I am trying to get started with differential geometry, and am having a difficult time wrapping my head around the concept of a manifold. One thing that would make it easier to understand would be if authors would give more examples of things that aren't manifolds. Anyway, here's one potential example that I came up with that I think will help me along quite a bit if I can understand it.
So, let's say we're in $\Bbb R^3$ and we have the unit sphere $x^2+y^2+z^2=1$. Even I can tell that this is a manifold. But now let's take the disc $z=0, x^2+y^2<2$, cut a hole out of it at $x^2+y^2<1$ and "attach" it to our sphere. Now, is the resulting object a manifold? Why or why not? What happens if we take our disc again and cut out the hole $x^2+y^2 \leq 1$?
I can't see how this object violates any explicit part of the definition of a manifold, but it just doesn't seem right.
Thanks for helping me get some sleep again.