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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, where it is shown that the conditions are preserved under perfect mappings. What is the status of this problem? Any references are appreciated.

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What does it mean for a space to have "a locally countable open refinement"$\:$? $\;\;$ – Ricky Demer Aug 6 '12 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 '12 at 22:43
What is U?....... – Steven Landsburg Aug 6 '12 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 '12 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 '12 at 23:40
up vote 4 down vote accepted

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

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