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

## Subject classifications