A pullback theorem for locally-equiconnected spaces (Q1077755)

A space X is locally equiconnected (L.E.C.) if the inclusion of the diagonal \(\Delta\) X in \(X\times X\) is a cofibration. This paper contains the following result. Given any fibration \(p: E\to B\) and map \(f: X\to B\) where E, B and X are all LEC, then the pullback space \(X\varsubsetneq E=\{(x,e):\quad f(x)=p(e)\}\) is also LEC. This can be applied to construct for fibrations with path connected fibres, translation functions between fibres that are base point preserving [c.f. the author, Pac. J. Math. 117, 267-289 (1985; Zbl 0571.55002)].