A note on Saalfrank’s generalizatio of absolute retract

Abstract

In a recent paper [4], Saalfrank has defined the concept of absolute homotopy retract for compact Hausdorff spaces. By using a technique of Hanner [2] which avoids the necessity of a universal imbedding space, we generalize Saalfrank's results to include Lindelof and fully normal spaces. Finally, an indication is given for the further generalization to include the metric cases. All spaces are Hausdorff and mapping will always mean continuous function. Let Q denote either the fully normal, Lindel6f or compact class. DEFINITION. X is an absolute homotopy retract relative to the class Q{AHR(Q) } if XGCQ and every homeomorph of X which is a subset of a Q space is a homotopy retract [4] of that space. DEFINITION. X is a homotopy extension space relative to the class Q HES(Q) } if each mapping f of a closed subset B of a Q space Y into X has a homotopy extensionl [4] to Y. Since all mappings into a contractible space are inessential, we have the following