Heine-Borel's theorem

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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 .


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:
  1. is limited and closed .
  2. Every open cover of contains a finite partial cover .

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

Note and counterexamples

The premise that the surrounding space is is essential. In general (coverage) compactness is not equivalent to closure and limitedness.

The discrete metric on an infinite set provides a simple counterexample . It is defined by

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

Further counterexamples are all infinitely dimensional normalized vector spaces .


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.

Web links

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