Projective limits of topological space II: Steenrod’s theorem

Last time, I explained some of the strange things that happen with projective limits of topological spaces: they can be empty, even if all the spaces in the given projective system are non-empty and all bonding maps are surjective, and they can fail to be compact, even if all the spaces in the projective system are compact.

Steenrod’s Theorem (as Fujiwara and Kato call it) shows that all those pathologies disappear if we work with compact sober spaces.  This rests on a lemma, according to which projective limits of non-empty compact sober spaces are non-empty, which is the subject of this month’s full post.  We will see how Steenrod’s Theorem follows… next time.