Two topologies coincide if they have the same convergent nets

Question

There are many similar questions to mine in the site, but I'm still not sure. Let $X$ be a vector space with $T_1$ and $T_2$ two topologies that make $X$ a TVS (Hausdorff). If I want to show that $T_1=T_2$, does it suffice to show that every converegent net $(x_{\lambda})_{\lambda}$ in $X$ to some $x\in X$ w.r.t. $T_1$ converges to $x$ w.r.t. $T_2$ and conversely?

Thank you

Answer 1

More generally: in any topological space, $U$ is an open set iff for every net $x_\lambda$ converging to $x \in U$, $x_\lambda \in U$ eventually. So if $T_1$ and $T_2$ agree with respect to convergence of nets, they have the same open sets and are the same topology.

Answer 2

Notice that closed sets are characterized by convergence of nets so topologies also