Conditions which ensure that a simple map does not raise dimension

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.