# Relationship between Convergence and Open sets

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: math.stackexchange.com/questions/69174/… – Mathemagician1234 Jan 29 '12 at 8:17
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 8:30

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.
It 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