Tychonoff’s theorem

Let (Xi)iI be a family of nonempty topological spacesMathworldPlanetmath. The product space (see product topology)


is compactPlanetmathPlanetmath if and only if each of the spaces Xi is compact.

Not surprisingly, if I is infiniteMathworldPlanetmath, the proof requires the Axiom of ChoiceMathworldPlanetmath. Conversely, one can show that TychonoffPlanetmathPlanetmath’s theorem implies that any productPlanetmathPlanetmath of nonempty sets is nonempty, which is one form of the Axiom of Choice.

Title Tychonoff’s theorem
Canonical name TychonoffsTheorem
Date of creation 2013-03-22 12:05:14
Last modified on 2013-03-22 12:05:14
Owner matte (1858)
Last modified by matte (1858)
Numerical id 12
Author matte (1858)
Entry type Theorem
Classification msc 54D30
Synonym Tichonov’s theorem
Related topic Compact