# closed set

closed under

Let $(X,\tau)$ be a topological space. Then a subset $C\subseteq X$ is closed if its complement $X\setminus C$ is open under the topology $\tau$.

Examples:

• In any topological space $(X,\tau)$, the sets $X$ and $\varnothing$ are always closed.

• Consider $\mathbb{R}$ 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).

• Consider $\mathbb{R}$ with the lower limit topology. Then $[0,1)$ is closed since its complement $(-\infty,0)\cup[1,\infty)$ is open.

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, $C^{\prime}\subseteq C$.

So the set $\{1,1/2,1/3,1/4,\ldots\}$ is not closed under the standard topology on $\mathbb{R}$ since $0$ is a cluster point not contained in the set.

Title closed set ClosedSet 2013-03-22 12:30:23 2013-03-22 12:30:23 yark (2760) yark (2760) 10 yark (2760) Definition msc 54-00 closed subset closed