Definitions
Sorry, no definitions found. You may find more data at heine-borel.
Etymologies
Sorry, no etymologies found.
Support
Help support Wordnik (and make this page ad-free) by adopting the word Heine-Borel.
Examples
-
For example, to prove that a continuous function is bounded, you either find a neighbourhood of each point on which by finitely many such neighbourhoods (by the Heine-Borel theorem), or you assume that, apply the Bolzano-Weierstrass theorem, and reach a contradiction.
Gowers's Weblog gowers 2010
-
If you want to prove something using the compactness of, then you can either prove it directly using the Heine-Borel theorem or you can prove it by contradiction by reformulating everything in terms of sequences and applying the Bolzano-Weierstrass theorem.
Gowers's Weblog gowers 2010
-
From Heine-Borel we know that a continuous function on a closed interval takes a global maximum and a minimum.
-
1984, “Heine-Borel does not imply the Fan Theorem”,
Category Theory Marquis, Jean-Pierre 2007
Comments
Log in or sign up to get involved in the conversation. It's quick and easy.