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)

If $\|\cdot\|$ is a norm on $X$, there is the induced metric $(x,y)\mapsto \|x-y\|$, which defines a topology with the collection of all sets of the form $$ \{y\in X : \|x-y\| < r\},$$ (with $r>0$) as a basis. A subset of $X$ is open with respect to this norm off it can be written as a union of basis elements.
If two norms generate the same topology, then the open sets will be the same regardless of which norm we are using.
