I've got this homework and have been looking at the trival and discrete topology for the two point space for now. But if it gets a bit more complex, it is not so easy to describe the continuous functions and deduce the singular chain complex and homotopy. For example, even looking at the Sierpinski space, I only know that $r^{-1}({a})$ has to open in the n-th simplex for a continuous function $r$, but that does not help me a lot. Any tips how to approach this with simple as methods (i.e. no homotopy invariance) ?
