Completed shell

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In topology and analysis , the closed envelope (also closure or closure ) of a subset of a topological or metric space is the smallest closed superset of .


If a topological space is , then the closed envelope or the closure of a subset is the intersection of all closed subsets of which contain. The set is itself closed, so it is the smallest closed superset of .

A point is called a point of contact or adherence point of if at least one element of is contained in every neighborhood of . consists exactly of the contact points of .

The conclusion as a set of limit values

If the first axiom of countability is fulfilled (this applies, for example, if is a metric space), then is the set of all limit values of convergent sequences whose terms lie in.

If any topological space is, then the closure of a subset is the set of limit values ​​of convergent networks , the terms of which are in.

Completion of spheres in metric spaces

Let it be a metric space with metrics . Note that in general the closed envelope of an open sphere

with radius and center is not the same as the closed sphere

Since the closed sphere is a closed set that contains the open sphere, it also contains its closure:

To give an example in which this inclusion is real, let X be a set (with at least two elements) on which a metric passes

is defined. Then applies to each :

In addition, there are also metric spaces in which, for a point x and a radius r, both inclusions are real at the same time:

An example is the set with the metric induced by Euclidean space . Here the specified inclusion condition is met :


  • Gabriele Castellini: Categorical Closure Operators. Birkhäuser, Boston MA et al. 2003, ISBN 0-8176-4250-1 .