1. A space $X$ is said to be star-Menger if for every sequence $(\mathcal{U}_n)$ of open covers of $X$ there exists a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a finite subset of $\mathcal{U}_n$ and $\cup_{n\in\mathbb N}\{St(V,\mathcal{U}_n) : V\in\mathcal{V}_n\}$ is an open cover of $X$.

  2. A space $X$ is said to be strongly star-Lindelof if for every open cover $\mathcal U$ of $X$ there exists a countable subset $A$ of $X$ such that $St(A,\mathcal U)=X$.

  3. A $P$-space is a space in which every countable intersection of open sets is open.

Give an example of a regular $P$-space which is strongly star-Lindelof but not star-Menger.

  • $\begingroup$ To start with, can you provide an example of a regular 𝑃-space that is Lindelof but not Menger? $\endgroup$ Sep 25, 2021 at 21:31
  • 1
    $\begingroup$ @Boaz Tsaban: For a regular $P$-space, Lindelof property and Menger property are equivalent (see Corollary 2.5 of doiserbia.nb.rs/ft.aspx?id=0354-51801501099K). $\endgroup$
    – Nur Alam
    Sep 26, 2021 at 5:23
  • $\begingroup$ Can't you use the same argument to establish the same result for your question? $\endgroup$ Sep 26, 2021 at 10:09


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