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Let $X$ be a connected topological space. Then it can be locally connected. At the other hand, a locally connected topological space can be connected. There is no implication in eather of the cases.

Is there a result (theorem), which makes, under some conditions, the one or the other presumable implication work, and under stronger conditions maybe the equivalence of both ?

Many thanks.

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    $\begingroup$ Thanks. But $(0,1)\cup(1,2)$ is open in $\mathbb{R}$ and locally connected but not connected. $\endgroup$
    – user249018
    Feb 10 '18 at 15:30
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    $\begingroup$ @CheerfulParsnip I think you are mixing up path-connectedness with local connectedness. $\endgroup$
    – Aloizio Macedo
    Feb 10 '18 at 17:06
  • $\begingroup$ Nothing very interesting along these lines, but you can say for instance that for subsets of ${\mathbb R}$, connected implies locally connected. $\endgroup$ Feb 10 '18 at 17:39
  • $\begingroup$ Oops. Deleted my comment. $\endgroup$ Feb 10 '18 at 18:37
  • $\begingroup$ @MoisheCohen. But not for subsets of $\Bbb R^2$. $\endgroup$ Feb 10 '18 at 23:19

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