Norms That Define An Open Set How can a norm define a set in a vector space. I don't understand for example how 2 different norms can define a same open set. 
It's not intuitive to me.
An open set doesn't need a norm to be open (or to even exist)

With a norm you have a metric and thus a notion of distance. From this you can create balls. B(x,r) is the set of all vectors distance less than r from x. B(x,r)=B(x,s) with (WLOG) r
