| Version 10 |
Version 9 |
| {\bf Definition} |
{\bf Definition} |
| 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 $X$. |
|
|
|
\subsubsection*{Properties}
|
\subsubsection{Properties}
|
| \begin{enumerate} |
\begin{enumerate} |
| \item If $A,B$ are separated in $X$, $f\colon X\to Y$ is a homeomorphism, |
\item If $A,B$ are separated in $X$, $f\colon X\to Y$ is a homeomorphism, |
| then $f(A)$ and $f(B)$ are separated in $Y$. |
then $f(A)$ and $f(B)$ are separated in $Y$. |
| \end{enumerate} |
\end{enumerate} |
|
|
|
\subsubsection*{Examples}
|
\subsubsection{Examples}
|
| \begin{enumerate} |
\begin{enumerate} |
| \item On $\R$, the intervals $(0,1)$ and $(1,2)$ are separated. |
\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 $B_s(y)$ are |
\item If if $d(x,y)\ge r+s$, then the open balls $B_r(x)$ and $B_s(y)$ are |
| separated \PMlinkname{(proof.)}{WhenAreBallsSeparated}. |
separated \PMlinkname{(proof.)}{WhenAreBallsSeparated}. |
| \item If $A$ is a clopen set, then $A$ and $A^\complement$ are separated. |
\item If $A$ is a clopen set, then $A$ and $A^\complement$ are separated. |
| This follows since $\overline{S}=S$ when $S$ is a closed set. |
This follows since $\overline{S}=S$ when $S$ is a closed set. |
| \end{enumerate} |
\end{enumerate} |
|
|
|
\subsubsection*{Remarks}
|
\subsubsection{Remarks}
|
| The above definition follows \cite{kelley}. In |
The above definition follows \cite{kelley}. In |
| \cite{jameson}, separated sets are called |
\cite{jameson}, separated sets are called |
| {\bf strongly disjoin{t}} sets. |
{\bf strongly disjoin{t}} sets. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \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{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} |
