separated
Separated
2005-05-17 14:02:11
2006-05-24 10:23:03
Definition
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{\bf Definition}
Suppose $A$ and $B$ are subsets of a topological space
$X$. Then $A$ and $B$ are {\bf separated} provided that
\[
\begin{array}{ccc}
\overline{A}\cap B &=& \emptyset, \\
A\cap \overline{B} &=& \emptyset,
\end{array}
\]
where $\overline{A}$ is the \PMlinkname{closure operator}{Closure} in $X$.
\subsubsection*{Properties}
\begin{enumerate}
\item If $A,B$ are separated in $X$, and $f\colon X\to Y$ is a homeomorphism,
then $f(A)$ and $f(B)$ are separated in $Y$.
\end{enumerate}
\subsubsection*{Examples}
\begin{enumerate}
\item On $\R$, the intervals $(0,1)$ and $(1,2)$ are separated.
\item If $d(x,y)\ge r+s$, then the open balls $B_r(x)$ and $B_s(y)$ are
separated \PMlinkname{(proof.)}{WhenAreBallsSeparated}.
\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.
\end{enumerate}
\subsubsection*{Remarks}
The above definition follows \cite{kelley}. In
\cite{jameson}, separated sets are called
{\bf strongly disjoin{t}} sets.
\begin{thebibliography}{9}
\bibitem{kelley}
J.L. Kelley, \emph{General Topology}, D. van Nostrand Company, Inc., 1955.
\bibitem{jameson} G.J. Jameson, \emph{Topology and Normed Spaces},
Chapman and Hall, 1974.
\end{thebibliography}