# Completed shell

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 . ${\ displaystyle U}$${\ displaystyle U}$

## definition

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 . ${\ displaystyle X}$ ${\ displaystyle {\ overline {U}}}$${\ displaystyle U \ subseteq X}$${\ displaystyle X}$${\ displaystyle U}$${\ displaystyle {\ overline {U}}}$${\ displaystyle U}$

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 . ${\ displaystyle b \ in X}$${\ displaystyle U}$${\ displaystyle b}$${\ displaystyle U}$${\ displaystyle {\ overline {U}}}$${\ displaystyle U}$

## 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. ${\ displaystyle X}$${\ displaystyle X}$${\ displaystyle {\ overline {U}}}$${\ displaystyle U}$

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. ${\ displaystyle X}$${\ displaystyle U \ subseteq X}$${\ displaystyle U}$

## 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 ${\ displaystyle X}$${\ displaystyle d}$${\ displaystyle {\ overline {B (x, r)}}}$

${\ displaystyle B (x, r) = \ {y \ in X \ mid d (x, y)

with radius and center is not the same as the closed sphere ${\ displaystyle r}$${\ displaystyle x \ in X}$

${\ displaystyle {\ overline {B}} (x, r) = \ {y \ in X \ mid d (x, y) \ leq r \}.}$

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

${\ displaystyle {\ overline {B (x, r)}} \ subseteq {\ overline {B}} (x, r)}$

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

${\ displaystyle d (x, y) = \ left \ {{\ begin {matrix} 1 & \ mathrm {f {\ ddot {u}} r} \ x \ not = y \\ 0 & \ mathrm {f {\ ddot {u}} r} \ x = y \ end {matrix}} \ right.}$

is defined. Then applies to each : ${\ displaystyle x \ in X}$

${\ displaystyle \ {x \} = B (x, 1) = {\ overline {B (x, 1)}} \ subsetneq {\ overline {B}} (x, 1) = X.}$

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:

${\ displaystyle B (x, r) \ subsetneq {\ overline {B (x, r)}} \ subsetneq {\ overline {B}} (x, r).}$

An example is the set with the metric induced by Euclidean space . Here the specified inclusion condition is met : ${\ displaystyle X = \ {(a, 0) | a \ in \ mathbb {R}, -1 \ leq a \ leq 1 \} \ cup \ {(0,1) \}}$${\ displaystyle \ mathbb {R} ^ {2}}$${\ displaystyle x = (0,0), r = 1}$

${\ displaystyle B (0,1) = \ {(a, 0) \ mid a \ in \ mathbb {R}, -1
${\ displaystyle {\ overline {B (0,1)}} = \ {(a, 0) \ mid a \ in \ mathbb {R}, -1 \ leq a \ leq 1 \} \ subsetneq}$
${\ displaystyle {\ overline {B}} (0,1) = \ {(a, 0) \ mid a \ in \ mathbb {R}, -1 \ leq a \ leq 1 \} \ cup \ {(0 , 1) \} = X}$

## literature

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