A product space is compact iff is compact for all . 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
This statement implies the axiom of choice, as
proven by Kelley (1950).
See alsoAxiom of Choice
, Product Space
Portions of this entry contributed by Margherita
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ReferencesKelley, J. L. "The Tychonoff Product Theorem Implies the Axiom of Choice." Fund. Math. 37, 75-76, 1950.
on Wolfram|AlphaTychonoff Theorem
Cite this as:
Barile, Margherita and Weisstein, Eric W. "Tychonoff Theorem." From MathWorld--A
Wolfram Web Resource. https://mathworld.wolfram.com/TychonoffTheorem.html