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Version 13 |
| 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 $A,B\in P(X)$, |
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} |
\begin{enumerate} |
| \item if $A\cap B\ne \varnothing$, then $A\delta B$; |
\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 if $A\delta B$, then $A\ne \varnothing$ and $B\ne \varnothing$; |
| \item (symmetry) if $A\delta B$, then $B\delta A$; |
\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 $(A_1\cup A_2)\delta B$ iff $A_1\delta B$ or $A_2\delta B$; |
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\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 $(A,B)\notin \delta$.
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\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 $(A,B)\notin \delta$.
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| \end{enumerate} |
\end{enumerate} |
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| 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 $\lbrace x\rbrace \delta \lbrace y \rbrace$. |
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 $\lbrace x\rbrace \delta \lbrace y \rbrace$. |
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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 $x\delta'B$.
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When $A\delta B$, we say that $A$ is \emph{near} $B$. $\delta$ is also called a \emph{proximity relation}. 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 $x\delta'B$.
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| \textbf{Definition}. A set $X$ with a proximity as defined above is called a \emph{proximity space}. |
\textbf{Definition}. A set $X$ with a proximity as defined above is called a \emph{proximity space}. |
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| 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 $X$: |
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} |
\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} |
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. |
This turns $X$ into a topological space. Thus any proximity space is a topological space induced by the closure operator defined above. |
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| A proximity space is said to be \emph{separated} if for any $x,y\in X$, $x\delta y$ implies $x=y$. |
A proximity space is said to be \emph{separated} if for any $x,y\in X$, $x\delta y$ implies $x=y$. |
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| \textbf{Examples}. |
\textbf{Examples}. |
| \begin{itemize} |
\begin{itemize} |
| \item |
\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. |
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 |
\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 $X$. |
\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 |
\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 $X$. |
\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} |
\end{itemize} |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{willard} S. Willard, \emph{General Topology}, |
\bibitem{willard} S. Willard, \emph{General Topology}, |
| Addison-Wesley, Publishing Company, 1970. |
Addison-Wesley, Publishing Company, 1970. |
| \bibitem{nw} S.A. Naimpally, B.D. Warrack, \emph{Proximity Spaces}, Cambridge University Press, 1970. |
\bibitem{nw} S.A. Naimpally, B.D. Warrack, \emph{Proximity Spaces}, Cambridge University Press, 1970. |
| \end{thebibliography} |
\end{thebibliography} |
