# 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. ${\ displaystyle \ mathbb {R} ^ {n}}$