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closed set (Definition)

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, $C'\subseteq C$.

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.



"closed set" is owned by yark. [ full author list (2) | owner history (2) ]
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Other names:  closed subset
Also defines:  closed
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Cross-references: cluster points, lower limit topology, open sets, union, standard topology, open, complement, subset, topological space
There are 129 references to this entry.

This is version 7 of closed set, born on 2002-03-02, modified 2007-02-06.
Object id is 2739, canonical name is ClosedSet.
Accessed 18223 times total.

Classification:
AMS MSC: 54-00 (General topology :: General reference works )
Pending Errata and Addenda
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