Originally Posted by

**Showcase_22** 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) $\displaystyle e^{x+y^2-z}-1-7x^2-7y^2+7z^3

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

\delta=\frac{1}{2} \left(e^{x+y^2-z}-1-7x^2-7y^2+7z^3 \right)

$ and centre (x,y,z), (the $\displaystyle B_{\delta}(x,y,z)$ that you wrote) then this will be an open ball that is always a subset of U.

Is there something wrong with my idea?