If you show that convergence of nets in a topological vector space $V$ with topology $\tau$ is equivalent to convergence of nets in a topological vector space $V$ With topology $\sigma$, does it necessarily follow that $\tau = \sigma$?
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A related question that may help in addition to the responses here illuminate your question: http://math.stackexchange.com/questions/69174/if-you-know-the-convergent-sequences-how-do-you-know-the-open-sets – Mathemagician1234 Jan 29 '12 at 08:17
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BTW-I'm going to fix my deleted answer below and repost. I still think it was instructive as it stood,but clearly it confused some people as it stood,so I'm going to try and get it into a form that's "acceptable" to them. That may not be possible,but I will try.......... – Mathemagician1234 Jan 29 '12 at 08:30
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Yes.
You may find this easier to prove by showing that $\sigma$ and $\tau$ have the same closed sets, since closed sets are easier to describe in terms of nets than open sets are.
Nate Eldredge
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2It may also be easier to realize that it is true in general for topological spaces, not just for topological vector spaces. If $A$ is a subset of a topological space $X$, then $A$ is closed if and only no net in $A$ converges to a point in $X\setminus A$. See for example Theorem 2 in Chapter 2 on page 66 of Kelley's Topology. – Jonas Meyer Jan 28 '12 at 22:39