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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 at 3:13

1 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.

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This certainly answers the question that was directly asked, but it may implicitly suggest that the reason the earth seems flat is because it is locally homeomorphic to the plane. It would be great if you could mention curvature. –  RghtHndSd May 30 at 3:20
    
@rghthndsd: I'm not sure what you expect me to say about curvature. –  Michael Albanese May 30 at 3:23
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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 at 3:29

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