Elementary chains and compact spaces with a small diagonal

Abstract

It is a well known open problem if, in ZFC, each compact space with a small diagonal is metrizable. We explore properties of compact spaces with a small diagonal using elementary chains of submodels. We prove that ccc subspaces of such spaces have countable \pi-weight. We generalize a result of Gruenhage about spaces which are metrizably fibered. Finally we discover that if there is a Luzin set of reals, then every compact space with a small diagonal will have many points of countable character

Reflecting Lindelöf and converging ω1-sequences

“…In fact it was the space X of Example 3.2 that was the motivation for Proposition 2.4 of [2]. The space X has copies of Cantor sets and ω 1 -sequences that co-countably converge to these.…”

“…While we do not know the answer to this question, let us remark that the space constructed in Example 3.2 does not have a small diagonal. In fact it was the space X of Example 3.2 that was the motivation for Proposition 2.4 of [2]. The space X has copies of Cantor sets and ω 1 -sequences that co-countably converge to these.…”

Measures and fibers

“…Metrizably-and finitely-fibered spaces were considered in the context of spaces with a small diagonal (see e.g. [Gru02], [DH12]) and of Rosenthal compacta (see e.g. [KM11]).…”