An open ball is an open set
How to prove that in any metric space an epsilon-neighborhood is an open set? solution: Suppose x belongs to V(p). Then d(x, p)
Not every open set can be written as a union of countably many epsilon neighborhoods. For example, take R with the discrete topology. Then any epsilon neighborhood is either R or a singleton set and so no proper uncountable set can be written as a countable union of these. However, R with its usual metric does have this property
is that correct?