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.