M-embedded subspaces of certain product spaces

A subspace Y of a space X is said to be M-embedded in X if every continuous f:Y→Z with Z metrizable extends to a continuous function .For topological spaces Xi (i∈I) and J⊆I, set XJ:=∏i∈JXi.The authors prove a general theorem concerning κ-box topologies and pseudo-(α,κ)-compact spaces, of which the following is a corollary of the special case κ=α=ω.Theorem – If Y⊆XI and πJ[Y]=XJ for all ∅≠J∈[I]<ω+, and if each XJ, for ∅≠J∈[I]<ω, is Lindelöf, then Y is M-embedded in XI.Remark – Several results in Chapter 10 of the book [W.W. Comfort, S. Negrepontis, Chain Conditions in Topology, Cambridge Tracts in Math., vol. 79, Cambridge Univ. Press, 1982] depend on Lemma 10.1, of which the given proof was incomplete. A principal contribution here is to furnish a correct proof, allowing the present authors to verify and unify all the results from Chapter 10 whose status had become questionable, and to extend several of these.