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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).

Thanks.

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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.

  • 0
    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.2011-08-26
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    If the topology is generated by a [family](http://en.wikipedia.org/wiki/Locally_convex#Seminorms) of [semi-norms](http://en.wikipedia.org/wiki/Absolute_convergence#Background) that satisfy $||x||_i \leq ||x+x||_i$ for all $x$ and $i$, then the group is locally convex.2011-08-26