Abstract
We obtain several results and examples concerning the general question “When must a space with a small diagonal have a G δ -diagonal?”. In particular, we show (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 δ -diagonal; (3) every first-countable hereditarily Lindelöf space with a small diagonal has a G δ -diagonal; (4) assuming CH, every Lindelöf Σ-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 δ -diagonal.
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