3
$\begingroup$

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

$\endgroup$
6
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Thank you. Now I feel confident :) $\endgroup$ – Kiko Feb 17 '17 at 18:36
3
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.