Total limitation

The concept of total limitation (or precompactness ) designates a certain limitation property of a metric space . One can show that a metric space is compact if and only if it is complete and totally bounded.

definition

A subset of a metric space is called totally bounded (or also precompact) if there is a finite set of points (a -net) for each such that ${\displaystyle A}$ ${\displaystyle \left(M,d\right)}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle x_{1},\ldots ,x_{n}\in A}$ ${\displaystyle \varepsilon }$

${\displaystyle A\subseteq \bigcup _{k=1}^{n}\{x\in M:d(x,x_{k})<\varepsilon \}}$

applies. That is, the subset is for any finite number of - balls covered. ${\displaystyle A}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \varepsilon }$

Equivalent definition

It can be shown that a metric space is totally bounded if and only if every sequence has a subsequence that is a Cauchy sequence .

properties

Although the two terms were developed independently of one another in different contexts, the equivalence applies:

A subset of a complete metric space is totally bounded if and only if it is relatively compact .

The motivation for independent consideration of total limitation lies in the following statement:

A metric space is compact if and only if it is complete and totally bounded.

In a way, this is a generalization of Heine-Borel's theorem , which states that a subset of is compact if and only if it is closed and bounded. ${\displaystyle \mathbb {R} ^{n}}$

Generalization to uniform spaces

Like many other terms from the theory of metric spaces, the term can also be totally restricted or precompactly generalized to the class of uniform spaces :

A subset of a uniform space is called precompact if there is a finite set of points for each such that ${\displaystyle A}$ ${\displaystyle \left(X,\Phi \right)}$ ${\displaystyle U\in \Phi }$ ${\displaystyle x_{1},\ldots ,x_{n}}$

${\displaystyle A\subseteq \bigcup _{k=1}^{n}\{x\in X:(x,x_{k})\in U\}}$ applies.

It is equivalent that every network has a Cauchy subnet .

However, a further generalization to any topological space is not possible. Total confinement or precompactness is not a topological property, for example the interval is homeomorphic to , but understood as a metric space, in contrast to the latter, precompact. ${\displaystyle (0,1)}$ ${\displaystyle \mathbb {R} }$

literature

• AV Arkhangel'skii: Totally-bounded space . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).