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closure (Definition)

The closure $ \overline{A}$ of a subset $ A$ of a topological space $ X$ is the intersection of all closed sets containing $ A$.

Equivalently, $ \overline{A}$ consists of $ A$ together with all limit points of $ A$ in $ X$ or equivalently $ x\in\overline{A}$ if and only if every neighborhood of $ x$ intersects $ A$. Sometimes the notation $ \operatorname{cl}(A)$ is used.

If it is not clear, which topological space is used, one writes $ \overline{A}^X$. Note that if $ Y$ is a subspace of $ X$, then $ \overline{A}^X$ may not be the same as $ \overline{A}^Y$. For example, if $ X=\mathbb{R}$, $ Y=(0,1)$ and $ A=(0,1)$, then $ \overline{A}^X=[0,1]$ while $ \overline{A}^Y=(0,1)$.



"closure" is owned by mathwizard. [ full author list (2) | owner history (1) ]
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See Also: closure axioms, interior

Keywords:  topology
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Cross-references: subspace, clear, neighborhood, limit points, closed sets, intersection, topological space, subset
There are 69 references to this entry.

This is version 5 of closure, born on 2002-01-03, modified 2006-12-08.
Object id is 1191, canonical name is Closure.
Accessed 8823 times total.

Classification:
AMS MSC: 54A99 (General topology :: Generalities :: Miscellaneous)
Pending Errata and Addenda
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