M-embedded subspaces of certain product spaces (Q952627)

Let \(\mathbf{M}\) be a class of metric spaces and \(X\) be a subspace of a space \(Y\). Then \(X\) is said to be \(\mathbf{M}\)-embedded in Y if for every continuous \(f:X\to Z\) with \(Z\in\mathbf{M}\) there exists a continuous function \(g:Y\to Z\) such that \(g|X=f\). The authors of the paper under review prove a general theorem concerning \(\mathbf{M}\)-embeddings, \(\kappa\)-box topologies and pseudo-\((\alpha,\kappa)\)-compact spaces. In the special case \(\kappa=\alpha=\omega\) the result reads as follows: If \(Y\subseteq \prod_{i\in I}X_i\) and \(\pi_J[Y]=\prod_{i\in J}X_i\) for all \(\emptyset\not =J\in[I]^{<\omega^+}\) and each \(\prod_{i\in I}X_i\) for \(\emptyset\not =J\in[I]^{<\omega^+}\) is Lindelöf then \(Y\) is \(\mathbf{M}\)-embedded in \(\prod_{i\in I}X_i\) . The authors give a correct proof of the of lemma 10.1 of the joint book [Chain conditions in topology. Cambridge Tracts in Mathematics, 79. Cambridge etc.: Cambridge University Press (1982; Zbl 0488.54002)] by the first author and \textit{S. Negrepontis}. Several interesting consequences of this are presented here. The paper contains also an extension of a Corollary 10.7(a) of the book cited above. In particular the authors present some conditions under which the product of topologically complete spaces is the Dieudonné completion of a subspace.