Tychonoff’s theorem, DYK Fact #611

[JaneFairfax]

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.

[algebraic topology]

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.

[JaneFairfax]

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.

[algebraic topology]

Did you know?

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

[shyvera]

Did you know? Andrey Nikolayevich Tychonoff (1906–93) was a Russian mathematician.

[shygeorge]

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