C*-, C- and P-embedded subsets in products and the undecidability of a certain property on N~(ω1)

摘要

Let X = Pi(alpha is an element of omega 1) X-alpha be a product of omega(1) many separable metric spaces. It is proved that every C-embedded subset in X is C-embedded in X under a certain settheoretic assumption. Combining a result of E. Pol and R. Pol for N-2 omega, it is undecidable in ZFC that there is a (closed) C*-embedded but not C-embedded subset in N-omega 1. It is also proved in ZFC that every C-embedded subset in X is P-embedded in X. Next, we discuss when C*- or C-embedding implies P-embedding in products of generalized metrizable spaces, such as M-spaces, Sigma-spaces and semistratifiable spaces. (C) 2020 Published by Elsevier B.V.