Give an example of a star-Menger space which is not star-$K$-Menger

Question

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.

Answer 1

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).