For this question,

Prove: for each x in R and $\displaystyle \varepsilon > 0$, $\displaystyle N^*(x , \varepsilon)$ is an open set.

why does it suffice to show that the set is open by showing that every point of the set is an interior point?

EDIT: sorry if this or any of some of my future questions are trivial, i think this one is, but my professor has been hard to reach these past 2 weeks.