Borel-Moore and ordinary homology agree for compact spaces

Every ordinary singular chain is locally finite: a sum with finite support has only finitely many distinct images, so any neighbourhood of any point meets at most those finitely many images. The inclusion $C_k(X;R)\hookrightarrow C_k^{\mathrm{B}\mathrm{M}}(X;R)$ is therefore defined, commutes with the coefficientwise boundary, and is injective.

Conversely let $c={\sum }_{\sigma }{a}_{\sigma }\sigma$ be locally finite on the compact $X$. Local finiteness provides an open cover of $X$ by neighbourhoods $U$ each of which meets only finitely many images of simplices with nonzero coefficient; by compactness [F2] finitely many such neighbourhoods $U_1,\dots ,U_m$ cover $X$. If $\sigma$ has $a_{\sigma }\mathrm{\ne }0$, then its image is nonempty, so it meets some $U_j$, and for each $j$ only finitely many such $\sigma$ meet $U_j$; hence only finitely many coefficients are nonzero and $c$ lies in the image of the inclusion.

By steps 1.1 and 1.2 the inclusion is a bijection on every chain group, hence an isomorphism of chain complexes; it induces isomorphisms on homology. For a continuous map $f:X\to Y$ of compact Hausdorff spaces, for every compact $K\subseteq Y$, Hausdorffness makes $K$ closed, continuity makes $f^{-1}(K)$ closed in the compact $X$, and this preimage is therefore compact by [F2]. Thus $f$ is proper, and the pushforward $f_{\ast }$ of [F1] restricts on finite chains to the ordinary singular chain map, so the isomorphisms are natural. No choice principle was used.