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

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. – Pete L. Clark Feb 15 '11 at 11:02

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 $\mathbb{R}$.
There is not just one cardinality that is "uncountably infinite;" after all, the various iterated power sets of the reals are all uncountable and infinite, but different sizes nonetheless. You should instead say that the desired cardinality here is "size continuum," or $\beth_1$, since that is what your argument is aimed at establishing. – JDH Feb 15 '11 at 5:16
@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. – Andrés E. Caicedo Feb 15 '11 at 5:57