A generalization of the Levin–Rubin–Schapiro Factorization Theorem

Given a map f:X→Y of compact Hausdorff spaces, the Mardešić Factorization Theorem provides us a factorization f=qj, j:X→Z, q:Z→Y through a compact Hausdorff space Z with and weight of Z being at most weight of Y. The theorem has been generalized several times in various contexts with the Levin–Rubin–Schapiro Factorization Theorem being one of the most notable developments.This paper introduces a new generalization in which the factoring space Z inherits the extension property for every map in the spirit of the Levin–Rubin–Schapiro Factorization Theorem. Such inheritance of extension properties is expressed by a new notion of extensional equivalence. Furthermore, we study the impact of such a generalization on the extension relation between maps, fτi.