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

It is clear from the above definitions that every star-$K$-compact space is starcompact. But we are unable to construct an example of a starcompact space which is not star-$K$-compact.

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A starcompact but not star-$K$-compact space is constructed in Example 2.2 in the paper “On $\mathcal K$-Starcompact Spaces” by Yan-Kui Song (Bull. Malays. Math. Sci. Soc. (2) 30:1 (2007), 59–64).

Note that what you call starcompact is called $1\frac{1}{2}$-starcompact in this paper and furthermore star-$K$-compact is $\mathcal K$-starcompact.

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