Topology induced by norm What is the meaning of topology induced by norm.
To me topology is a collection of subsets satisfying certain rules.
How can a norm induce a topology...?
For example how can $\|\cdot\|_{2}$ induce a topology on $\mathbb{R}^{2}$ ?
 A: Norms induce metrics, and metrics induce topologies.
More specifically, the basic open sets are $B_r(x)=\{y: \|x-y\|<r\}$ for some real number $r>0$.
You could look closely and note that this is the standard topology in the case of $\Bbb R^2$ and $\|\cdot\|_2$. And it is a nontrivial result that every norm on $\Bbb R^2$ will induce the same topology (this is generally true for finite dimensional spaces).
However in the case of infinite dimensional spaces, different norms will general result in different topologies. For example $\ell_2$ and $\ell^\infty$ are both isomorphic as vector spaces (both have the same dimension), but one norm defines a separable metric space and another defines a non-separable metric space.
A: A norm very clearly induces a metric $d(x,y)=||x-y||$ (you can check that this satisfies the definition of a metric based on the axioms for a norm). Then the norm topology is just the metric topology. If you like, this is the topology with the basis of all open balls in this metric.
