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Let $X$ be a Hausdorff locally convex topological vector space whose topology is generated by a family of continuous seminorms on $X$. For each continuous seminorm $p$ on $X$, let $$U_p=\{x\in X: p(x)\le 1\}.$$ Well, I know that $U_p$ is convex, absorbing and balanced.

Question. Is $U_p$ closed? If yes, can you please explain. I need some help on this.

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I've been making silly mistakes today, so correct me if I'm wrong, but isn't the complement of $U_p$ the set $\{ x \in X : p(x) > 1 \} = p^{-1}(1, + \infty)$, which is open by the continuity of $p$?

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    $\begingroup$ Yeah, but actually I noticed (after I posted my question) that$U_p=p^{-1}([0,1])$ and hence a closed set beccause $p$ is continuous.:) $\endgroup$ – Juniven Aug 9 '13 at 17:29

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