Geometry of the Earth Is this true that every simple closed curve on the earth can be deformed continuously to a point without leaving the earth?
Is the earth compact?
Now if we consider the earth as a 2-manifold, can we say that the earth is a sphere by the classification of 2-manifolds? 
What is the reason?
Is there a practical way to detect the above properties for the earth?
Edit: With the assumption that the earth is simply connected (without hole), the earth must be a sphere or a plane (up to the compactedness of the earth).

We typically take the Earth to be a 2-manifold.
If we want to be as truthful as possible, it's best modeled as a 3 dimensional shape, upon which we can define a metric which lets us define a 2-manifold.  For all intents and purposes, that assumption is pretty good.
However, sometimes the most straightforward answer is a counterxample:

