# Is the following space $X$ star-$K$-compact?

Let $$\kappa$$ be an infinite cardinal and $$D=\{d_\alpha : \alpha<\kappa\}$$ be a discrete space of cardinality $$\kappa$$. Let $$aD=D\cup\{\infty\}$$ be the one point compactification of $$D$$. In the product space $$aD\times(\omega+1)$$, replace the local base of the point $$(\infty,\omega)$$ by the family $$\{U\setminus(D\times\{\omega\}) : (\infty,\omega)\in U\;\text{and}\; U\;\text{is an open set in}\; aD\times(\omega+1)\}$$. Let $$X$$ be the space obtained by such replacement.

Then $$X$$ is starcompact. But we are unable to show that $$X$$ is star-$$K$$-compact.

Note that

1. 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 $$St(\cup\mathcal V,\mathcal U)=X$$.

2. A space $$X$$ is said to be star-$$K$$-compact if for every open cover $$\mathcal U$$ of $$X$$ there exists a compact subset $$K$$ of $$X$$ such that $$St(K,\mathcal U)=X$$.