Tychonoff Theorem

A product space product_(i in I)X_i is compact iff X_i is compact for all i in I. In other words, the topological product of any number of compact spaces is compact. In particular, compactness is a productive property. As a consequence, every Hilbert cube is compact.

This statement implies the axiom of choice, as proven by Kelley (1950).

See also

Axiom of Choice, Compact Space, Product Space

Portions of this entry contributed by Margherita Barile

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Kelley, J. L. "The Tychonoff Product Theorem Implies the Axiom of Choice." Fund. Math. 37, 75-76, 1950.

Referenced on Wolfram|Alpha

Tychonoff Theorem

Cite this as:

Barile, Margherita and Weisstein, Eric W. "Tychonoff Theorem." From MathWorld--A Wolfram Web Resource.

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