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 Tychonoff’s theorem, DYK Fact #611
JaneFairfax
Posted: 19:19 Thursday 06 May 2010


The Enlightenment


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Did you know? The topological product of an arbitrary collection of compact topological spaces is compact. This is known as Tychonoff’s theorem.

Tychonoff’s theorem has applications in modern analysis. It is also used to prove the Heine–Borel theorem, namely that all closed and bounded subspaces of are compact.

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algebraic topology
Posted: 06:13 Friday 07 May 2010


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Did you know?

Tychonoff’s theorem in the infinite case is equivalent to the axiom of choice. For the finite case, however, the theorem can be proved without the axiom of choice – and that is all that is needed for the proof of the Heine–Borel theorem.

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JaneFairfax
Posted: 17:21 Tuesday 11 May 2010


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Did you know? The converse to Tychonoff’s theorem also holds – because if is compact, then each , being a continuous projection of X, is compact as well. The converse to the Heine–Borel theorem is also true: all compact subspaces of are closed and bounded. This is in fact true of all metric spaces, not just Euclidean n-spaces.

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algebraic topology
Posted: 18:22 Tuesday 11 May 2010


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Did you know?

More generally, all compact subspaces of a metric space are bounded, and all compact subspaces of a Hausdorff space are closed.

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shyvera
Posted: 20:53 Tuesday 11 May 2010


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Did you know? Andrey Nikolayevich Tychonoff (1906–93) was a Russian mathematician.
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shygeorge
Posted: 22:44 Tuesday 11 May 2010


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Did you know? Eduard Heine (1821–81) was a German mathematician and Émile Borel (1871–1956) was a French mathematician.
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