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 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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References
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. https://mathworld.wolfram.com/TychonoffTheorem.html
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