Countability of the homology of an open Euclidean subset
= Countability of the homology of an open Euclidean subset
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.