## Zero-dimensional space

Hey there, I´m having trouble with this problem:

A topological space X is zero-dimensional if it has a basis consisting on both open and closed sets.
Prove that if X is zero-dimensional then every point of X has a (local) basis of open sets with empty boundary.

My quiestion is, is it enough to prove that every set that´s both open and closed has empty boundary? Or does the fact that I´m working with a basis for the topology has something to do?

Thanks!