Any ordinary homology theory has a cellular chain complex on a cw pair

Given: The objects and hypotheses in the statement above.

Let $h$ be an ordinary theory with coefficient group $G=h_0(\ast )$. For every $n\ge 0$ and $k\in \mathbb{Z}$, $h_k(D^n,S^{n-1})\cong \{\begin{array}{ll}G& k=n,\\ 0& k\mathrm{\ne }n\mathrm{.}\end{array}$ At $n=0$ the pair means $(\ast ,\mathrm{\varnothing })$. Also ${\tilde{h}}_k(S^n)\cong G$ for $k=n$ and zero otherwise, including $n=0$. Choose the disk identifications by ordered orientations and iterated cone boundaries, so they commute with these boundaries. (Any ordinary homology theory computes relative cell groups from its coefficient group)

Ordinary unreduced theories on CW pairs and reduced ordinary theories on based CW spaces with vertex basepoints determine one another, naturally and compatibly with morphisms and coefficients. For $A\mathrm{\ne }\mathrm{\varnothing }$ the correspondence gives $h_n(X,A)\cong {\tilde{h}}_n(X\mathrm{/}A)$; for $A=\mathrm{\varnothing }$ it gives $h_n(X)\cong {\tilde{h}}_n(X_{+})$, where $X_{+}=X\bigsqcup \{\ast \}$. Under this correspondence, pair boundaries are cofiber boundaries followed by inverse suspension, and arbitrary disjoint-sum additivity corresponds to arbitrary wedge additivity. For a CW triple $B\subset A\subset X$ there is a natural exact sequence $\cdots \to h_n(A,B)\to h_n(X,B)\to h_n(X,A)\to h_{n-1}(A,B)\to \cdots$, whose last map is the pair boundary followed by $h_{n-1}(A)\to h_{n-1}(A,B)$. (Unreduced pair and reduced quotient axioms are equivalent on cw pairs)

Write $X^n$ for the union of cells of dimension at most $n$, with $X^{-1}=\mathrm{\varnothing }$. A CW subcomplex $A\subseteq X$ is a union of open cells such that, whenever $A$ contains an open cell $e$, it contains the whole closure $\bar{e}$. A relative CW complex $(X,A)$ is formed from the subcomplex $A$ by attaching cells in stages; thus $A\hookrightarrow X$ is a cellular inclusion. (Skeleta, CW subcomplexes, and relative CW complexes)