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

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

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 : ${\displaystyle X=\{(a,0)|a\in \mathbb {R} ,-1\leq a\leq 1\}\cup \{(0,1)\}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle x=(0,0),r=1}$

${\displaystyle B(0,1)=\{(a,0)\mid a\in \mathbb {R} ,-1<a<1\}\subsetneq }$

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

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

literature

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