More about spaces with a small diagonal

Alan Dow¹ ; Oleg Pavlov¹

¹ Department of Mathematics UNC-Charlotte 9201 University City Blvd. Charlotte, NC 28223-0001, U.S.A.

Fundamenta Mathematicae, Tome 191 (2006) no. 1, pp. 67-80

Résumé

Hušek defines a space $X$ to have a small diagonal if each uncountable subset of $ X^2$ disjoint from the diagonal has an uncountable subset whose closure is disjoint from the diagonal. Hušek proved that a compact space of weight $\omega_1$ which has a small diagonal will be metrizable, but it remains an open problem to determine if the weight restriction is necessary. It has been shown to be consistent that each compact space with a small diagonal is metrizable; in particular, Juhász and Szentmiklóssy proved that this holds in models of CH. In the present paper we prove that this also follows from the Proper Forcing Axiom (PFA). We furthermore present two (consistent) examples of countably compact non-metrizable spaces with small diagonal, one of which maps perfectly onto~$\omega_1$.

Détail

DOI : 10.4064/fm191-1-5

Keywords: defines space have small diagonal each uncountable subset disjoint diagonal has uncountable subset whose closure disjoint diagonal proved compact space weight omega which has small diagonal metrizable remains problem determine weight restriction necessary has shown consistent each compact space small diagonal metrizable particular juh szentmikl ssy proved holds models present paper prove follows proper forcing axiom pfa furthermore present consistent examples countably compact non metrizable spaces small diagonal which maps perfectly omega