# Does there exist a regular $P$-space which is strongly star-Lindelof but not star-Menger?

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.

• To start with, can you provide an example of a regular 𝑃-space that is Lindelof but not Menger? Sep 25, 2021 at 21:31
• @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). Sep 26, 2021 at 5:23
• Can't you use the same argument to establish the same result for your question? Sep 26, 2021 at 10:09