| Version 6 |
Version 5 |
|
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 $\tau$.
|
Let $(X,\tau)$ be a topological space. Then a subset $C\subseteq X$ is \emph{closed} if its complement $X\backslash C$ is open under the topology $\tau$.
|
|
|
| Example: |
Example: |
| \begin{itemize} |
\begin{itemize} |
| \item In any topological space $(X,\tau)$, the sets $X$ and $\emptyset$ are always closed. |
\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 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. |
\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} |
\end{itemize} |
|
|
| Closed subsets can also be characterized as follows: |
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, $C'\subseteq C$. |
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. |
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. |
