Let \(\mathcal U\) and \(\mathcal V\) be open covers of a space \(X\). \(\mathcal V\) is a shrinkable refinement of \(\mathcal U\) [the reviewer with \textit{M. P. Berri} and \textit{R. M. Stephenson jun.}, Proc. Kanpur Topol. Conf. 1968, 93--114 (1971; Zbl 0235.54018)] if for each \(V\in{\mathcal V}\), there is a \(U\in{\mathcal U}\) such that \(\text{cl }V\subseteq U\). A space is \(U(i)\) or quasi-\(U\)-closed [\textit{C. T. Scarborough}, Pac. J. Math. 27, 611--617 (1968; Zbl 0189.23104)] if every open cover with shrinkable refinement has a finite subfamily whose closures cover. The authors introduce the concept of \(R\)-compactness; a space is \(R\)-compact if every open cover with shrinkable refinement has a finite subcover. It follows that a quasi-\(H\)-closed space is \(R\)-compact and an \(R\)-compact space is quasi-\(U\)-closed. Many characterizations and some mapping results of \(R\)-compact are obtained.