Munkres 3.25 exercise 8 says:

Let $\displaystyle p:{X}\rightarrow{Y}$ be a quotient map. Show that if $\displaystyle X$ is locally connected then $\displaystyle Y$ is locally connected.

To do this we consider a component $\displaystyle C$ of the open set $\displaystyle U$ of $\displaystyle Y$. We should show that $\displaystyle p^{-1}(C)$ is a union of components of $\displaystyle p^{-1}(U)$.

Anybody have any ideas about how to show that $\displaystyle p^{-1}(C)$ is a union of components of $\displaystyle p^{-1}(U)$?