Retraction of spaces with a \(\sigma\)-almost locally finite base (Q788309)

Let \({\mathcal P}\) denote the class of all spaces with a \(\sigma\)-almost locally finite base. (These spaces are \(M_ 1\)-spaces but the converse remains open.) For each \(X\in {\mathcal P}\) the author constructs a space Z(X)\(\in {\mathcal P}\) such that X is (homomorphic to) a subspace of Z(X) and X is an A\(R({\mathcal P})\) \((ANR({\mathcal P}))\) iff X is a retract (nbhd retract) of Z(X).