Theorem. Let $\phi:X\rightarrow Y$ be a quasi-isometry between two (Gromov) hyperbolic spaces $X$ and $Y$. If $X$ and $Y$ are proper, then ϕ induces a homeomorphism between their boundaries.


The proof of the above statement is well-written in Bridson and Haefliger's book.


My question is that `can we drop the condition that $X$ and $Y$ are proper?'. In some papers about boundaries of hyperbolic spaces, the authors usually say that the above theorem is true without mentioning that $X$ and $Y$ are proper. If you know the answer or any references, then let me know.



Induced homeomorphism from a quasi-isometry between hyperbolic spaces
Properness is already needed to have a well-defined boundary at infinity, i.e., with a topology not depending on the chosen base point. This is Proposition III.3.7 in Bridson-Haefliger, which builds on some previous lemmata that are applications of the Arzela-Ascoli theorem. To apply the Arzela-Ascoli theorem one needs bounded sets to be compact.


