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Sep8-09, 09:47 PM
P: 92
Reading Analysis on Manifolds by Munkres

Quote Quote by slider142 View Post
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 Rn. 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 Quote by ╔(σ_σ)╝ View Post
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(x0, ε) = {x | d(x, x0) < ε}