Heine–Borel theorem

The set of Heine- Borel, also called covering theorem, named after the mathematicians Eduard Heine and Émile Borel, is a set of the topology of metric spaces.

Statement

The theorem states that two different definitions of compactness are equivalent in finite dimensional real vector spaces.

This theorem can be applied specifically to subsets of the set of real numbers.

Counter-examples

It is important that the surrounding space is the Euclidean metric. In general ( deadband ) compactness not equivalent to closedness and boundedness.

A simple counter-example is the discrete metric on an infinite set. The discrete metric is defined by

• ,
• For, .

In this metric is every subset of is closed and bounded, but only the finite subsets are compact.

Other counterexamples are all infinite-dimensional normed vector spaces.

Generalization

However, true for general metric spaces that compact sets are those that are completely and totally bounded. This is therefore a generalization, because a subset of exactly then is complete if it is complete and since it is exactly then totally bounded if it is bounded.