singular homology of the two point spaces For the discrete topology, it is easy to compute the singular chain complex and its homology. But for the trival topology, I can compute $H_0$ and that's it. Does anyone has an hint how to approach the problem, how to approach that problem?

A nonempty space with the trivial topology is actually contractible. Let $X$ be such a space, choose some base point $x_0 \in X$, and define a homotopy $H$ by:
$$\begin{align}
H : X \times [0,1] & \to X \\
(x,t) & \mapsto \begin{cases}
x & \text{if } t = 0, \\
x_0 & \text{if } t > 0.
\end{cases}
\end{align}$$
This is a continuous map (simply because all maps into a trivial space are continuous), and it satisfies $H(-,0) = \operatorname{id}_X$, $H(-,1) = \operatorname{cst}_{x_0}$, and $H(x_0, t) = x_0$.
In particular the space $X$ satisfies $H_0(X) = \mathbb{Z}$, $H_i(X) = 0$ for $i > 0$.
