# How is local connectedness related to connectedness

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.

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