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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. $\endgroup$
    – Joseph Van Name
    Jun 2, 2016 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, 2016 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. $\endgroup$
    – Joseph Van Name
    Jun 6, 2016 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. The final version of the example is now available on Arxiv.org and in Topology and its Applications

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