# 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 .