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

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 immediate that $\mathbb R$ with cocountable topology is starcompact. But we don't know whether $\mathbb R$ with cocountable topology is star-$K$-compact. It is clear from the definitions that every star-$K$-compact space is starcompact.