A proof of the Edwards-Walsh resolution theorem without Edwards-Walsh CW-complexes

In the paper titled “Bockstein basis and resolution theorems in extension theory” (Tonić, 2010 [10]), we stated a theorem that we claimed to be a generalization of the Edwards-Walsh resolution theorem. The goal of this note is to show that the main theorem from Tonić (2010) [10] is in fact equivalent to the Edwards-Walsh resolution theorem, and also that it can be proven without using Edwards-Walsh complexes. We conclude that the Edwards-Walsh resolution theorem can be proven without using Edwards-Walsh complexes.