Singular homology satisfies dimension and arbitrary additivity

Given: The objects and hypotheses in the statement above.

A CW pair is $(X,A)$ with $A$ a CW subcomplex of $X$, as in def-skeleta-cw-subcomplex-and-relative-cw-complex. Morphisms are all continuous maps of pairs, not just cellular maps. An ordinary unreduced homology theory assigns covariant functors $h_n$ from CW pairs to abelian groups, for every $n\in \mathbb{Z}$, and natural homomorphisms $\mathrm{\partial }:h_n(X,A)\to h_{n-1}(A)$, where $h_n(X)=h_n(X,\mathrm{\varnothing })$, satisfying: - Homotopic maps of pairs induce equal homomorphisms. - The inclusion maps and $\mathrm{\partial }$ form an exact sequence $\cdots \to h_n(A)\to h_n(X)\to h_n(X,A)\to h_{n-1}(A)\to \cdots$. - For CW subcomplexes $U,V$ of $X=U\cup V$, inclusion induces $h_n(U,U\cap V)\cong h_n(X,V)$. - For a point $\ast$, $h_n(\ast )=0$ when $n\mathrm{\ne }0$; write $G=h_0(\ast )$. - For every set-indexed family of CW pairs, including the empty family, the inclusions induce ${⨁}_{\alpha }{h}_n(X_{\alpha },A_{\alpha })\cong h_n({⨆}_{\alpha }{X}_{\alpha },{⨆}_{\alpha }{A}_{\alpha })$. Thus $h_n(\mathrm{\varnothing })=0$. No finite-dimensionality or finite-cell restriction is implicit. (Unreduced homology theory on cw pairs)

For every fixed abelian group $G$, singular homology $H_n(-,-;G)$, extended by zero in negative degrees, satisfies homotopy invariance, pair exactness, naturality of the connecting maps, and CW excision in def-unreduced-homology-theory-on-cw-pairs. (Singular homology satisfies homotopy exactness and excision)

For a topological space $X$ and an abelian group $G$, the singular chain groups and boundary maps of def-singular-boundary-operator form the singular chain complex $C_{\bullet }(X;G):=(\cdots \to C_n(X;G)\stackrel{{\mathrm{\partial }}_n}{\to }{C}_{n-1}(X;G)\to \cdots ),$ because thm-the-singular-boundary-squares-to-zero gives ${\mathrm{\partial }}_{n-1}{\mathrm{\partial }}_n=0$. Its degree-$n$ cycles and boundaries are $Z_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(X;G):=\ker {\mathrm{\partial }}_n,B_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(X;G):=\mathrm{im}{\mathrm{\partial }}_{n+1},$ in the sense of def-cycle-and-boundary-subobjects-of-a-complex. The $n$th singular homology group is the homology object of this chain complex: $H_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(X;G):=Z_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(X;G)/B_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(X;G),$ equivalently $H_n(C_{\bullet }(X;G))$ in the notation of def-homology-object-of-a-chain-complex. When the coefficient group is $\mathbb{Z}$, write simply $C_n(X)$ and $H_n(X)$ when no confusion can arise. (The singular chain complex and singular homology)

Let $X={⨆}_{\alpha \in A}{X}_{\alpha }$ be a disjoint union of topological spaces, and let $G$ be an abelian group. Then for every $n\ge 0$, $H_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(X;G)\cong {⨁}_{\alpha \in A}{H}_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(X_{\alpha };G)\mathrm{.}$ (The singular homology of a disjoint union is the direct sum)