1. The problem statement, all variables and given/known data What is the torus excluding a disc homeomorphic to? What is the boundary of a torus (excluding a disc)? 3. The attempt at a solution RP^2 X RP^2? As a guess.
Excluding a disk? You mean you slice a disk out of the torus? What's left is simply connected and looks to me like it is homeomorhic to a ball.
Yes, slice out a disk. A torus is a surface so it hollow? A ball is a solid. The torus still has a hole in it. How can it be homeomorphic to a ball? I'd say it is homeomorphic to a proper torous which is homeomorphic to what?
What is it homeorphic to? Infinitely many things, obviously. But I don't immediately see them as being interesting. Now, what is it homotopic to, there is an interesting question. The boundary of a torus excluding a (closed) disc is obvious, surely. What do you think happens to an object without a boundary if we remove something like a disc?