I have formalized an answer at Homeomorphisms and "mod finite" by @VilleSalo, in a moderately-sized Lean file.

Outcome: his claims turned out to be 100% accurate.

(The only stumbling block was that for me, it was easier to prove continuity of the inverse directly rather than appealing to general theory of compact and Hausdorff spaces.)

Has anyone verified other MathOverflow answers?