\(M_ 0\)-spaces are \(\mu\)-spaces (Q761650)

X is a \(\mu\)-space if X is embeddable in a countable product of \(F_{\sigma}\)-metrizable paracompact spaces. \(M_ 3\)-\(\mu\)-spaces play a central role in dimension theory. The question of which \(M_ 3\) spaces are \(\mu\)-spaces is partially answered. Theorem. Let X be an \(M_ 3\)- space with a peripherally compact \(\sigma\)-closure preserving quasi-base. Then X is embedded in the countable product of \(F_{\sigma}\)-metrizable \(M_ 3\)-spaces and is therefore a \(\mu\)-space.