  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.
