= Ideal boundary
{title2=$\partial_\infty\mathbb H^n\cong S^{n-1}$}
The ideal boundary of <hyperbolic space> $\mathbb H^n$ records directions in which geodesics escape to infinity. Its points can be represented by <geodesic rays>, identifying rays that stay a bounded distance apart. With the usual boundary topology it is a sphere $S^{n-1}$.
In the unit-ball model it is the surrounding unit sphere; in the upper-half-space model it is $\mathbb R^{n-1}\cup\{\infty\}$. For $n=3$ this is the <Riemann sphere>. These are ideal endpoints, not points at finite hyperbolic distance. For instance, the vertical metric element $dy/y$ makes the distance from height one to height $\varepsilon$ equal to $|\log\varepsilon|$, which diverges as $\varepsilon\to0$.
A complete hyperbolic geodesic has two distinct ideal endpoints. Hyperbolic <isometries> act on the ideal boundary, translating geometric questions about geodesics into questions about their endpoint pairs. The boundary topology and the hyperbolic metric play different roles: the closed-ball compactification adds ideal points without assigning them finite distance.
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