Spaces having a small diagonal

Gary Gruenhage

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arXiv · math/9910153

Spaces having a small diagonal

Gary Gruenhage

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Abstract

We obtain several results and examples concerning the general question ``When must a space with a small diagonal have a G_delta-diagonal?". In particular, we show (1) every compact metrizably fibered space with a small diagonal is metrizable; (2) there are consistent examples of regular Lindelof (even hereditarily Lindelof) spaces with a small diagonal but no G_delta-diagonal; (3) every first-countable hereditarily Lindelof space with a small diagonal has a G_delta-diagonal; (4) assuming CH, every Lindelof Sigma-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 G_delta diagonal.

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Gary Gruenhage. 1999-10-27. Spaces having a small diagonal. https://arxiv.org/abs/math/9910153

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