\(D\)-property, monotone monolithicity and function spaces
In the papers [\textit{O. T. Alas, V. V. Tkachuk} and \textit{R. G. Wilson}, Acta Math. Hungar. 125(4), 369--385 (2009)]; [\textit{V. V. Tkachuk}, Topology Appl. 156, No. 4, 840--846 (2009; Zbl 1165.54009); ibid. 159, No. 6, 1529--1535 (2012; Zbl 1245.54022)] several interesting questions, related to monotonically monolithic spaces and spaces having the Collins-Roscoe property, were raised. In order to answer these questions, the authors prove the following interesting results in this paper. 1. \(C_p(C_p(X))\) is monotonically monolithic if and only if \(X\) is so. 2. If \(X\) is a \(\Sigma_{\kappa}\)-product of a family of spaces with countable network weight, then \(C_p(X)\) is monotonically monolithic. 3. If \(X\) is the product of Lindelöf \(\Sigma\)-spaces, then \(C_p(X)\) is monotonically monolithic. 4. If \(C_p(X)\) is monotonically monolithic and \(Y\) is a cozero subset of \(X\), then \(C_p(Y)\) is monotonically monolithic. 5. Every compact \(\omega\)-monolithic space is monotonically monolithic. 6. If \(X\) is monotonically \(< \kappa\)-monolithic with \(\kappa \geq nw(X)\), then \(X\) is hereditarily D. 7. If \(X\) is pseudocompact and \(nw(X) \leq \omega_1\), then \(C_p(X)\) is hereditarily D. This is a difficult, but highly interesting paper for all researchers interested in \(C_p\)-theory.
monotonically \(\kappa\)-monolithic space
monotonically \(< \kappa\)-monolithic space
scientific article; zbMATH DE number 6092983
scientific article