A direct proof that $[0,\omega_1)$ is strongly star-Hurewicz In their article Star-Hurewicz and related properties, Bonanzinga, Cammaroto and Kočinac state in passing that the ordinal space $[ 0 , \omega_1 )$ is strongly star-Hurewicz (what they abbreviate SSH):

Let us mention that there is a SSH space $X$ and a Lindelöf space $Y$ such that the product $X \times Y$ is not SSH. Take $X = [ 0 , \omega_1 )$ with the usual order topology and $Y$ to be the one-point Lindelöfication of $X$.

This paper also states that strongly starcompact spaces are strongly star-Hurewicz. In Star covering properties, van Douwen, Reed, Roscoe and Tree show that all countably compact spaces are strongly starcompact.
Since $[ 0 , \omega_1 )$ is widely known to be countably compact (see, for example, Steen and Seebach's Counterexamples in Topology), this string of implications yields that $[ 0 , \omega_1 )$ is strongly star-Hurewicz, however, I want a direct proof of this fact.

Definitions
Let $\mathcal P$ be a collection of subsets of $X$. Then the star of a set $A\subseteq X$ with respect to $\mathcal P$ is denoted by $\operatorname{St} (A,\mathcal P)$ and defined by $$\operatorname{St} (A,\mathcal P)=\bigcup\{B\in\mathcal P : B\cap A\neq\emptyset\}$$

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*An open cover $\mathcal U$ of a space $X$ is said to be a $\gamma$-cover of $X$ if $\mathcal U$ is infinite and each $x\in X$ belongs to all but finitely many members of $\mathcal U$.


*A space $X$ is said to be strongly starcompact if for every open cover $\mathcal U$ of $X$ there exists a finite $F\subseteq X$ such that $X=\operatorname{St} (F,\mathcal U)$.


*A space $X$ is said to be strongly star-Hurewicz if for each sequence $(\mathcal U_n)$ of open covers of $X$ there exists a sequence $(F_n)$ of finite subsets of $X$ such that $\{\operatorname{St} (F_n,\mathcal U_n) : n\in\mathbb N\}$ is a $\gamma$-cover of $X$.
 A: I think your definition of "strongly star-Hurewicz" may be slightly incorrect. Instead of requiring that $\{ \operatorname{St} ( F_n , \mathcal{U}_n ) : n \in \mathbb{N} \}$ is a $\gamma$-cover (which will be false for irrelevant reasons if each $\operatorname{St} ( F_n , \mathcal{U}_n )$ is equal to $X$), you should instead simply require that each $x \in X$ is contained in $\operatorname{St} ( F_n , \mathcal{U}_n )$ for all but at most finitely many $n \in \mathbb{N}$. This is the definition used by Bonanzinga, Cammaroto and Kočinac in their article Star-Hurewicz and related properties. Other articles, such as Kočinac's Star selection principles: a survey, bypass this problem by allowing $\gamma$-covers to be finite.

Using the above corrected definition of "strongly star-Hurewicz", it suffices to show that $\omega_1 = [ 0 , \omega_1 )$ is strongly star-compact. (Since then for each sequence $\{ \mathcal{U}_n : n \in \mathbb{N} \}$ of open covers of $x$ there is for each $n$ a finite $F_n \subset \omega_1$ such that $\operatorname{St} ( F_n , \mathcal{U}_n ) = \omega_1$, so clearly each $\xi \in \omega_1$ is an element of $\operatorname{St} ( F_n , \mathcal{U}_n )$ for all but at most finitely many $n \in \mathbb{N}$.
Recall that for any $\xi \in \omega_1$, $\xi \neq 0$ and any open neighborhood $U$ of $\xi$ there is a $\alpha < \xi$ such that $( \alpha , \xi ] \subset U$.
Take any open cover $\mathcal{U}$ of $\omega_1$. We will first find a finite $A \subset \omega_1$ (in fact, $A$ will be a singleton) such that $\operatorname{St} ( A , \mathcal{U} )$ includes a finial segment of $\omega_1$, and then apply compactness to find a finite $B \subset \omega_1$ such that $\operatorname{St} ( B , \mathcal{U} )$ covers whatever remains.

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*For each $\xi \in \omega_1 \setminus \{ 0 \}$, pick some $\alpha_\xi < \xi$ with the property that there is a $U \in \mathcal{U}$ such that $( \alpha_\xi , \xi ] \subset U$. In this manner we define a function $f : \omega_1 \setminus \{ 0 \} \to \omega_1$. Note that this function is regressive. Applying Fodor's Lemma to the function $f$ we obtain a stationary $S \subseteq \omega_1$ and a $\alpha < \omega_1$ such that $f ( \xi ) = \alpha$ for each $\xi \in S$. In particular, the set $S$ is unbounded in $\omega_1$. Now given any $\eta \in ( \alpha , \omega_1 )$ there is a $\xi \in S$ with $\eta < \xi$, and since $f ( \xi ) = \alpha$ by the choice of the function $f$ it follows that there is a $U \in \mathcal{U}$ such that $( \alpha , \xi ] \subseteq U$, and so in particular $\alpha + 1 \in U$ and $\eta \in U$. From this it follows that $[ \alpha + 1 , \omega_1 ) \subset \operatorname{St} ( \{ \alpha + 1 \} , \mathcal{U} )$.


*Using the fact that $[ 0 , \alpha ]$ is compact there are finitely many $U_1 , \ldots , U_n \in \mathcal{U}$ such that $[ 0 , \alpha ] \subset U_1 \cup \cdots \cup U_n$, and picking an $\alpha_i \in U_i$ for each $i$ it follows that $[ 0 , \alpha ] \subset \operatorname{St} ( \{ \alpha_1 , \ldots , \alpha_n \} , \mathcal{U} )$.
Putting these together we have that $\operatorname{St} ( \{ \alpha + 1 , \alpha_1 , \ldots , \alpha_n \} , \mathcal{U} ) = \omega_1$, and so $\omega_1$ is strongly star-compact, and hence strongly star-Hurewicz.
