Difference between revisions of "Tychonoff theorem"
(New page: Tychonoff's theorem states that the product of arbitrary collection of compact topological spaces is compact. Tychonoff's theorem is equivalent to the axiom of choice, and can be proved u...) |
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| − | Tychonoff's theorem states that the product of arbitrary collection of compact topological | + | '''Tychonoff's theorem''' states that the product of arbitrary collection of compact [[topological space]]s is compact. Tychonoff's theorem is equivalent to the [[axiom of choice]], and can be proved using [[Proof by induction|transfinite induction]]. |
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| + | Stated another way, '''Tychonoff's theorem''' holds that a nonempty product space is compact if and only if each factor space is compact. | ||
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| + | [[Category:Topology]] | ||
Latest revision as of 20:53, July 13, 2016
Tychonoff's theorem states that the product of arbitrary collection of compact topological spaces is compact. Tychonoff's theorem is equivalent to the axiom of choice, and can be proved using transfinite induction.
Stated another way, Tychonoff's theorem holds that a nonempty product space is compact if and only if each factor space is compact.