On Locally Connected and Related Sets

Abstract

In our investigations on the fixed point problem' we came across a noteworthy type of sets belonging to a class later investigated at length by K. Borsuk2 under the name of retracts. We considered in various places in our book certain generalized locally connected sets and indicated in outline their bearing on the fixed point problem. Our chief object in the present paper is to study more completely the mutual relations of those two classes of sets, and incidentally to exhibit more clearly their relation to the fixed point problem. Let A be a compact metric space. We call a complex K on A consisting partly of abstract, partly of singular cells a semi-singular complex K on A (a more precise definition will be found in No. 3). We shall show that all absolute neighborhood retracts A are intrinsically characterized by the following property (a sort of uniform local connectedness): for every e there is a a such that every finite semi-singular complex K on A of mesh p being zero. This enables us to complete a result of Topology regarding the fixed point problem. Our treatment rests upon the following noteworthy result: given a closed subset A of a Hilbert parallelotope A, we can construct an infinite complex K and a deformation II of & into K such that I1 = 1 on A and ll(t A) = K.