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)
 A: Are you talking about open balls or open sets?  If $|| \cdot ||$ is a norm on a vector space $V$, then an open ball in $V$ (with respect to the norm $|| \cdot ||$) is a set of the form $$ \{ v \in V : ||v - v_0|| < r \}$$ for a fixed $v_0 \in V, 0 < r \in \mathbb{R}$.  An open set (with respect to $|| \cdot ||$) is a set which is equal to a union of open balls in the sense of $|| \cdot ||$.  
If $V$ is a finite dimensional real vector space, then $V$ can be identified with a finite product of $\mathbb{R}$s, which gives $V$ a topology of open sets (product topology).   As you say, these open sets are defined independently of any norm.
But it is a standard result any two norms $|| \cdot ||_1, ||\cdot ||_2$ on such a space $V$ are equivalent, in the sense that any open set in the sense of $|| \cdot ||_1$ is also an open set in the sense of $|| \cdot ||_2$, and vice versa.  In fact, the open sets from either of these norms are the exactly the same as the open sets of $V$ from the product topology.
This is not saying that open balls in the sense of one norm are also opens ball in the sense of the other.  Rather, any set $S \subseteq V$ which is equal to a union of various open balls in the sense of $|| \cdot ||_1$, can also be expressed as a union of other various open balls in the sense of $|| \cdot ||_2$, and vice versa.
A: 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.
A: 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
