Originally Posted by

**Defunkt** I realize that, but I still can't seem to solve it.

Given a collection of closed subsets in X that has the FIP, its image (of p) also has the FIP since p is closed. Now, since Y is compact, there exists some $\displaystyle t \in \bigcap_{T \in \mathcal{T}}T$ where $\displaystyle \mathcal{T}$ is the collection of images of p. But we may have that $\displaystyle p^{-1}(\{t\})$ contains more than one element "spread" over the collection of closed subsets in X, and then I don't see how we are sure that there is an element in the intersection of that collection.

Sorry for being a bother, I just can't seem to see how to get the desired result. I, of course, appreciate your help.