Singular homology of cofinite topology space

Question

Suppose $X$ is an infinite set equipped with cofinite topology, what are its singular homology groups? For example, $X=\mathbf{CP}^1$ equipped with Zariski topology (the cofinite topology), what are its singular homology groups? It is connected, so $H^0(X)\cong\mathbf{Z}$.

Answer 1

I adapted the answer to this question (which deals with the case of the fundamental group). The crucial part is that a map into $X$ is continuous iff the preimage of any point is closed (directly from the definition). The following works as soon as $X$ has at least the cardinality of the continuum (so in particular when $X = \mathbf{CP}^1$):

First, $X$ is path-connected: let $x,y \in X$, then $\gamma : [0,1] \to X$ is to be any injection between $[0,1]$ and $X$ such that $\gamma(0) = x$ and $\gamma(1) = y$ (these exist since $X$ has cardinality greater than $|[0,1]|$). The preimage of any point is either a singleton or empty, which are closed in $[0,1]$, hence $\gamma$ is continuous and $X$ is path-connected.

Now pick any base point $x_0 \in X$. I will show that $\pi_n(X,x_0) = 0$ for all $n > 0$, and therefore $X$ is weakly contractible and has therefore trivial homology. Let $f : S^n \to X$ be a continuous map such that $f(u_0) = x_0$ (where $u_0$ is the base point of $S^n$). Then define a homotopy $H : S^n \times [0,1] \to X$ by: $$H(u,t) = \begin{cases} f(u) & t = 0 \\ x_0 & t = 1 \text{ or } u = u_0 \\ g(u,t) & \text{otherwise} \end{cases}$$ Where $g$ is a any injection between $(S^n \setminus \{u_0\}) \times (0,1) \to X$ (again, cardinality assumption on $X$). Then an easy check shows that the preimage of any point of $X$ is a finite subset of $S^n \times [0,1]$, hence closed. Therefore $f$ is homotopic to the identity (as based maps).