 Homology of a directed union

Suppose $X={\bigcup }_a{X}_a$ 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 $C_{\ast }(X)={\underrightarrow{\lim }}_a{C}_{\ast }(X_a)$. Exactness of filtered colimits gives