A version of Zabrodsky’s lemma

Abstract:

Zabrodsky’s Lemma says: Suppose given a fibrant space $Y$ and a homotopy fiber sequence $F\to E\to X$ with $X$ connected. If the map $Y=\operatorname {map} (*,Y)\to \operatorname {map} (F,Y)$ which is induced by $F\to *$ is a weak equivalence, then $\operatorname {map} (X,Y)\to \operatorname {map} (E,Y)$ is a weak equivalence. This has been generalized by Bousfield. We improve on Bousfield’s generalization and give some applications.