Homology of a directed union
= Homology of a directed union
{title2=$H_i(\bigcup_aX_a)\cong\varinjlim_aH_i(X_a)$}
Suppose $X=\bigcup_aX_a$ is directed by inclusion and every compact subset of $X$ lies in some $X_a$. Every finite <singular chain> then lies in one $X_a$, so $C_*(X)=\varinjlim_aC_*(X_a)$. Exactness of <filtered colimits> gives
$$
H_i(X)\cong\varinjlim_aH_i(X_a).
$$