Prove that the topologies being same is equivalent to convergence with respect to nets being same I am trying to solve the following problem.
If $\mathcal{T}_1,\mathcal{T}_2$ are two topologies on a set $X$, prove that the following two statements are equivalent:
a) $\mathcal{T}_1=\mathcal{T}_2$
b) If $\{x_i\}_{i \in I}$ is a net in $E$ and $x \in X$, then
$$
x_i \rightarrow x \mbox{ with respect to } \mathcal{T}_1 \Leftrightarrow x_i \rightarrow x \mbox{ with respect to } \mathcal{T}_2
$$
I could prove a) $\Rightarrow$  b). How do I prove the other way? Can anyone give useful hints?

Suppose b). To see a), note it suffices to see that the closed sets coincide. Can you prove (using the convergence of nets property) that $E \subset X$ is closed in $T_1$ if and only if it is closed in $T_2$?
