Abstract
Given a Lindelöf space X and a regular uncountable cardinal κ, we establish that κ is a caliber of Cp(X) if and only if the diagonal of the space X is κ-small. We show that the Lindelöf property of X cannot be omitted in this result because there exists a pseudocompact space X with a Gδ-diagonal such that ω1 is not a caliber of Cp(X). We also construct an example of a pseudocompact space X such that ω1 is a caliber of Cp(X) but not a caliber of Cp(βX)=Cp(υX); this solves two published open questions.
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