O ct 1 99 9 SPACES HAVING A SMALL DIAGONAL
نویسندگان
چکیده
We obtain several results and examples concerning the general question “When must a space with a small diagonal have a Gδ-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 Gδ-diagonal; (3) every first-countable hereditarily Lindelöf space with a small diagonal has a Gδ-diagonal; (4) assuming CH, every Lindelöf Σ-space with a small diagonal has a countable network; (5) whether countably compact spaces with a small diagonal are metrizable depends on your set theory; (6) there is a locally compact space with a small diagonal but no Gδ diagonal.
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