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'clopen subset'
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| Title of object: |
clopen subset |
| Canonical Name: |
ClopenSubset |
| Type: |
Definition |
| Created on: |
2003-02-06 18:56:00 |
| Modified on: |
2004-11-13 16:46:50 |
| Classification: |
msc:54D05 |
| Synonyms: |
clopen subset=clopen set
clopen subset=clopen
clopen subset=closed and open |
Revision comment (for changes between this and next version):
| Changes for correction #10498 ('no proof!'). |
Preamble:
%\documentclass{amsart}
\usepackage{amsmath}
%\usepackage[all,poly,knot,dvips]{xy}
%\usepackage{pstricks,pst-poly,pst-node,pstcol}
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% THEOREM Environments --------------------------------------------------
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\newtheorem{lem}[thm]{Lemma}
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\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
\numberwithin{equation}{subsection}
%--------------------- Greek letters, etc -------------------------
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\newcommand{\GQ}{\Theta}
\newcommand{\Gi}{\iota}
\newcommand{\Gk}{\kappa}
\newcommand{\Gl}{\lambda}
\newcommand{\GL}{\Lamda}
\newcommand{\Gm}{\mu}
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\newcommand{\GX}{\Xi}
\newcommand{\Gp}{\pi}
\newcommand{\GP}{\Pi}
\newcommand{\Gr}{\rho}
\newcommand{\Gs}{\sigma}
\newcommand{\GS}{\Sigma}
\newcommand{\Gt}{\tau}
\newcommand{\Gu}{\upsilon}
\newcommand{\GU}{\Upsilon}
\newcommand{\Gf}{\varphi}
\newcommand{\GF}{\Phi}
\newcommand{\Gc}{\chi}
\newcommand{\Gy}{\psi}
\newcommand{\GY}{\Psi}
\newcommand{\Gw}{\omega}
\newcommand{\GW}{\Omega}
\newcommand{\Gee}{\epsilon}
\newcommand{\Gpp}{\varpi}
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\newcommand{\Gss}{\varsigma}
\def\co{\colon\thinspace} |
Content:
A subset $C$ of a topological space $X$ is called \emph{clopen} if it is both
open and closed.
\begin{thm}
The clopen subsets form a Boolean algebra under the operation of
union, intersection and complement. In other words:
\begin{itemize}
\item $X$ and $\emptyset$ are clopen,
\item the complement of a clopen set is clopen,
\item finite unions and intersections of clopen sets are clopen.
\end{itemize}
\end{thm}
\begin{proof}
The first follows by the definition of a topology, the second by noting that complements of open sets are closed, and vice versa, and the third by noting that this property holds for both open and closed sets.
\end{proof}
One application of clopen sets is that they can be used to describe connectness. In particular, a topological space is connected if and only if its only clopen
subsets are itself and the empty set. (\PMlinkname{proof}{CharacterizationsOfConnectedness})
If a space has finitely many connected components then each
connected component is clopen. This may not be the case if there
are infinitely many components, as the case of the rational numbers
demonstrates. |
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