Extraordinary dimension theories generated by complexes

We study the extraordinary dimension function dimL introduced by Ščepin. An axiomatic characterization of this dimension function is obtained. We also introduce inductive dimensions indL and IndL and prove that for separable metrizable spaces all three coincide. Several results such as characterization of dimL in terms of partitions and in terms of mappings into n-dimensional cubes are presented. We also prove the converse of the Dranishnikov–Uspenskij theorem on dimension-raising maps.