Tychonoff’s theorem, DYK Fact #611
 JaneFairfax Posted: 19:19 Thursday 06 May 2010 The Enlightenment Group: Moderators Posts: 673 Member No.: 20 Joined: 03 Mar 2007 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 $\mathbb{R}^n$ are compact.
 algebraic topology Posted: 06:13 Friday 07 May 2010 Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 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 Posted: 17:21 Tuesday 11 May 2010 The Enlightenment Group: Moderators Posts: 673 Member No.: 20 Joined: 03 Mar 2007 Did you know? The converse to Tychonoff’s theorem also holds – because if $X=\prod_{\lambda\,\in\,\Lambda}X_\lambda$ is compact, then each $X_\lambda$, being a continuous projection of X, is compact as well. The converse to the Heine–Borel theorem is also true: all compact subspaces of $\mathbb{R}^n$ are closed and bounded. This is in fact true of all metric spaces, not just Euclidean n-spaces.
 algebraic topology Posted: 18:22 Tuesday 11 May 2010 Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 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 Posted: 20:53 Tuesday 11 May 2010 The Enlightenment Group: Moderators Posts: 512 Member No.: 150 Joined: 29 Mar 2009 Did you know? Andrey Nikolayevich Tychonoff (1906–93) was a Russian mathematician.
 shygeorge Posted: 22:44 Tuesday 11 May 2010 The Enlightenment Group: Moderators Posts: 534 Member No.: 161 Joined: 11 Dec 2009 Did you know? Eduard Heine (1821–81) was a German mathematician and Émile Borel (1871–1956) was a French mathematician.
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