ProximitySpace 2007-03-06 02:43:37 2008-06-06 11:09:07 Definition nearness relation separated proximity space discrete proximity indiscrete proximity \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics \usepackage{xypic} \usepackage{pst-plot} \usepackage{psfrag} % define commands here \newtheorem{prop}{Proposition} \newtheorem{thm}{Theorem} \newtheorem{ex}{Example} \newcommand{\real}{\mathbb{R}} Let $X$ be a set. A binary relation $\delta$ on $P(X)$, the power set of $X$, is called a \emph{nearness relation} on $X$ if it satisfies the following conditions: for $A,B\in P(X)$, \begin{enumerate} \item if $A\cap B\ne \varnothing$, then $A\delta B$; \item if $A\delta B$, then $A\ne \varnothing$ and $B\ne \varnothing$; \item (symmetry) if $A\delta B$, then $B\delta A$; \item $(A_1\cup A_2)\delta B$ iff $A_1\delta B$ or $A_2\delta B$; \item $A\delta'B$ implies the existence of $C \subseteq X$ with $A\delta'C$ and $(X-C)\delta'B$, where $A\delta'B$ means $(A,B)\notin \delta$. \end{enumerate} If $x,y\in X$ and $A\subseteq X$, we write $x\delta A$ to mean $\lbrace x\rbrace \delta A$, and $x\delta y$ to mean $\lbrace x\rbrace \delta \lbrace y \rbrace$. When $A\delta B$, we say that $A$ is \emph{$\delta$-near}, or just \emph{near} $B$. $\delta$ is also called a \emph{proximity relation}, or \emph{proximity} for short. Condition 1 is equivalent to saying if $A\delta'B$, then $A\cap B=\varnothing$. Condition 4 says that if $A$ is near $B$, then any superset of $A$ is near $B$. Conversely, if $A$ is not near $B$, then no subset of $A$ is near $B$. In particular, if $x\in A$ and $A\delta' B$, then $x\delta'B$. \textbf{Definition}. A set $X$ with a proximity as defined above is called a \emph{proximity space}. For any subset $A$ of $X$, define $A^c=\lbrace x\in X\mid x\delta A\rbrace$. Then $^c$ is a closure operator on $X$: \begin{proof} Clearly $\varnothing^c=\varnothing$. Also $A\subseteq A^c$ for any $A\subseteq X$. To see $A^{cc}=A^c$, assume $x\delta A^c$, we want to show that $x\delta A$. If not, then there is $C\subseteq X$ such that $x\delta'C$ and $(X-C)\delta'A$. The second part says that if $y\in X-C$, then $y\delta'A$, which is equivalent to $A^c \subseteq C$. But $x\delta'C$, so $x\delta'A^c$. Finally, $x\in (A\cup B)^c$ iff $x\delta (A\cup B)$ iff $x\delta A$ or $x\delta B$ iff $x\in A^c$ or $x\in B^c$.\end{proof} This turns $X$ into a topological space. Thus any proximity space is a topological space induced by the closure operator defined above. A proximity space is said to be \emph{separated} if for any $x,y\in X$, $x\delta y$ implies $x=y$. \textbf{Examples}. \begin{itemize} \item Let $(X,d)$ be a pseudometric space. For any $x\in X$ and $A\subseteq X$, define $d(x,A):=\inf_{y\in A} d(x,y)$. Next, for $B\subseteq X$, define $d(A,B):=\inf_{x\in A} d(x,B)$. Finally, define $A\delta B$ iff $d(A,B)=0$. Then $\delta$ is a proximity and $(X,d)$ is a proximity space as a result. \item \emph{discrete proximity}. Let $X$ be a non-empty set. For $A,B\subseteq X$, define $A\delta B$ iff $A\cap B\ne\varnothing$. Then $\delta$ so defined is a proximity on $X$, and is called the \emph{discrete proximity} on $X$. \item \emph{indiscrete proximity}. Again, $X$ is a non-empty set and $A,B\subseteq X$. Define $A\delta B$ iff $A\ne \varnothing$ and $B\ne \varnothing$. Then $\delta$ is also a proximity. It is called the \emph{indiscrete proximity} on $X$. \end{itemize} \begin{thebibliography}{9} \bibitem{willard} S. Willard, \emph{General Topology}, Addison-Wesley, Publishing Company, 1970. \bibitem{nw} S.A. Naimpally, B.D. Warrack, \emph{Proximity Spaces}, Cambridge University Press, 1970. \end{thebibliography}