Unreduced homology theory on cw pairs

A CW pair is $(X,A)$ with $A$ a CW subcomplex of $X$, as in Skeleta, CW subcomplexes, and relative CW complexes. 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.