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Assume that $(G,+)$ is an Abelian topological group (maybe locally compact, if necessary) and assume that $V$ is an open connected neighbourhood of zero. Does there exist an open "convex" neighborhood of zer0 $W$ such that $W \subset V$?

A subset $A$ of an Abelian group $G$ we call convex if for each $x\in G$ the condition $2x\in A+A$ implies that $x \in A$. (It is a generalization of notion of convex set in a real linear space).


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up vote 2 down vote accepted

If $V=G$ then $G$ is such a neighborhood of zero.

If $(G,+)$ is the $\mathbb{Z}/2\mathbb{Z}$ with the discrete topology and $V = \{\operatorname{zero}\}$,
then there is no such neighborhood of zero.
(Since $[1]+[1] = \operatorname{zero} \;$ and $\; [1]\not\in V \:$ .)

Therefore whether or not there is such a neighborhood
of zero depends on things which you did not specify.

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Thanks. I have not experience in topological groups. If I assume that group is "locally convex", that is it has a basis in zero consisting of convex open subsets, then the answer will be true (assumption that $V$ is connected in this case is unnecessary). Maybe you know another additional conditions on $G$ or on $V$ under which my question has positive answer. – Richard Aug 26 '11 at 10:46
If the topology is generated by a family of semi-norms that satisfy $||x||_i \leq ||x+x||_i$ for all $x$ and $i$, then the group is locally convex. – Ricky Demer Aug 26 '11 at 22:35

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