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~~

Neither connected nor locally connected implies the other, nor do their negations.  Four examples:
$\Bbb R$ is connected and locally connected.
$[1,2] \cup [3,4]$ is locally connected but not connected
The topologist's sine curve is connected but not locally connected.
$\Bbb Q$ is neither
