Let $(X,\tau)$ and $(X,\tau')$ are both metrizable topological vector space, and let $(x_{n})\subset X$ and $x\in X$ and $x_{n}\rightarrow x$ in $\tau$ topology if and only if $x_{n}\rightarrow x$ in $\tau'$ topology . Prove then two topologies coincide on $X$.
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@user244115 You have asked this exact same question three times within the last three hours: http://math.stackexchange.com/questions/1367445/coincide-topologies http://math.stackexchange.com/questions/1367608/when-two-topologies-are-same please do not spam the same question across multiple posts. People will answer your questions, just give them time. – TomGrubb Jul 20 '15 at 15:35