Show that $\displaystyle S = \{ (x,y) \in \mathbb {R}^2 : xy > 1 \} $ and $\displaystyle C = \{ x \in \mathbb {R}^n : d(x,y) < 1 \ , \ y \in B $, B being any set, are open.

I understand that for both problems I need to pove that $\displaystyle D(x, t ) $ are subsets of S and C for some positive number t. But, in the first one, I don't know how to get the y involve.

For the second one, I have w in D(x,a), then |w-y| = |w-x+x-y| < |w-x| + |x-y| = |w-x| + 1, but then... how do I get it < 1?