Elementary chains and compact spaces with a small diagonal (Q441367)

A~space~\(X\) is said to have a~small diagonal, if every uncountable subset of~\(X^2\) that is disjoint from the diagonal has an uncountable subset whose closure is disjoint from the diagonal. By a~csD-space is meant a~compact Hausdorff space with a~small diagonal. The authors explore properties of csD-spaces using elementary chains of substructures. They prove that ccc subspaces of csD-spaces have countable \(\pi\)-weight, they generalize a~result of Gruenhage about spaces which are metrizably fibered, and they prove that if there is a~Luzin set of reals, then every csD-space has many points of countable character. The main question on csD-spaces, namely whether they are metrizable, remains open.