TITLE: Are any two open connected subsets of $\Bbb{R}^n$ homeomorphic?
QUESTION [7 upvotes]: Are any two open, connected subsets of $\Bbb{R}^n$ homeomorphic?  This seems intuitively true.

REPLY [19 votes]: 1 $\neq$ 0 $\neq$ 8
(Think of these as being open subsets of the plane. The set consists of the area that is colored black. Remove the infinitely thin boundary)

REPLY [7 votes]: When $n>1$, an open annulus 
$$\{x\in\mathbb{R}^n\mid a<|x|<b\},\quad 0<a<b$$
and an open ball
$$\{x\in\mathbb{R}^n\mid |x|<r\},\quad 0<r$$
are open connected subsets of $\mathbb{R}^n$ that are not homeomorphic.
When $n=1$ or $0$, it is true that any open connected subsets of $\mathbb{R}^n$ are homeomorphic. This is because any open connected subset of $\mathbb{R}$ is an open interval, and $\mathbb{R}^0$ is just a point.

REPLY [6 votes]: Nope ! For example $\rm B(0, 2)$ is not homeomorphic to $\rm B(0,2) \backslash \rm \overline{B(0,1)}$.