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Hereditarily monotonically Sokolov spaces have countable network weight. (English) Zbl 1429.54022

Let \(X\) be a Tychonoff space and \(C_{p}(X)\) the set of all real-valued continuous functions on \(X\) equipped with the pointwise convergence topology. A space \(X\) is called monotonically retractable if we can assign to any countable subset \(A\) of \(X\) a continuous retraction \(r_{A}:X\rightarrow X\) and a countable network \(\mathcal{N}(A)\) of \(X\) modulo \(r_{A}\) such that \(A\subset r_{A}(X)\) and the assignment \(\mathcal{N}\) is \(\omega\)-continuous. A space \(X\) is monotonically Sokolov if we can assign to any countable family \(\mathcal{F}\) of closed subsets of \(X\) a continuous retraction \(r_{\mathcal{F}}:X\rightarrow X\) and a countable external network \(\mathcal{N}(\mathcal{F})\) of \(r_{\mathcal{F}}(X)\) in \(X\) such that \(r_{\mathcal{F}}(F)\subset F\) for each \(F\in \mathcal{F}\) and the assignment \(\mathcal{N}\) is \(\omega\)-continuous.
In the paper under review, the authors provide an example of a monotonically Sokolov and monotonically retractable space admitting a continuous image which is neither monotonically Sokolov nor monotonocally retractable. They prove that: (1) hereditarily monotonically Sokolov and hereditarily monotonically retractable spaces are cosmic spaces. (2) The Sokolov property and the monotone Sokolov property are preserved by Lindelöf \(\Sigma\)-subspaces. (3) For the Alexandrov double \(AD(X)\) of a space \(X\), the space \(C_p(AD(X))\) is Lindelöf \(\Sigma\) if and only if \(X\) and \(C_p(X)\) are Lindelöf \(\Sigma\).
In the last section of the paper, the authors prove that if \(X\) is an \(L\Sigma(\leq c)\)-space and \(Y\) is a Lindelöf \(\Sigma\)-subspace of \(C_p(X)\) then \(C_{p,n}(Y)\) is an \(L\Sigma(\leq \omega)\)-space for any \(n\in \omega\). They also prove that if \(X\) is a Lindelöf \(\Sigma\)-space and \(C_p(X)\) is an \(L\Sigma(\leq c)\)-space, then \(X\) is an \(L\Sigma(\leq \omega)\)-space.

MSC:

54C15 Retraction
54C35 Function spaces in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54E20 Stratifiable spaces, cosmic spaces, etc.
54D30 Compactness
54G20 Counterexamples in general topology
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