CONDITIONS WHICH ENSURE THAT A SIMPLE MAP DOES NOT RAISE DIMENSION BY

The present paper deals with those continuous maps from compacta into metric spaces which assume each value at most twice. Such maps are called here, after Borsuk and Molski (1958) and as in our previous paper (1990), simple. We investigate the possibility of decomposing a simple map into essential and elementary factors, and the so-called splitting property of simple maps which raise dimension. The aim is to get insight into the structure of those compacta which have the property that simple maps from them do not raise dimension. In what follows a map means a continuous map, unless explicitly stated otherwise. A space is, except in some general lemmas, understood to be metrizable. A compactum means a compact metric space. 1. Outline of the problem. If f: X → Y is a simple map from a compactum, then the dimension of f(X) can exceed that of X by at most 1. This follows from the formula dim f(X) − dimX ≤ k − 1, established by Hurewicz (1933) for maps from compacta assuming each value at most at k points. There are compacta no simple map from which raises dimension. For instance, as proved by Hurewicz (1933), this property is enjoyed by any compactum satisfying the following condition: (α) subcompacta of full dimension have non-empty interiors. This condition is rather restrictive. It is satisfied by manifolds, in par-ticular by intervals of the reals. If a one-dimensional compactum satisfies (α), its non-degenerate subcon-tinua cannot be nowhere dense. If it is connected, it is hereditarily locall