I've been solving problems from my Topology course, and still don't get clear how to prove certain sets are $n$-manifolds when it's not very obvious how to pick the neighbourhoods. Consider the following exercise I'm trying:
Consider the following sets: $$M_1=\{(x,y,z)\in\mathbb R^3 \mid x^2+y^2+(z-1)^2=1\}$$ $$M_{-1}=\{(x,y,z)\in\mathbb R^3 \mid x^2+y^2+(z+1)^2=1\}$$ $$M=M_1\cup M_{-1}\phantom{a}, \phantom{aa} N=M\setminus\{(0,0,0)\}$$ Determine which of them are 2-manifolds, which are surfaces and which are compact surfaces.
Of course the topological space where those sets live is $\mathbb R^3$ with the Euclidean topology.
At first, I already know in my mind how those sets are, clearly $M_1$ and $M_{-1}$ are spheres whith just the coordinates origin in common, and the form of $M$ and $N$ is deduced from that. So from that information I in fact have spoiled me that both $M_1$ and $M_{-1}$ are compact surfaces.
For $N$, I intuitively see (correct me if I'm wrong) that it's 2-manifold, but it's not surface (hence not compact surface) because it's not connected.
For $M$, I think isn't even a 2-manifold, since $(0,0,0)$ don't has a neighbourhood homeomorphic to a 2-dimensional euclidean ball.
So I can intuitively see the properties of the subspaces, but my problem is rigurously proving it. How can I prove, for example, that any point of $M_1$ has a neighbourhood homeomorphic to a 2-dimensional euclidean ball? How can I prove $(0,0,0)$ hasn't it in $M$?