I was letting my mind wander when I came up with this interesting topological figure:
It's a torus with a hole punched through its side towards its center. I then tried to determine whether or not this figure is "topologically equivalent" (homeomorphic) to a double torus. I suspect that they aren't, but how do I prove this? Do I need to know a lot of topology to prove something like this?
To clarify: The torus is solid and the hole does not lead to the actual interior of the torus; it does have a "skin" connecting the surfaces of the torus on either side of the hole. For example, if you had a point next to the hole on the "outer ring" of the torus and a point next to a hole on the "inner ring", you could connect the two with a path going through the hole without going over the top or bottom of the torus.
(As I mentioned in a comment I think the picture is ambiguous. Below is one interpretation based on the OP's comment to a deleted answer that not all of you might be able to see.)
A punctured torus is homotopy equivalent to what is called a bouquet of two circles (two circles meeting at a point): basically the idea is to keep making the hole bigger and bigger. This is not homotopy equivalent to a genus $2$ surface (and hence a punctured torus is not homotopy equivalent, and so not homeomorphic, to a genus $2$ surface); the two can be distinguished most simply by their Euler characteristic, although it takes some effort to prove that the Euler characteristic is a homotopy invariant depending on how it is defined.