algebraic topology - hatcher 3.3 exercise 17 The following is a question from Hatcher's "Algebraic Topology" 
“show that homology commutes with direct limits. “
I have tried to solve this problem  but I can’t .

Here is another easy counterexample: take $S^1$ and consider $\{S_i\subset S^1:S_i\  \text {is countable}\}.$ Then, $S_i$ is totally disconnected, so $H_1(S_i)=0$. And  $\varinjlim S_i=S$ but $H_1(S)=\mathbb Z$.
