Compact space

Tikhonov's theorem

Tikhonov's1 theorem states that the topological product of compact spaces is itself compact. In its full form, it is equivalent to the Axiom of Choice over .2

Proof from Alexander subbasis theorem

Let have the product topology, which by construction bares the subbasis

Now let be an open subbasic cover of . Then

is inhabited for some , so invoking the Axiom of Choice we may fix some and get a subcover . But this induces an open cover of , which by compactness has an open subcover such that is a subcover of .

Corollaries

#state/tidy | #lang/en | #SemBr

Footnotes

  1. Usually transcribed Tychonoff. ↩

  2. See Wikipedia for a proof. ↩