TITLE: Poincaré Duality for non-compact manifolds without Zorn's Lemma
QUESTION [7 upvotes]: Does exists a proof of the Poincaré Duality version for non-compact manifolds without using the Zorn's Lemma? I know that there is a proof using the Whitney embedding theorem, but I don't know this theorem's proof, so I don't know if it uses the Zorn's Lemma too.

REPLY [3 votes]: A slight modification of Hatchers Proof gives you what you want:
Hatcher relies on the fact that if $M$ is a union of opens totally ordered by inclusion, which satisfy PD, then $M$ does, too. It is easy to see that you can weaken that to the opens only being directed. Also, another case tells you that all finite intersections of coordinate balls satisfy PD, so all finite unions do too. Then, at the end instead of using Zorn's Lemma you only have to look at the cover of all finite unions of coordinate balls, which are obviously directed w.r.t. inclusion.