Spaces having a small diagonal (Q1612245)

It is known that a space with a \(G_\delta\)-diagonal has a small diagonal. The question is, for what classes of spaces does small diagonal imply \(G_\delta\)-diagonal? In the present paper, the author offers several nice results and interesting examples concerning the above question. More precisely, the author shows: (1) every compact metrizably fibered space with a small diagonal is metrizable; (2) there are consistent examples of regular Lindelöf (even hereditarily Lindelöf) spaces with a small diagonal but no \(G_\delta\)-diagonal; (3) every first-countable hereditarily Lindelöf space with a small diagonal has a \(G_\delta\)-diagonal; (4) assuming CH, every Lindelöf \(\Sigma\)-space with a small diagonal has a countable network; (5) the statement ``Countably compact spaces with a small diagonal are metrizable'' is consistent with and independent of ZFC; (6) there is in ZFC a locally compact space with a small diagonal but no \(G_\delta\)-diagonal.