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Let $X$ be a Lindelöf space. If every cover $\{G_M : M\in\mathcal{M}(X)\}$ of $X$, where $G_M$ is a $G_\delta$ subset of $X$ with $M\subseteq G_M$, has a subcover of cardinality $\mathfrak{b}$, then is $X$ 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. $\mathcal{M}(X)$ is the collection of all star-Menger subspaces of $X$.
  3. $\mathfrak{b}$ is the minimum cardinality of an unbounded subset of $\mathbb{N}^\mathbb{N}$.
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