# Give an example of a star-Menger space which is not star-$K$-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 star-$$K$$-Menger if for every sequence $$(\mathcal{U}_n)$$ of open covers of $$X$$ there exists a sequence $$(K_n)$$ of compact subsets of $$X$$ such that $$\{St(K_n,\mathcal{U}_n) : n\in\mathbb N\}$$ is an open cover of $$X$$.

It is well known that every star-$$K$$-Menger space is star-Menger. Give an example of a star-Menger space which is not star-$$K$$-Menger.

Let $$P=\{x_\alpha : \alpha<\mathfrak{c}\}$$, $$Q=\{y_n : n\in\mathbb{N}\}$$ and $$Y=\{(x_\alpha, y_n) : \alpha<\mathfrak{c}, n\in\mathbb{N}\}$$, and let $$X=Y\cup P\cup\{p\}$$ where $$p\notin Y\cup P$$.
We define a topology on $$X$$ as follows: every point of $$Y$$ is isolated, a basic neighbourhood of a point $$x_\alpha\in P$$ for each $$\alpha<\mathfrak{c}$$ is of the form $$U_{x_\alpha}(n)=\{x_\alpha\}\cup\{(x_\alpha, y_m) : m>n\}$$ for $$n\in\mathbb{N}$$ and a basic neighbourhood of $$p$$ is of the form $$U_p(A)=\{p\}\cup\{(x_\alpha, y_n) : x_\alpha\in P\setminus A, n\in\mathbb{N}$$} for a countable subset $$A$$ of $$P$$.
Then $$X$$ is a star-Menger space which is not star-$$K$$-Menger (see Example 2.3 of Song - Remarks on star-K-Menger spaces).