Homology of a directed union ($H_i({\bigcup }_a{X}_a)\cong {\underrightarrow{\lim }}_a{H}_i(X_a)$)

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

Every open subset $U\subseteq {\mathbb{R}}^N$ is the directed union of finite unions of closed rational cubes contained in $U$. Every compact subset of $U$ lies in one such finite polyhedron, whose homology is finitely generated. The homology of a directed union therefore expresses each $H_i(U;\mathbb{Z})$ as a direct limit over a countable family of countable groups, so it is countable.