Let $A$ be a nonempty open connected subset of a (real) topological vector space $X$.
Question. Is it true that there exists a nonempty open connected set $B\subseteq A$ such that $B$, in addition, is symmetric (i.e., $B=-B$)?
2026-03-26 14:19:27.1774534767
Existence of symmetric subsets
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$(0,1)$ is a nonempty open connected subset of $\mathbb R$ and it does not contain any nonempty symmetric set.