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?
What is the cardinality of a subset of the hyperbolic upper half plane?
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set-theory
hyperbolic-geometry
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1You 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
1 Answers
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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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0@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