Quote by slider142
What is the topology of the set that U is a subset of? A subset is open if and only if it is an element of the topology; without a topology, or a basis for the topology, the quality of being open is undefined.

I believe it to be general for
R^{n}. But there is nothing else in the question, except before the problems it states: "Throughout, let X be a metric space with metric d."
Quote by ╔(σ_σ)╝
I don't know what else to say that would help,partly because I'm unsure of your notation.
Is U the union sign?
Anyway I assume that with the definition provided above you can show that your set is open. I guess a prove by contradiction would be appropriate, you can show that there is no point is your set that does not have an epsilon neighborhood.

U is not the union sign, U(x
_{0}, ε) = {x  d(x, x
_{0}) < ε}