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 $Z F$ .²

Proof from Alexander subbasis theorem

Let $𝑋 = \prod 𝛼 \in 𝐴 𝑋 𝛼$ have the product topology, which by construction bares the subbasis

A 𝑋 = {𝜋 - 1 𝛼 𝑈 : 𝑈 \in T 𝛼 : 𝛼 \in 𝐴}

Now let $C$ be an open subbasic cover of $𝑋$ . Then

C 𝛼 = {𝑉 \in C : (\exists 𝑈 \in T 𝛼) [𝜋 - 1 𝛼 𝑈 = 𝑉]}

is inhabited for some $𝛼$ , so invoking the Axiom of Choice we may fix some $𝛼$ and get a subcover $C 𝛼 \subseteq C$ . But this induces an open cover of $𝑋 𝛼$ , which by compactness has an open subcover $D 𝛼$ such that $𝜋 - 1 D 𝛼$ is a subcover of $C 𝛼 \subseteq C$ .

Corollaries

Heine-Borel theorem

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

Usually transcribed Tychonoff. ↩
See Wikipedia for a proof. ↩

Tikhonov's theorem

Corollaries

Footnotes