Open Set and Interior Point
- Yes, because if $u\in U$ then, since $U$ is open, then, by definition of open set, there is a $\varepsilon>0$ such that $(u-\varepsilon,u+\varepsilon)\subset U$. And this means that $u$ is an interior point of $U$.
- No. Take $U=[-1,1]$. Then $U$ is not open (because there is no $\varepsilon>0$ such that $(1-\varepsilon,1+\varepsilon)\subset U$), but $0$ is an interior point of $U$.
(1) Yes; (2) No.
It is true in general that a set is open if and only if it contains only interior points.