The Complex plane, set of all $z=x+iy$ where $x$ and $y$ are real, surface area equals cross product of $x$ and $y$ equals aleph-something (that's not the question). Projecting the plane onto a sphere, and adding the complex infinity to the set (without a value for $\arg(z)$), gives the Riemann sphere, which, being a sphere, is compact.
Does the addition of infinity, as some sort of bounder, make the complex plane compact? is the Riemann sphere without infinity compact, since it has an open boundary? also, if the hyperbolic plane can be shown in a disk, what does that mean? The answer to this question might primarily clarify the definitions in my new studies of topology of manifolds, etc. thanks.