotopology2}{(5)} and $\mathcal{T}$ regular $\implies$ \hyperlink{regular2}{(2)}:} Let $x\in X$ and $U$ be a $\mathcal{T}$-open containing $x$. By regularity of $\mathcal{T}$ we can find disjoint $\mathcal{T}$-opens $V' \ni x$ and $U' \supset \tilde{U}$ ($\tilde{U}$ denotes the [[complement]]). Due to property \hyperlink{cotopology2}{(5)} the closed set $\widetilde{U'}$ contains a $\mathcal{T}^*$-closed neighborhood of $x$. This is the neighborhood we sought.

\textbf{\hyperlink{regular1}{(1)} $\implies$ \hyperlink{cotopology2}{(5)}:} Given some $\mathcal{T}$-closed $\mathcal{T}$-neighborhood $V$ of some point $x$ choose a neighborhood base according to property \hyperlink{regular1}{(1)} and take an element $U$ therein that is contained in $V$. $\backslash$end\{proof\}

$\backslash$begin\{definition\}$\backslash$label\{regular\} Let $(X, \mathcal{T}, \mathcal{T}^*)$ be a bitopological space. The topology $\mathcal{T}$ is \textbf{regular with respect to} $\mathcal{T}^*$ if one of the two equivalent conditions \hyperlink{regular1}{(1)} and \hyperlink{regular2}{(2)} from proposition \ref{implications} holds. A bitopological space $(X, \mathcal{T}, \mathcal{T}^*)$ is called \textbf{pairwise regular} if $\mathcal{T}$ is regular with respect to $\mathcal{T}^*$ and vise versa. $\backslash$end\{definition\}

$\backslash$begin\{definition\}$\backslash$label\{coupled\} Let $(X, \mathcal{T}, \mathcal{T}^*)$ be a bitopological space. The topology $\mathcal{T}^*$ is \textbf{coupled to} $\mathcal{T}$ if one of the two equivalent conditions \hyperlink{coupled1}{(3)} and \hyperlink{coupled2}{(4)} from proposition \ref{implications} holds. $\backslash$end\{definition\}

Not that is if $\mathcal{T}^*$ is coupled to a finer topology $\mathcal{T} \supset \mathcal{T}^*$ then $\mathcal{T}^*$ is coupled to every topology coarser than $\mathcal{T}$ due to property \hyperlink{coupled1}{(3)}. Moreover in this case also $\mathcal{T}$ is coupled to $\mathcal{T}^*$ (again a direct consequence of property \hyperlink{coupled1}{(3)}).

$\backslash$begin\{definition\}$\backslash$label\{cotopology\} Let $(X, \mathcal{T}, \mathcal{T}^*)$ be a bitopological space. The topology $\mathcal{T}^*$ is called a \textbf{[[cotopology]] of} $\mathcal{T}$ if $\mathcal{T}^* \subseteq \mathcal{T}$ and property \hyperlink{cotopology2}{(5)} from proposition \ref{implications} holds. The space $(X, \mathcal{T}^*)$ is also called a \textbf{cospace} of $(X, \mathcal{T}$. $\backslash$end\{definition\}

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

It is interesting and perhaps surprising that many advanced topological notions can be described using bitopological spaces, even when you would not na\"i{}vely think that there are two topologies around. (At least, that's my vague memory of what they were good for. I think that this was in some article by Isbell.)

\hypertarget{related_articles}{}\subsection*{{Related Articles}}\label{related_articles}

\begin{itemize}%
\item [[cotopology]]
\item [[topologically complete space]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item \href{http://en.wikipedia.org/wiki/Bitopological_space}{Wikipedia entry}


\item [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], \emph{Abstract and Concrete Categories: The Joy of Cats}, Dover New York 2009. (\href{http://katmat.math.uni-bremen.de/acc/acc.pdf}{pdf}) pp. 59-60, 278


\item B. Dvalishvili, \emph{Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures and Applications}, Elsevier Amsterdam 2005.


\item [[Peter Johnstone]], \emph{Collapsed Toposes as Bitopological Spaces}, pp. 19-35 in \emph{Categorical Topology}, World Scientific Singapore 1989.


\item O. K. Klinke, A. Jung, A. Moshier, \emph{A bitopological point-free approach to compactications} (2011). (\href{http://www.cs.bham.ac.uk/~axj/pub/papers/compactifications.pdf}{preprint})


\item R. Kopperman, \emph{Asymmetry and duality in topology}, Topology Appl. \textbf{66} no. 1 (1995) pp. 1-39.



\end{itemize}
The idea naturally appeared first in the context of quasi-metric spaces

\begin{itemize}%
\item Wallace Alvin Wilson, \emph{On Quasi-Metric Spaces}, American Journal of Mathematics (1931), vol. 53, no. 3, pp. 675-684.

\end{itemize}
The notions of separation axioms were introduced in

\begin{itemize}%
\item J. D. Weston, \emph{On the comparison of topologies} 1956, Journal of the London Mathematical Society, vol. s1-32 no. 3, pp. 342-354,


\item J. C. Kelly, \emph{Bitopological spaces}, Proc. London Math. Soc. \textbf{13} no.3 (1963) pp. 71-89.



\end{itemize}
Only Kelly introduced the concept in its nowadays formulation of a set equipped with two topologies. The Russian school contributed the following comprehensive overviews of this and related topics

\begin{itemize}%
\item A. A. Ivanov, \emph{Problems of the Theory of Bitopological Spaces}, 1990, Journal of Soviet Mathematics, vol. 52, Issue 1, pp. 2759-2790. Originally published as \emph{Проблематика теории битопологических пространств} in Zap. Nauchn. Sem. POMI, 1988, vol. 167 (\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=znsl&paperid=5560&option_lang=en}{Russian version}).


\item A. A. Ivanov, \emph{Problems of the Theory of Bitopological Spaces 2}, 1996, Journal of Math. Sciences, vol. 81, Issue 2. Originally publishes as \emph{Проблематика теории битопологических пространств. 2} in Zap. Nauchn. Sem. POMI, 1993, Volume 208, pp. 5–67 (\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=znsl&paperid=5828&option_lang=eng}{Russian version}).


\item A. A. Ivanov, \emph{Problems of the Theory of Bitopological Spaces 3}, 1998, Journal of Math. Sciences, vol. 91, Issue 6, pp 3339–3364. Originally published as \emph{Проблематика теории битопологических пространств. 3} in Zap. Nauchn. Sem. POMI, 1995, Volume 231, pp. 9–54 (\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=znsl&paperid=3739&option_lang=eng}{Russian version}).



\end{itemize}
as well as a more introductory text book

\begin{itemize}%
\item \emph{Битопологические пространства}, 1997. Исследования по топологии. 9, Zap. Nauchn. Sem. POMI, 242, editor A. A. Ivanov (\href{http://www.mathnet.ru/php/archive.phtml?jrnid=znsl&wshow=issue&bshow=contents&series=0&year=1997&volume=242&issue=&option_lang=rus&bookID=335}{Russian version}).

\end{itemize}
[[!redirects bitopological space]] [[!redirects bitopological spaces]] [[!redirects Bitop]] [[!redirects BiTop]] [[!redirects Bi Top]]



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