Research Article
Functionally countable subspaces of nice products
DOI:
10.2989/16073606.2025.2467230
Keywords:
Primary: 54C35, Secondary: 54C25, 54H11, 54G12, Scattered space, Lindelöf space, P-space, Kurepa hypothesis, dense subspace, lc-scattered space, functionally countable space,
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.
Get new issue alerts for Quaestiones Mathematicae
