LOCALLY CONNECTED GENERALIZED ABSOLUTE NEIGHBORHOOD RETRACTSl

ing generalization of absolute neighborhood retracts. Definition. A compact subset X of a metric space Y is an approximative neighborhood retract of Y in the sense of Noguchi if there is a neighborhood U of X in Y such that for every positive real number E there is a map r from U into X such that d(r\X,idX) < E [Gml, p. 61], [Gm2, p. 9], [Gr, pp. 16-17], [Fi, p. 42]. Definition. A compact metric space X is an approxima tive absoZute neighborhood retract in the sense of Noguchi (AAN~) provided it is an approximative neighborhood retract in the sense of Noguchi of any metric space in which it can be embedded. A further generalization which allows the neighborhood to vary with £ as well as the map r was given in 1970 by Michael Clapp. These spaces are called approximative absolute neighborhood retracts in the sense of CZapp (AANR) [Cl, p.C 118], [Fi, p. 43]. The importance of these generalizations is that they satisfy certain fixed point prope~ties which are well known for the case of ANR's. Such properties are as follows: 1Th"1S paper appears as a part of the author's Ph.D. dis sertation done under Professor Leonard R. Rubin at the Uni versity of Oklahoma