Tychonoff's theorem

For other theorems named after Tychonoff, see Tychonoff's theorem (disambiguation).

In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is transcribed Tychonoff), who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1935 article by Tychonoff, "Über einen Funktionenraum".[1]

Tychonoff's theorem is often considered as perhaps the single most important result in general topology (along with Urysohn's lemma).[2] The theorem is also valid for topological spaces based on fuzzy sets.[3]

  1. ^ Tikhonov, Andrey Nikolayevich (1935), "Über einen Funktionraum", Mathematische Annalen (in German) (111): 762–766
  2. ^ Willard, Stephen (2004), General Topology, Dover, p. 120, ISBN 978-0-486-43479-7
  3. ^ Goguen, Joseph (September 1973), "The Fuzzy Tychonoff Theorem", Journal of Mathematical Analysis and Applications, 43 (3): 734–742