Are the sets $\{(x, y): x^2 + y^2 < 1\}$ and $\{(x, y): 0 < x^2 + y^2 < 1\}$ homeomorphic?

Question

Are the sets $A = \{(x, y): x^2 + y^2 < 1\}$ and $B = \{(x, y): 0 < x^2 + y^2 < 1\}$ homeomorphic?

My guess is that they are not, but I do not know how to prove this. I have studied some topology only in the context of metric spaces, and I know that connectedness and compactness are topological properties, but I don't know if that applies here.

Answer 1

No, they are not. The set $A$ has this property: for every compact subset $K$ of $A$, there is a compact subset $K^*$ of $A$ such that $K\subset K^*\subset A$ and that $A\setminus K^*$ is connected. But this doesn't hold for $B$; take$$K=\left\{(x,y)\in\mathbb{R}^2\,\middle|\,x^2+y^2=\frac14\right\}.$$

Answer 2

$A$ is a simply connected space and $B$ is not...