APPROXIMATE POLYHEDRA AND SHAPE THEORY

In recent years a number of generalizations of the notion of absolute neighborhood retract (ANR) have appeared. Among these are approximate absolute neighbor hood retracts (AANR) due to M. H. Clapp [4] and the equiva lent notion of NE-sets due to K. Borsuk [3]. S. Mardesic [11] has recently given a generalization called approximate polyhedra (AP) which applies to topological spaces. Mardesic's AP's agree with AANR's in the compact metric case so for brevity we use his notation throughout this paper. Although the AP's form a much larger class than ANR's, Clapp showed that they possess many of the fixed point properties of ANR's. Moreover, he showed that the AP's form a quite natural class in the sense that they are precisely the limits of polyhedra in the metric of con tinuity