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(repost from the topology Q&A board)

I have a (T_1), Normal, countably paracompact space X. I would like to know if every locally countable open cover of X (i.e. an open cover such that every x in X has a neighbourhood which intersects only countably many members of the cover) has a locally finite refinement.

My suspicion is that the answer is a resounding no, but every time I try to construct a counterexample it starts to seem more plausible.

If the answer does turn out to be yes I'd love to know if it generalises from aleph_0 to arbitrary cardinals.

flag	asked Mar 29 2010 at 23:40 David R. MacIver 539●2●10

Henno Brandsma · Accepted Answer · 2010-03-30 06:09:05Z UTC

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In Caryn Navy's thesis under Mary Ellen Rudin she constructed several spaces that are normal, countably paracompact and paralindelöf (every cover has a locally countable refinement) but not paracompact. All such spaces provide counterexamples (we can refine a cover without a locally finite refinement to a locally countable one and then we cannot continue...)

I'm not sure (as I do not have access to the PhD-thesis in question, and I only know it from references like http://www1.elsevier.com/homepage/sac/opit/10/article.pdf) whether these examples are all under extra set-theoretic assumptions (like MA + non-CH) or whether there are absolute ones as well.

answered Mar 30 2010 at 6:09

Henno Brandsma
2,216●6●18

Perfect, thanks. Even if there are only examples under extra hypotheses it serves to nicely close down several lines of reasoning that I was pursuing into dead ends (and actually is sufficiently strong sounding that one of the counterexamples might suffice to close down the entire conjecture, which would be interesting). I've (hopefully - details were difficult to find) dropped Caryn an email to see if she has a copy of the thesis she can send me. If she does, want me to forward you a copy? – David R. MacIver Mar 30 2010 at 8:12

Yes, I would be interested. I have always liked these covering properties. Hopefully a digital copy or a published paper of these results exists. – Henno Brandsma Mar 30 2010 at 13:52

ok. Will let you know if I manage to track down a copy. As far as I can tell this was never published - I saw reference to an "upcoming" paper, but it doesn't seem to have ever come. I did find a reference in the topology proceedings from 1980 (puzzling given that the thesis was published a few years later. Oh well) referencing "Caryn Navy's construction of assorted para-Lindelof, non paracompact spaces, both in ZFC and under the axiom that there is a Q-set (implied by MA + ,CH)", so if this statement is accurate no extra axioms are needed. – David R. MacIver Mar 30 2010 at 19:51

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UMI will sell you a copy (paper) for USD44. disexpress.umi.com/dxweb and search for "Caryn Navy". – Nate Eldredge Mar 30 2010 at 21:34

So they will, thanks. I'll wait and see if either of the feelers I sent out turn up a hint and then buy a copy off there. – David R. MacIver Mar 30 2010 at 22:12

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Countable paracompactness, normality and locally countable open covers

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