Sphere homeomorphic to plane?

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?

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They are not homeomorphic, but if you remove one point from the sphere, it becomes homeomorphic to the plane. – Tunococ May 30 '14 at 3:13

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.
How about: Being locally homeomorphic to a plane says nothing about how flat an object locally appears. This is instead measured by curvature. For a sphere of radius $r$, its curvature is given by $1/r^2$. For a large radius, this is close to zero and makes the surface of the sphere seem flat. – RghtHndSd May 30 '14 at 3:29