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 $\tau = \sigma$?
Relationship between Convergence and Open sets
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general-topology
convergence
topological-vector-spaces
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0A 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 – 2012-01-29
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0BTW-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.......... – 2012-01-29