Totally bounded space

The notion of total boundedness (or Präkompaktheit ) designate a certain boundedness property of a metric space. One can show that a metric space is compact if it is complete and totally bounded.

Definition

A subset of a metric space is called totally bounded (or precompact ) if, for every finite set of points ( a network), so that

Applies. That is, the portion is finally covered balls for each of the many.

Equivalent definition

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

Properties

Although the two concepts were developed independently in different contexts, the equivalence holds:

The motivation for independent consideration of the total boundedness is located in the following statement:

This is in some ways a generalization of the Heine- Borel, which says that a subset of is compact if and only if it is closed and bounded.

Generalization to uniform spaces

Like many other concepts from the theory of metric spaces, the term can also totally bounded or precompact generalized to the class of uniform spaces:

A subset of a uniform space is called precompact if, for every finite set of points, so that

Is equivalent in that each network has a Cauchy subnet.

However, a further generalization to arbitrary topological areas is not possible. Total boundedness, or Präkompaktheit is not a topological property, such as the interval is indeed homeomorphic to, as a metric space, however, be construed as opposed to the latter precompact.