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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 $St(K,\mathcal U)=X$.

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