If $p: E\to B$ is a fibration, is the map $q:M_p \to B$ from the mapping cylinder of $p$ also a fibration?

I know that it is if $p$ is trivial, or locally trivial; and I know (from Strøm's "The Homotopy Category is a Homotopy Category") that the topology of $M_p$ can be slightly modified to make $q$ a fibration.

I wonder

whether the modification is necessary: does anyone know of a fibration $p$ such that $q$ is not a fibration, and

whether the modification is necessary if we assume the spaces are compactly generated (Strøm works with all spaces).

EDIT: There is a natural injective map from mapping cylinder $M_f$ of the map $f: X\to Y$ to the join $X * Y$; the topology on $M_f$ is modified so that it coincides with the subspace topology.

everywhere, then your two mapping cylinders will be homeomorphic (and also metrizable). The preprint about this is not on the arxiv yet (delayed by other sections) but I can share it. In fact, the two mapping cylinders areuniformlyhomeomorphic. So is there a nice model category structure onmetrizableuniform spaces and u.c. maps? Unfortunately, not all pushouts are metrizable, though all adjunction spaces are. I wonder if some trick can save pushouts. – Sergey Melikhov Dec 17 '10 at 2:07