# Does there exist a starcompact space which is not star-$K$-compact?

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.

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.