Is connectedness $\implies $ local connectedness? Let $X$ be a connected topological space.
Then I wonder that $X $ can be locally connected or not.
I think the title of my question is NOT correct considering a Euclidean space like $\mathbb R^n$.
Do you think so??
Also, conversely, if $X$ is a locally connected, and if some conditions are given additionally, Can $X$ be a connected space?
I think the latter case may be possible (not certainly), but don't know the necessary condition although it is true.
Please somebody tell me answers of them~~

The Topologist's Curve is connected but not locally connected. Let $I_0, I_1\subseteq \mathbb{R}$ two disjoint intervals, $I_0\neq\emptyset\neq I_1$. Then the union $I_0\cup I_1$ is locally connected, but not connected.
