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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?

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    You might as well just ask: what is the cardinality of an open subset of $\mathbb{R}^2$? This is more general, and makes clear that there is not really any hyperbolic geometry in the question.2011-02-15

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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 $\mathbb{R}$.

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    @anon: Sure, whatever $\aleph_\epsilon$ the number $\beth_1$ happens to be. The issue is that the usual axioms of set theory do not suffice to answer the question of what this is. The continuum hypothesis says that $\epsilon=1$. Other assumptions give different values.2011-02-15