Assume that $V$ is a vector space and two norms $||.||$ and $||.||'$ are defined on $V$. We say $||.||$ and $||.||'$ are equivalent if there exist $M,m \gt 0$ such that:
$\forall x \in V \space\space m||x||\le ||x||' \le M||x||$
Assume that $V$ is an arbitrary vector space (not necessarily of finite dimension) and $||.||$ and $||.||'$ are two equivalent norms on it.
Prove that $A \subseteq V$ is open with respect to $||.||$ iff its open with respect to $||.||'$. Finally, Conclude that the topology induced by $||.||$ is the same as the one induced by $||.||'$.
I don't know what a topology is and how to show that a set is open. Our teacher didn't define these two and assumed that we've learned them in other courses. So, I can't even get started. Any idea? help?