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Lindelöf σ-property in Cp(X) and p(Cp(X)) = ω do not imply countable network weight in X

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We prove that there are Tychonoff spaces X for which p(Cp(X)) =ϖ and Cp(X) is a Lindelöf Σ-space while the network weight of X is uncountable. This answers Problem 75 from [4]. An example of a space Y is given such that p(Y)=ϖ and Cp(Y) is a Lindelöf Σ-space, while the network weight of Y is uncountable. This gives a negative answer to Problem 73 from [4]. For a space X with one non-isolated point a necessary and sufficient condition in terms of the topology on X is given for Cp(X) to have countable point-finite cellularity.

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Okunev, O.G., Tkachuk, V.V. Lindelöf σ-property in Cp(X) and p(Cp(X)) = ω do not imply countable network weight in X . Acta Mathematica Hungarica 90, 119–132 (2001). https://doi.org/10.1023/A:1006796010091

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