Two norms $\|.\|_1$ , $\|.\|_2$ defined on a normed linear space N are such that N forms a Banach space with respect to both the norms.Then $||.||_1$ and $||.||_2$ are equivalent if and only if $\tau_{||.||_1}$ $\subseteq$ $\tau_{||.||_2}$ where $\tau_{||.||_1}$ and $\tau_{||.||_2}$ are topologies induced by $||.||_1$ and $||.||_2$ respectively.
I presume that this could be a consequence of OPEN MAPPPING THEOREM, but I am having problem in showing the incomparability of the topologies.
[$\Rightarrow$] Let $(N,\|.\|_1)$ be equivalent to $(N,\|.\|_2)$. Then it follows that the identity map $I: (N,\|.\|_1) \rightarrow (N,\|.\|_2)$ is a homeomorphism: it clearly is linear and bijective, and the condition of equivalent norms is equivalent to $I$ and $I^{-1}$ being bounded. Consequently, $$I(\tau_{\|.\|_1}) \subseteq \tau_{\|.\|_2}.$$
[$\Leftarrow$] Let $\tau_{||.||_1} \subseteq \tau_{||.||_2}$. Then, $I^{-1}$ as defined above continuous and therefore, bounded. By the open mapping theorem, it is a homeomorphism, which implies that $I$ is bounded.