There is a homology theory called "Borel-Moore" or "locally finite" homology, which can either be constructed by using locally-finite chains or by more advanced sheaf-theoretic methods. Here are two useful facts about it:

- If $X$ is locally compact and $\tilde X$ is a reasonable compactification, then $H^{lf}(X) \cong H(X, \tilde X \setminus X)$ agrees with the ordinary homology relative to infinity.
- If $X$ is an orientable manifold with boundary (maybe weaker hypotheses suffice), not necessarily closed, then $H^{lf}_k(X)$ is dual to $H_{n-k}(X, \partial X)$.

Either of these results might be stated slightly wrong, which is partially what my question is about.

I am looking for a reference that constructs $H^{lf}$ and proves one or both of these results, but using less technical machinery than last chapter of Bredon's *Sheaf Theory* (which I think has what I want, but I don't know enough sheaf theory to understand it.) Hatcher only mentions $H^{lf}$ in an exercise, and Bredon's *Topology and Geometry* doesn't discuss it at all.

I see that there are some related questions (here and here) that suggest there are no good references, so maybe this is a lost cause, but I notice neither mentions the duality property (2).

Homology and Cohomology Theory. He doesn't call it Borel-Moore, but he cites their paper and constructs some kind of locally-finite cohomology. I don't know how much he does. $\endgroup$ – Tyrone Dec 6 '20 at 12:52