Is there a subset of $\mathbb{R}^2$ that is bounded but not compact? How to find a subset of $\mathbb{R}^2$ that is bounded but not compact. Does the open disk satisfies this? 

The open unit disk
$$D=\{(x,y):x^2+y^2<1\}$$
is such a set. Indeed, it is covered by the open disks
$$U_n=\{(x,y):x^2+y^2<(1-1/n)^2\},$$
of radius $1-1/n$, but there is no finite subcollection of $\{U_n\}_{n=1}^\infty$ which covers $D$.
