Let $M$ be an $n$-dimensional topological manifold, and let $(U_k)_{k \in \mathbb{N}}$ be an increasing sequence of open sets $U_k \subset M$ such that for each $k \in \mathbb{N}$, $U_k$ is homeomorphic to $\mathbb{R}^n$. Is it necessarily the case that $\,U\!:=\bigcup_{k=1}^\infty U_k\,$ is homeomorphic to $\mathbb{R}^n$?
If so, does anyone know any nice reference for this fact (either as a theorem/lemma/etc. or as an exercise)?