Let $X$ be a continuous field. Let $\gamma$ be a transversal section of $X$. Then, consider $$B(\gamma)=\{\sigma_{x_0}(t): x_0 \in \gamma \ \ and \ \ t \in (-\alpha, \alpha)\},$$ where $\sigma_{x_0}(t), \, t \in (-\alpha,\alpha),$ is a solution of $x'=X(x)$ and $x(0)=x_0$. Is it true that $B(\gamma)$ is an open set?
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1$\begingroup$ Without any more regularity on $X$ than continuity, the flow is not necessarily uniquely defined. $\endgroup$– Fan ZhengSep 2, 2017 at 12:12
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$\begingroup$ Yes, but the set $B(\gamma)$ could be an open set?! $\endgroup$– Wagner RanterSep 2, 2017 at 12:36
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$\begingroup$ This question could use a more informative title. $\endgroup$– j.c.Sep 2, 2017 at 13:44
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$\begingroup$ The add is that $X(x)\neq 0$ for every $x$. $\endgroup$– Wagner RanterSep 2, 2017 at 17:24
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