Hi, I want to find out the homology groups (over some commutative ring K) of the 2 element spaces. Up to homeomorphism, there are only

1) the discrete 2 element space

2) the trivial one

3) the intermediate one with 3 open sets

So far I only know the definiton of homology of X, i.e. we look at n simplices and form a chain complex out of continuous functions into X.

1) The discrete space is easy, as only constant functions are continuous which leads to H_0 = K^2 and H_n = 0 for n > 0

For the other 2 cases I cant figure out what to do. In the trivial topology all functions are continuous, which leads to a huge chain complex, in the intermediate topology only some are, well you see I need some help...