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Tychonoff's theorem

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Tychonoff's theorem in topology states that the product of any collection of compact topological spaces is compact.

For finite collections of compact spaces, this is not very surprising and is easy to prove. The statement is in fact true for infinite collections of arbitrary size, depends heavily on the peculiar definition of the product topology in this case, and is equivalent to the axiom of choice.

proof sketch would be nice

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