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 ZF.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

• Heine-Borel theorem

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Footnotes

1. Usually transcribed Tychonoff. ↩

2. See Wikipedia for a proof. ↩