|
|
|
Viewing Version
6
of
'proximity space'
|
[ view 'proximity space'
|
back to history
]
| Title of object: |
proximity space |
| Canonical Name: |
ProximitySpace |
| Type: |
Definition |
| Created on: |
2007-03-06 02:43:37 |
| Modified on: |
2007-03-10 22:50:16 |
| Classification: |
msc:54E05 |
| Defines: |
nearness relation, separated proxmity space |
| Synonyms: |
proximity space=near
proximity space=proximity |
Preamble:
\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}} |
Content:
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 $C\in P(X)$ with $A\delta'C$ and $(X-C)\delta'B$, where $A\delta'B$ is meant $(A,B)\notin \delta$.
\end{enumerate}
When $A\delta B$, we say that $A$ is \emph{near} $B$. $\delta$ is also called a proximity. Condition 1 is equivalent to saying if $A\delta'B$, then $A\cap B=\varnothing$. Condition 3 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 $B$.
\textbf{Definition}. A set $X$ with a proximity as defined above is called a \emph{proximity space}.
Let $X$ be a proxmity space. For any $x,y\in X$ and $A\subseteq X$, we write $x\delta A$ to mean $\lbrace x\rbrace \delta A$, and $x\delta y$ to mean $\lbrace x\rbrace \delta \lbrace y \rbrace$.
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 $x\delta y$ implies $x=y$.
\textbf{Example}. Let $(X,d)$ is 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 proxmity space as a result.
\begin{thebibliography}{9}
\bibitem{willard} S. Willard, \emph{General Topology},
Addison-Wesley, Publishing Company, 1968.
\bibitem{nw} S.A. Naimpally, B.D. Warrack, \emph{Proximity Spaces}, Cambridge University Press, 1970.
\end{thebibliography} |
|
|
|
|
|
