Let $X$ be a topological space and $\sim$ be an equivalence relation on $X$ such that the quotient space $X/\sim$ is connected and each equivalence class of $\sim$ is connected (as a subspace of $X$ ). Then how to prove that $X$ is connected ?
I was trying to go by contradiction assuming $X$ has a non-trivial cl-open subset say $A$ and then trying to get a non-trivial cl-open subset of $X/\sim$ from that, but with no luck. Please help.
Suppose that $O$ is non-empty clopen in $X$.
Note that for any $[x] \in X/\sim$, if $x \in O$, then $[x] \subseteq O$ by connectedness of the set $[x]$ (it intersects $O$ (in $x$) so cannot intersect $X\setminus O$ or it would disconnect the set $[x]$. Similarly reasoning for $X\setminus O$, we get that $x \in X\setminus O$ implies $[x] \subseteq X\setminus O$.
The above reasoning in fact shows that $q^{-1}[q[O]] = O$, where $q(x) = [x]$ is the standard quotient map (the left to right inclusion of it, in fact).
This shows, by the definition of the quotient topology that $q[O]$ is clopen as well. So $q[O] = X/\sim$ by connectedness and so $O = X$ and we're done.