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Is there a Lusin space (in the sense Kunen) $X$ such that

  1. $X$ is Tychonoff;
  2. $X$ is a $\gamma$-space ?

Note that if $X$ is metrizable and a $\gamma$-space then it is not Lusin.

In mathematics, a Luzin space (or Lusin space), named for N. N. Luzin, is an uncountable topological $T_2$ space without isolated points in which every nowhere-dense subset is countable. (Kunen, Kenneth (1977), "Luzin spaces", Topology Proceedings, Vol. I (Conf., Auburn Univ., Auburn, Ala., 1976), pp. 191–199)

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    $\begingroup$ Define, in the question, "Lusin space in the sense of Kunen" $\endgroup$
    – Boaz Tsaban
    Commented Jan 21, 2023 at 20:31
  • $\begingroup$ Did you check the proof that gamma-sets of reals are meager? What do you need to assume about the space for this proof to work? Of course, if the proof worked for all spaces this would answer your question. $\endgroup$
    – Boaz Tsaban
    Commented Mar 9, 2023 at 10:07

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