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Does anyone know of an example of a (Tychonoff) $F$-space which is zero-dimensional but not strongly zero-dimensional?

By an $F$-space we mean every cozeroset is $C^*$-embedded.

By zero-dimensional we mean has a base of clopen sets.

By strongly zero-dimensional we mean every cozeroset is a countable union of clopen sets.

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  • $\begingroup$ +1. If I remember correctly, I have at a conference heard Alan Dow refer to this problem as an open problem. $\endgroup$ Jun 2 '16 at 5:26
  • $\begingroup$ Yes to my knowledge it is. As I am new here I figured this might be a good place to say it to see if anybody has any ideas about it. (Let me know if this is not appropriate.) $\endgroup$
    – W.McGovern
    Jun 5 '16 at 20:43
  • $\begingroup$ I personally think this is a very good question for mathoverflow and it is very interesting to me as are most questions about zero-dimensional spaces which are not strongly zero-dimensional. By the way, welcome to mathoverflow. $\endgroup$ Jun 6 '16 at 0:06
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Alan Dow and I constructed an example. It can be found by following this link. The construction is based on the locally compact modification of Dowker's example in this answer. A complete(r) version of the example is now available on Arxiv.org

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