On questions of strict contractibility

A space X is strictly contractible to a point x0∈X if there exists a homotopy H :X×[0,1]→X starting at the identity such that H(x,t)=x0 if and only if either x=x0 or t=1. Michael proved that if E is a locally compact and non-compact space then E×0 is a perfect retract of the product E×[0,1) if and only if the one-point compactification E∗=E∪x∗ of E is strictly contractible to x∗. We answer some questions posed by Michael [Closed retracts and perfect retracts, Topology Appl., to appear] and we characterize strictly contractible ANRs in shape-theoretic terms.