Let $ X $ and $ E $ be topological spaces, and let $ p\colon E\to X $ be continuous. We can think of the data $ p\colon E\to F $ as an $ X $-indexed family of "fibers" $ p^{-1}(\{x\}) $, one for each $ x\in X $, "glued together" by the topology of $ X $.
Set theoretically the set $ E $ is isomorphic to the disjoint union $ \coprod_{x\in X}p^{-1}(\{x\}) $. However, it is almost never the case that $ E $ as a space is isomorphic to the disjoint union of the $ p^{-1}(\{x\}) $.
I am wondering under what circumstances $ E $ is homeomorphic to the topological disjoint union of its fibers. It is sufficient for this to happen that $ X $ is discrete?