# Concerning Problem 108 in “Open Problems in Topology”

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.

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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