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.
This is a great question. You don't say whether you're thinking of topological manifolds or of smooth manifolds, but it doesn't really matter.
Let's reduce the dimension by $1$, and consider the unit circle $x^2+y^2=1$, together with a whisker, $y=0$, $x\ge 1$. Does the point $(1,0)$ have a neighborhood that is homeomorphic (diffeomorphic) to an interval $I=\{|t|<\epsilon\}\subset\Bbb R$? Suppose you had such a homemorphism, and suppose it sends $(1,0)$ to $0\in I$. Can you see an elementary reason that this is impossible?
The point-set topology won't quite work in higher dimensions, and you need to use a bit more. But in the world of smooth manifolds, it's easy to see things won't work, using linear algebra.