Math Is Fun Forum

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Tychonoff’s theorem

Tychonoff’s theorem: http://z8.invisionfree.com/DYK/index.php?showtopic=900

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.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Tychonoff’s theorem

More about Tychonoff’s theorem.

Tychonoff’s theorem in the infinite case is equivalent to the axiom of choice. The converse to Tychonoff’s theorem is also true because if

is compact, then each

, being a continous projection of

, must be compact as well.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Tychonoff’s theorem

Oh, and Andrey Nikolayevich Tychonoff (1906–93) was a Russian mathematician.

http://z8.invisionfree.com/DYK/index.ph … &p=4451424