Heine-Borel's theorem

The Heine-Borel Theorem , also known as the coverage theorem , named after the mathematicians Eduard Heine (1821-1881) and Émile Borel (1871-1956), is a theorem of the topology of metric spaces .

statement

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

For a subset of the (the metric space of all real n-tuples with the Euclidean metric ) the following two statements are equivalent: ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \mathbb {R} ^{n}}$

1. ${\displaystyle {\mathcal {M}}}$is limited and closed .
2. Every open cover of contains a finite partial cover . ${\displaystyle {\mathcal {M}}}$

This theorem can be applied specifically to subsets of the set of real numbers . ${\displaystyle \mathbb {R} }$

Note and counterexamples

The premise that the surrounding space is is essential. In general (coverage) compactness is not equivalent to closure and limitedness. ${\displaystyle \mathbb {R} ^{n}}$

The discrete metric on an infinite set provides a simple counterexample . It is defined by ${\displaystyle X}$

${\displaystyle d(x,y):={\begin{cases}0&\mathrm {f{\ddot {u}}r} \ x=y\\1&\mathrm {f{\ddot {u}}r} \ x\neq y.\end{cases}}}$

In this metric, every subset of is closed and bounded, but only the finite subsets are compact. ${\displaystyle X}$

Further counterexamples are all infinitely dimensional normalized vector spaces .

generalization

For general metric spaces, however, the compact sets are those which are complete and totally bounded . This is a generalization because a subset of is complete if and only if it is closed and because it is totally bounded if and only if it is bounded. ${\displaystyle \mathbb {R} ^{n}}$

Web links

• Heine Borel (Video illustrating a proof of Heine-Borel's Theorem.)