Sphere homeomorphic to plane?

Question

I just took a course in general topology about a month back, and I was wondering whether it was possible to explain why the Earth seems flat from our point of view but is in fact a sphere using the concept of a homeomorphism? Is it the fact that the sphere and plane are homeomorphic to each other the reason for this?

Answer 1

The sphere and the plane are not homeomorphic to each other; the sphere is compact, the plane isn't.

The sphere is locally homeomorphic to the plane. That is, for each $p \in S^2$, there is an open neighbourhood $U$ of $p$ such that $U$ is homeomorphic to $\mathbb{R}^2$.

The sphere is an example of a two-dimensional topological manifold, often called a (topological) surface. In general, a topological space which is Hausdorff, second countable, such that each point has a neighbourhood homeomorphic to $\mathbb{R}^n$ is called an $n$-dimensional topological manifold.