| Version 2 |
Version 1 |
| {\bf Definition} |
{\bf Definition} \cite{rudin, kelley, ward}
|
| Suppose $A$ and $B$ are subsets of a topological space |
Suppose $A$ and $B$ are subsets of a topological space |
| $X$. Then $A$ and $B$ are {\bf separated} provided that |
$X$. Then $A$ and $B$ are {\bf separated} provided that |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \overline{A}\cap B &=& \emptyset, \\ |
\overline{A}\cap B &=& \emptyset, \\ |
| A\cap \overline{B} &=& \emptyset, |
A\cap \overline{B} &=& \emptyset, |
| \end{eqnarray*} |
\end{eqnarray*} |
| where $\overline{A}$ is the \PMlinkname{closure operator}{Closure} in $X$. |
where $\overline{A}$ is the \PMlinkname{closure operator}{Closure} in $X$. |
| When the ambient topological space is clear from the context, |
When the ambient topological space is clear from the context, |
| the notation $A\mid B$ indicates |
the notation $A\mid B$ indicates |
| that $A$ and $B$ are separated sets \cite{ward}. |
that $A$ and $B$ are separated sets \cite{ward}. |
|
|
| \subsubsection{Examples} |
|
| \begin{enumerate} |
|
| \item On $\R$, the intervals $(0,1)$ and $(1,2)$ are separated. |
|
| \item If if $d(x,y)\ge r+s$, then the open balls $B_r(x)$ and $B_s(x)$ are |
|
| separated \PMlinkname{(proof.)}{WhenAreBallsSeparated}. |
|
| \item If $A$ is a clopen set, then $A$ and $A^\complement$ are separated. |
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| This follows since $\overline{S}=S$ when $S$ is a closed set. |
|
| \end{enumerate} |
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| \end{enumerate} |
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|
| \subsubsection{Remarks} |
\subsubsection{Remarks} |
| In \cite{jameson}, separated sets are called |
In \cite{jameson}, separated sets are called |
|
{\bf strongly disjoin{t}} sets.
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{\bf strongly disjoint} sets.
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{rudin} |
\bibitem{rudin} |
| W. Rudin, \emph{Principles of Mathematical Analysis}, McGraw-Hill Inc., 1976. |
W. Rudin, \emph{Principles of Mathematical Analysis}, McGraw-Hill Inc., 1976. |
| \bibitem{kelley} |
\bibitem{kelley} |
| J.L. Kelley, \emph{General Topology}, D. van Nostrand Company, Inc., 1955. |
J.L. Kelley, \emph{General Topology}, D. van Nostrand Company, Inc., 1955. |
|
\bibitem{ward} L.E. Ward, \emph{Topology, An Outline for a First Course}, |
|
Marcel Dekker, Inc., 1972. |
| \bibitem{jameson} G.J. Jameson, \emph{Topology and Normed Spaces}, |
\bibitem{jameson} G.J. Jameson, \emph{Topology and Normed Spaces}, |
| Chapman and Hall, 1974. |
Chapman and Hall, 1974. |
| \end{thebibliography} |
\end{thebibliography} |
