On weak external \(Q\)-hyperconvexity (Q2346165)

Let \((X,d)\) be a \(q\)-hypercovex quasi-pseudometric space. The following are the main results in this paper: Theorem 1. Under the precise setting, \(D\subseteq X\) is weakly externally \(q\)-hyperconvex iff \(D\) is a proximinal nonexpansive retract of \(D\cup F\), for any finite \(F\subseteq X\setminus D\). Theorem 2. Suppose, in addition, that \(X\) is a \(T_0\)-quasi-metric space, and let \(A\subseteq X\) be weakly externally \(q\)-hyperconvex. Then, for each \(\varepsilon_1, \varepsilon_2> 0\), \(N_{\varepsilon_1,\varepsilon_2}(A)\) is also weakly externally \(q\)-hyperconvex. Some other aspects occasioned by these results are also discussed.