### A lattice of conditions on topological spaces II

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Borsuk's quasi-equivalence relation on the class of all compacta is considered. The open problem concerning transitivity of this relation is solved in the negative. Namely, three continua X, Y and Z lying in ℝ³ are constructed such that X is quasi-equivalent to Y and Y is quasi-equivalent to Z, while X is not quasi-equivalent to Z.

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...