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

Elementary chains and compact spaces with a small diagonal