Is a path connected version of this question Equivalence relation on topological space such that each equivalence class and the quotient space is connected true ? That is let $X$ be a topological space and $\sim$ be an equivalence relation on $X$ such that each equivalence class is path connected (as a subspace of $X$) and the quotient space $X/\sim$ is also path connected. Then is $X$ path connected ?
Definitely $X$ is connected. If $X$ is not path connected in general, then does some good additional condition forces $X$ to be path connected ?