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In the book Open Problems in Topology by Jan Van Mill and George M. Reed, the following problem was presented: 108. Is there a para- Lindelof Dowker space? Recall that a para-Lindelof Dowker space has a locally countable open refinement, satisfies Axiom T4, and is not countably paracompact. Some results on this problem are in http://topology.auburn.edu/tp/reprints/v11/tp11203.pdf, where it is shown that the conditions are preserved under perfect mappings. What is the status of this problem? Any references are appreciated.

flag	asked Aug 6 at 22:24 Jaivir Baweja 104●8

	What does it mean for a space to have "a locally countable open refinement"$\:$? $\;\;$ – Ricky Demer Aug 6 at 22:33
	@Ricky If a space has a locally countable open refinement, then it locally has the same cardinality of the natural numbers in a refinement $V$ (a cover such that for every $v\inV$ there exists $u\inU$ such that $V\subsetU$). – Jaivir Baweja Aug 6 at 22:43
	What is U?....... – Steven Landsburg Aug 6 at 23:17
	.....and what does it mean for two sets to have "locally the same cardinality" ("in a refinement" or otherwise)? – Steven Landsburg Aug 6 at 23:20
	Steven's first question is a good question. $\;\;$ Whatever the answer to that is, what you have in parentheses seems to be equivalent to $\: V\subseteq U \:$. $\;\;\;\;$ – Ricky Demer Aug 6 at 23:40

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Ramiro de la Vega · Accepted Answer · 2012-08-08 18:23:31Z UTC

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It is now problem 502 in "Open problems in topology II", I would guess it is still open.

answered Aug 8 at 18:23

Ramiro de la Vega
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Concerning Problem 108 in “Open Problems in Topology”

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