I'm trying to show that my set is a union of open balls so it's open. I'm not sure if it's the easiest way of doing it, but I think it might work.

I'm afraid that my question does not say. Is it possible to prove it for every metric? I not, it probably means Euclidean.

I was thinking that for every point (x,y,z)

will produce a single number. If I create a ball of radius

and centre (x,y,z), (the

that you wrote) then this will be an open ball that is always a subset of U.

Is there something wrong with my idea?