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).
 A: 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:

A: Near the village where I grew up, Wookey Hole, there is a cave system called (rather unimaginatively) the Wookey Hole Cave. The areas mapped so far look like this:

The part of the cave between sump 12 on the left and chamber 25 on the right has not been mapped but we know the two side are connected because dye poured into Swildon's Hole emerges at the resurgence.
Anyhow the existence of this cave system means the Earth is not topologically equivalent to a 2 sphere since a circle drawn around either cave entrance cannot be contracted to a point, not that I have tried this since the authorities take a dim view of people painting circles around the cave entrance.
A: If you're considering the Earth as a 2-manifold from the beginning, yeah, one can deform continuously any closed simple curve into a point. If you are really taking something as similar as possible to earth surface with it's imperfections, then no, we can't, because it's not smooth.
The Earth would be topologically equivalent to a sphere, by being a compact smooth manifold with no boundary and no holes. But it's more like a revolution ellipsoid (Ref: https://en.wikipedia.org/wiki/Figure_of_the_Earth). It has non-zero quadrupole gravitational moment, which would be zero for a uniform sphere.
