In a recent paper [4] , Saalfrank has defined the concept of absolute homotopy retract for compact Hausdorff spaces. By using a technique of H anner [2] which avoids the necessity of a universal imbedding space, we generalize Saalfrank's results to include Lindelöf 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,

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... elöf or compact class. Definition. X is an absolute homotopy retract relative to the class Q{ AHR(Q)} if XEQ 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{ HES02)} if each mapping / of a closed subset B of a Q space Y into X has a homotopy extension [4] to Y. Since all mappings into a contractible space are inessential, we have the following Lemma. If X is contractible, then each mapping of a subset B of Y into X has a homotopy extension to Y. Theorem. The following are equivalent if XEQ. (a) X is an AHR(0. (b) X is an HES(0. (c) X is contractible. Proof. (a)=>(b). Suppose that (Y, B) is a Q pair and/is a mapping of B into X. Let Z be the identification space obtained from XKJ Y by identifying yEB with f(y)EX. Take j:X-+Z to be given by j(x)=x and ¿: F->Z to be given by k(y)=f(y) if yEB and k(y)=y otherwise. A set GEZ is defined to be open if and only if j_I(G) and ¿_1(G) are open in X and Y, respectively. k\(Y -B) is a homeomorphism onto Z -X; hence X is closed in Z. Hanner [2] has shown that Z is a Q space for the Q classes under consideration whenever X and Y are Q spaces. Then (Z, X) is a Q pair; therefore, there exists r: Z-+X continuously with r| X~z. Since rk: Y-*X and rk\B = r/~/, F=rk is the desired homotopy extension of/.