Homology of $\mathbb{CP}^{\infty}$

Question

I know that $H_q(\mathbb{CP}^{n}) = \begin{cases} \mathbb{Z} & q \equiv 0 \hspace{0.2cm} & 0 \leq q \leq 2n \\ 0 \end{cases}$

Which follows from the construction, i.e. adding just one cell in the even dimensions.

I was wondering, what about the infinity case $H_q(\mathbb{CP}^{\infty})$? Since I'm unfamiliar with the construction, I don't know whether those constructions could be replicated.

Any help or reference would be appreciated.

Answer 1

You can still use $CW-$homology for infinite dimensional $CW-$complexes to compute homology. In particular this would imply that $H_{2i}(\mathbb C P ^{\infty}) = \mathbb Z$ and $H_{2i+1}(\mathbb C P ^{\infty}) = 0$ for $i \geq 0$ since $\mathbb C P ^{\infty}$ has one cell in each even dimension and zero cells in every odd dimension.