I'm trying to teach myself differential geometry from the internet, and I've hit a snag in proving homeomorphisms. First, show that [tex]\Re^n[/tex] is homeomorphic to any open ball in [tex]\Re^n[/tex]. (I'm not sure how to write the conventional "R" using Latex.) I'm trying to prove this statement, but I am having issues proving the continuity of the map and its inverse. It's clear that there exists a bijective map because the cardinality of both the open ball and the space itself are the same, but how should I go about proving continuity? The next problem is to prove that a sphere with a point removed is homeomorphic to [tex]\Re^2[/tex]. I have absolutely no idea how to do this. If someone could show me how to do this without skipping any steps, I would appreciate it.