Research Article

Functionally countable subspaces of nice products

Published in: Quaestiones Mathematicae
Volume 48 , issue 8 : Entrepreneurship and Africa’s Cultural Context , 2024 , pages: 1153–1164
DOI: 10.2989/16073606.2025.2467230
Author(s): Vladimir V. Tkachuk Universidad Autónoma Metropolitana, México,
Keywords: Primary: 54C35, Secondary: 54C25, 54H11, 54G12, Scattered space, Lindelöf space, P-space, Kurepa hypothesis, dense subspace, lc-scattered space, functionally countable space,
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Abstract

We establish that, for any Tychonoff space X of weight ≤ ω 1, if Xδ is the Gδ -modification of X , then Cp (Xδ ) has a dense functionally countable subspace. Furthermore, the space has a dense functionally countable subspace of cardinality ω 2 if and only if Kurepa Hypothesis holds. We also provide a consistent example of a scattered Lindelöf P-space X such that Cp (X) has no dense lc-scattered sub-space. If there exists a strongly inaccessible cardinal, then in some model of ZFC, (a) there is a scattered Lindelöf P-space X for which Cp (X) has no dense functionally countable subspace and (b) for any uncountable cardinal κ > ω 1 , if Iκ is the Gδ -modification of the Tychonoff cube , then Cp (Iκ ) has no dense functionally countable subspace. Our results give consistent answers to several published open questions.
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