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Has a Serre fibration $f:E\to B$ with $B$ a connected space isomorphic fibers over different points of $B$?

If $f$ is a fiber bundle, then all fibers are isomorphic. Hence, a possible counterexample would be a Serre fibration which is not a fiber bundle. It would be nice, if someone could provide a counterexample where $E$ and $B$ are non-pathological spaces, e.g. CW complexes.

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    $\begingroup$ By isomorphic, do you mean homeomorphic? If $X$ is a connected CW-complex, then $PX\rightarrow X$, where $PX$ is the path space, will only have homotopy equivalent fibers. $\endgroup$ – Joe Johnson 126 Dec 2 '11 at 13:28
  • $\begingroup$ Yes, I mean homeomorphic, isomorphic in the category of topological spaces. $\endgroup$ – Daniel Dreiberg Dec 2 '11 at 13:38
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No, not always. In a counterexample below most fibers are intervals, but the fiber over one point is a point.

Fibration

Fibers are always homotopy equivalent, though.

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