(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.



Countable paracompactness, normality and locally countable open covers
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.


