What exactly is the support of a cycle, and what is the intuition for it to be closed (Borel-Moore homology)? Borel-Moore homology is often referred to as homology with closed support, however it is not clear to me from the two definitions I use (via locally finite chains, and via Poincaré duality) how this follows. In particular, what is the support of a cycle in homology, and how is it closed?
I think I have seen it in the context of the definition via sheaf cohomology, but I would like to avoid all of that if possible. I have also seen cohomology with compact support, but the support of a cocycle seems to me more intuitive than any possible definition of the support of a cycle. Even when passing through the duality, the resulting constraints make no intuitive sense to me. A definition here and a brief explanation of the geometry/picture behind it would be a good answer.
 A: If you think about singular homology, then the support is quite easy to understand : for a chain $\sum n_i\sigma_i$ where the $n_i$ are non zero integers, the support is simply the union of the images of $\sigma_i$. Now since the standard simplexes are compact, so are their images and so is a finite union. It follows that the support of a cycle in simplicial homology is compact. Note however that the support of two homologous cycles are not necessarily identical.
The same hold Borel-Moore cycles. This is the union of of the images of all simplexes. The fact that it is locally finite implies that the support is closed, but not necessarily compact.
About terminology, Borel-Moore homology is sometimes called homology with compact support. This historical terminology is wrong, this is really the usual homology which has compact support. This is clear with the notion of support above but also in the light of Poincaré duality. Indeed, you have isomorphisms $H_i(X)\simeq H^{d-i}_c(X)$ and $H_i^{BM}(X)\simeq H^{d-i}(X)$ which also shows that homology has compact support (since it is isomorphic to cohomology with compact support) and Borel-Moore homology has non compact support (since it is isomorphic to cohomology). You also have intersection pairings $$H_i\otimes H^j\to H_{i-j}$$
$$H_i^{BM}\otimes H^j\to H_{i-j}^{BM}$$
$$H_i^{BM}\otimes H^j_c\to H_{i-j}$$
$$H_i\otimes H^j_c\to H_{i-j}$$
You can see that the intersection of something (cycle or cocycle) compact with something non-compact gives a compact cycle as expected (see the third for example). Only two give rise to perfect pairing : the first and the third, that is when we pair something compact with something non-compact. This is well known in functional analysis.
