Behaviour of the Lindelöf Σ-property in iterated function spaces

We prove that if Cp(X) is a Lindelöf Σ-space, then Cp,2n+1(X) is a Lindelöf Σ-space for every natural n. As a consequence, it is established that υCpCp(X) has the Lindelöf Σ-property. This answers Problem 47 of Arhangelʹskiı̆ (Recent Progress in General Topology, Elsevier Science, 1992). Another consequence is that only the following distribution of the Lindelöf Σ-property is possible in iterated function spaces: (1) Cp,n+1(X) is a Lindelöf Σ-space for every n∈ω; (2) Cp,n+1(X) is a Lindelöf Σ-space only for odd n∈ω; (3) Cp,n+1(X) is a Lindelöf Σ-space only for even n∈ω; (4) for any n∈ω the space Cp,n+1(X) is not a Lindelöf Σ-space. application of the developed technique, we prove that, if X is a Tychonoff space such that ω1 is a caliber for X and Cp(X) is a Lindelöf Σ-space, then X has a countable network. This settles Problem 69 of Arhangelʹskiı̆ (Recent Progress in General Topology, Elsevier Science, 1992).