The compact neighborhood extension property and local equi-connectedness

Abstract

Abstract. It is shown that any cr-compact metrizable space is an AR (ANR) if and only if it is (locally) equi-connected and has the compact (neighborhood) extension property. IntroductionIn this paper, all spaces are metrizable unless otherwise stated. An AR (or ANR) means an absolute retract (or absolute neighborhood retract). A space X is an AR or ANR if and only if X is an AE (= absolute extensor) or ANE (= absolute neighborhood extensor) for metrizable spaces, respectively. It is said that a space X has the compact extension property (CEP) [K] provided for any space Y and any compactum A in Y, each map /: A -> X can be extended over Y. If /: A -> X can be extended over a neighborhood of A in Y, we say that X has the compact neighborhood extension property (CENP). As is easily observed, X has the CEP if X is contractible and has the CNEP. Since any compactum can be embedded in a compact AR (e.g., the Hilbert cube), it follows that X has the CNEP (or the CEP) if and only if X is an ANE (or AE) for compacta. Hence every compactum with the CNEP (or the CEP) is an ANR (or an AR). J. van Mill [vM] constructed a separable metrizable space which has the CEP but is not an ANR. In [CM], this example was improved to be completely metrizable by using an example of Edwards, Walsh, and Dranishnikov [E, Wa, Dr] of dimension-raising cell-like maps. In the light of the example of van Mill, the following problem arises:Problem [We, Problem ANR 13]. Is every cr-compact space X with the CEP an ANR?