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

I think the Isbell–Mrówka space $\Psi(\mathcal A)$ with $\lvert\mathcal A\rvert=\omega_1$ is not starcompact. But I can't prove it. Give some explanations on it.

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

I think the Isbell–Mrówka space $\Psi(\mathcal A)$ with $\lvert\mathcal A\rvert=\omega_1$ is not starcompact. But I can't prove it.

A space $X$ is said to be starcompact if for every open cover $\mathcal U$ of $X$ there exists a finite subset $\mathcal V\subseteq\mathcal U$ such that $St(\cup\mathcal V,\mathcal U)=X$.

I think the Isbell-Mr$\rm\acute{o}$wka space $\Psi(\mathcal A)$ with $|\mathcal A|=\omega_1$ is not starcompact. But I can't prove it. Give some explanations on it.

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

I think the Isbell–Mrówka space $\Psi(\mathcal A)$ with $\lvert\mathcal A\rvert=\omega_1$ is not starcompact. But I can't prove it. Give some explanations on it.