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Tychonoff ’s theorem∗
matte†
2013-03-21 13:01:47
Let (Xi )i∈I be a family of nonempty topological spaces. The product space
(see product topology)
Y
Xi
i∈I
is compact if and only if each of the spaces Xi is compact.
Not surprisingly, if I is infinite, the proof requires the Axiom of Choice.
Conversely, one can show that Tychonoff’s theorem implies that any product of
nonempty sets is nonempty, which is one form of the Axiom of Choice.
∗ hTychonoffsTheoremi created: h2013-03-21i by: hmattei version: h31168i Privacy
setting: h1i hTheoremi h54D30i
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