What is the cardinality of a subset of the hyperbolic upper half plane?

Question

Given a subset of the hyperbolic upper half plane, say an ideal triangle (so with vertices on the boundary), what is the cardinality of all points contained in the interior?

Answer 1

Uncountably infinite. For example, considered as a subset of the plane, it contains the set $\{(x+t,y) | 0 \leq t \leq \epsilon\}$ for some $x,y \in \mathbb{R}$ and $\epsilon>0$. Such a set can be easily put in bijection with $\mathbb{R}$.