D. Doitchinov defined a filter \({\mathcal F}\) on a quasi-uniform space \((X,{\mathcal U})\) to be a \(D\)-Cauchy filter provided there exists a filter \({\mathcal G}\) on \(X\) such that for each \(U\in {\mathcal U}\) there exists a \(G\in {\mathcal G}\) and an \(F\in {\mathcal F}\) such that \(G\times F \subset {\mathcal U}\). J. Isbell defined a filter \({\mathcal F}\) on a uniform space to be stable provided that for each \(U \in {\mathcal U}\), \(\bigcap \{U[S]:S \in {\mathcal F}\} \in {\mathcal F}\). The authors call a quasi-uniform space \((X,{\mathcal U})\) stable provided each \(D\)-Cauchy filter is \({\mathcal U}\)-stable and co-stable provided each \({\mathcal U}^{-1}\)-\(D\)-Cauchy filter is \({\mathcal U}\)-stable. The principal result of their paper is that in a class of spaces comprising the quiet spaces of D. Doitchinov, for those spaces that are co-stable, the properties of \(D\)-completeness, strong \(D\)-completeness and bicompleteness coincide. The authors show that a mixed symmetric quiet space is both locally symmetric and co-stable and they give an example of a quiet co-stable space that is not locally symmetric. Thus, their principal result improves upon the corresponding theorem for mixed symmetric spaces [Proposition 6, Nonsymmetric topology, Bolyai Math. Studies 4, 303-338 (1993)]. If \((X, {\mathcal U})\) is co-stable and \((X,{\mathcal U}^{-1})\) is stable, the authors say that \((X,{\mathcal U})\) is strongly co-stable. They establish that for a quiet quasi-uniform space the following statements are equivalent: \((X,{\mathcal U})\) is co-stable; \((X,{\mathcal U})\) is strongly co-stable; the \(D\)-completion of \((X,{\mathcal U})\) is co-stable; the \(D\)-completion of \((X,{\mathcal U})\) is strongly co-stable; the \(D\)-completion of \((X,{\mathcal U})\) is point symmetric.