Imposing some reasonable conditions on our spaces (I think semilocally-simply-connected ought to do), one works through

**Exercise 1** $\mathbb{Z}[X]$, the free topological $\mathbb{Z}$-module continuously generated by a convenient space $X$ is an $E^\infty$ space; the maps $\mathbb{Z}[X] \to \mathbb{Z}[Y]$ induced by $ X \to Y \to \mathbb{Z}[Y]$ make this construction continuously functorial; these induced maps are again $E^\infty$ maps.

**Exercise 2** a weak homotopy equivalence of spaces $X \simeq X'$ induces a weak homotopy equivalence of $\mathbb{Z}$-modules.

**Exercise 3** For a cofibration $X \to Y$, there is a pullback square
$$ \begin{array}{c} \mathbb{Z}[X] & \to & \mathbb{Z}[Y] \\ \downarrow & & \downarrow \\ \mathbb{Z} & \to & \mathbb{Z}[Y/X] \end{array}$$

**Exercise 4** $\pi_0 \mathbb{Z}[*] \simeq \mathbb{Z}$; otherwise $\pi_k \mathbb{Z}[*] \simeq 0$.

**Exercise 5** the functor $X\mapsto \mathbb{Z}[X]$ preserves colimits of  sequences of cofibrations.

**Corollary** We have verified that the functors $\pi_k \mathbb{Z}[X]$ satisfy the Eilenberg-Steenrod axioms for ordinary homology.

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Again using the natural map $\mathbb{Z}[X] \to \mathbb{Z}$; write $\tilde{\mathbb{Z}}[X]$ for its kernel. To complete the exercises,  მამუკა ჯიბლაძე's cogent remark explains why the natural map $SP^\infty X \to \tilde{\mathbb{Z}}[X]$ is an equivalence for connected $X$s.

But yes, it really is magical!