Extension property and ANR-systems

The Kuratowski notation YτK means that every map g :A→K defined on a closed subset A of a space Y can be extended to a map ḡ :Y→K. We define the extension property YτS, where S is an ANR-system. We investigate some properties of the relation YτS. In particular, the sum theorem is proved for the class of spaces Y such that YτS. The relation YτS can be used for the definition of the counterpart of the Dranishnikov extension dimension.