Let $p:{X}\rightarrow{Y}$ be a quotient map. Show that if $X$ is locally connected then $Y$ is locally connected.
To do this we consider a component $C$ of the open set $U$ of $Y$. We should show that $p^{-1}(C)$ is a union of components of $p^{-1}(U)$.
Anybody have any ideas about how to show that $p^{-1}(C)$ is a union of components of $p^{-1}(U)$?