Borel-Moore homology by locally finite chains

Let $X$ be a locally compact Hausdorff topological space and let $R$ be a commutative ring. For $k\ge 0$ let $S_k(X)$ be the set of continuous maps ${\mathrm{\Delta }}^k\to X$ from the standard simplex (Singular simplices and singular chain groups with coefficients); the convention of In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular supplies neighbourhood bases by open sets with compact closure. A formal sum $c={\sum }_{\sigma \in S_k(X)}{a}_{\sigma }\sigma$ with $a_{\sigma }\in R$ is locally finite if every point $x\in X$ has an open neighbourhood meeting only finitely many of the images $\sigma ({\mathrm{\Delta }}^k)$ of the simplices with $a_{\sigma }\mathrm{\ne }0$ (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right for compactness). The locally finite $k$-chains are the $R$-module $C_k^{\mathrm{B}\mathrm{M}}(X;R)=\{c:\text{ the sum is locally finite}\},C_k^{\mathrm{B}\mathrm{M}}(X;R)=0(k<0),$ with addition and scalar multiplication defined coefficientwise. The boundary $\mathrm{\partial }\left({\sum }_{\sigma }{a}_{\sigma }\sigma \right)={\sum }_{\sigma }{a}_{\sigma }\mathrm{\partial }\sigma$ is again locally finite, because every face of $\sigma$ has image contained in the image of $\sigma$, and ${\mathrm{\partial }}^2=0$ because ${\mathrm{\partial }}^2\sigma =0$; thus $(C_{\ast }^{\mathrm{B}\mathrm{M}}(X;R),\mathrm{\partial })$ is a chain complex and its homology is the Borel-Moore homology $H_k^{\mathrm{B}\mathrm{M}}(X;R)$ (The singular chain complex and singular homology for the ordinary chain complex, which is the special case of sums with finite support).

A proper continuous map $f:X\to Y$ (inverse images of compact sets are compact) induces $f_{\ast }:C_{\ast }^{\mathrm{B}\mathrm{M}}(X;R)\to C_{\ast }^{\mathrm{B}\mathrm{M}}(Y;R)$ by $\sigma \mapsto f\circ \sigma$: for $y\in Y$ choose a compact neighbourhood $K$ of $y$; the compact set $f^{-1}(K)$ is covered by finitely many of the neighbourhoods furnished by local finiteness of $c$, by A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, so only finitely many simplices meet $f^{-1}(K)$, and a neighbourhood of $y$ contained in $K$ then meets only finitely many images $f(\sigma ({\mathrm{\Delta }}^k))$. This is a chain map because $f\circ \mathrm{\partial }\sigma =\mathrm{\partial }(f\circ \sigma )$, so it induces $f_{\ast }:H_{\ast }^{\mathrm{B}\mathrm{M}}(X;R)\to H_{\ast }^{\mathrm{B}\mathrm{M}}(Y;R)$. Coefficients and degree are otherwise unused below. No choice principle is used in the definition; a homeomorphism and, more generally, a proper continuous map act functorially.

• Locally finite versus finite. The ordinary singular chain group $C_k(X;R)$ is the subgroup of sums with finite support, and it is contained in $C_k^{\mathrm{B}\mathrm{M}}(X;R)$ because a finite family of images has every point meeting only finitely many of them. For compact $X$ the inclusion is an isomorphism, as proved on this page.
• Functoriality. Proper maps induce pushforwards on locally finite chains; an arbitrary continuous map need not. Properness is a sufficient condition, not a necessary condition for every individual map. The class maps of the companion theorem use the guaranteed functoriality for proper morphisms.