Hey there, Im 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 thats both open and closed has empty boundary? Or does the fact that Im working with a basis for the topology has something to do?

Thanks!