ClosedSet 2002-03-02 01:13:01 2007-02-06 05:19:18 Definition closed \usepackage{amsfonts} \usepackage{amssymb} \def\R{\mathbb{R}} \def\emptyset{\varnothing} \PMlinkescapephrase{closed under} \PMlinkescapeword{contains} \PMlinkescapeword{contained} Let $(X,\tau)$ be a topological space. Then a subset $C\subseteq X$ is \emph{closed} if its complement $X\setminus C$ is open under the topology $\tau$. Examples: \begin{itemize} \item In any topological space $(X,\tau)$, the sets $X$ and $\emptyset$ are always closed. \item Consider $\R$ with the standard topology. Then $[0,1]$ is closed since its complement $(-\infty,0) \cup (1,\infty)$ is open (being the union of two open sets). \item Consider $\R$ with the lower limit topology. Then $[0,1)$ is closed since its complement $(-\infty,0)\cup[1,\infty)$ is open. \end{itemize} Closed subsets can also be characterized as follows: A subset $C\subseteq X$ is closed if and only if $C$ contains all of its cluster points, that is, $C'\subseteq C$. So the set $\{1,1/2,1/3,1/4,\ldots\}$ is not closed under the standard topology on $\R$ since $0$ is a cluster point not contained in the set.