A metrizability theorem for compact spaces
by
Vladimir Tkachuk
Universidad Autonoma Metropolitana de Mexico
Coauthors: B. Cascales, J. Orihuela

Given a space M, a family of subsets A of a space X is ordered by M if A={A_K:K is a compact subset of M} and K ⊂ L implies A_K ⊂ A_L. We study the class M of spaces which have compact covers ordered by a second countable space. We prove that a space C_p(X) belongs to M if and only if it is a Lindelöf S-space. Under MA(w₁), if X is compact and (X×X)\D has a compact cover ordered by a Polish space then X is metrizable; here D = {(x, x):x ∈ X} is the diagonal of the space X. Besides, if X is a compact space of countable tightness and X²\D belongs to M then X is metrizable in ZFC.

We also consider the class M^* of spaces X which have a compact cover F ordered by a second countable space with the additional property that, for every compact set P ⊂ X there exists F ∈ F with P ⊂ F. It is a ZFC result that if X is a compact space and (X×X)\D belongs to M^* then X is metrizable. We also establish that, under CH, if X is compact and C_p(X) belongs to M^* then X is countable.

Date received: January 12, 2010