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)?
Yes, this is correct. You could prove it as a corollary of the annulus theorem (sketch of a sketch: at each stage the closure $\overline U_n$ is an open $n$-ball, and you're attaching an annulus, and the limit of this procedure is $\Bbb R^n$), but this wasn't known in all dimensions until the 80s. An early proof of this theorem was given in 1961 by Morton Brown here; it appears to use a modification of the above idea and fact, though I have not at all read it in detail. It appears that this was been known slightly earlier as a corollary of Mazur's proof of the topological Schoenflies theorem.
