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5
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'separated'
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| Title of object: |
separated |
| Canonical Name: |
Separated |
| Type: |
Definition |
| Created on: |
2005-05-17 14:02:11 |
| Modified on: |
2005-05-18 02:55:52 |
| Classification: |
msc:54-00, msc:54D05 |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage{mathrsfs}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
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% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}
\usepackage{bbm}
\newcommand{\Z}{\mathbbmss{Z}}
\newcommand{\C}{\mathbbmss{C}}
\newcommand{\F}{\mathbbmss{F}}
\newcommand{\R}{\mathbbmss{R}}
\newcommand{\Q}{\mathbbmss{Q}}
\newcommand*{\norm}[1]{\lVert #1 \rVert}
\newcommand*{\abs}[1]{| #1 |}
\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary} |
Content:
{\bf Definition}
Suppose $A$ and $B$ are subsets of a topological space
$X$. Then $A$ and $B$ are {\bf separated} provided that
\begin{eqnarray*}
\overline{A}\cap B &=& \emptyset, \\
A\cap \overline{B} &=& \emptyset,
\end{eqnarray*}
where $\overline{A}$ is the \PMlinkname{closure operator}{Closure} in $X$.
\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.
This follows since $\overline{S}=S$ when $S$ is a closed set.
\end{enumerate}
\subsubsection{Remarks}
The abvoe definition follows \cite{kelley}. In
\cite{jameson}, separated sets are called
{\bf strongly disjoin{t}} sets.
\begin{thebibliography}{9}
\bibitem{rudin}
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} |
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