Small diagonal of X and calibers of C_p(X)

摘要

Given a Lindelof space X and a regular uncountable cardinal kappa, we establish that kappa is a caliber of C-p(X) if and only if the diagonal of the space X is kappa-small. We show that the Lindelof property of X cannot be omitted in this result because there exists a pseudocompact space X with a G(delta)-diagonal such that wi is not a caliber of C-p(X).We also construct an example of a pseudocompact space X such that omega(1) is a caliber of C-p(X) but not a caliber of C-p(beta X) = C-p(upsilon X); this solves two published open questions. (C) 2018 Elsevier B.V. All rights reserved.