Scalene metric spaces (Q1063894)

Let (X,\(\rho)\) be a metric space. For every a,b\(\in X\) let \(I_{\rho}(a,b)=\{a\}\) if \(a=b\) and \(I_{\rho}(a,b)=\{c\in X;\quad \forall x\in X \rho (x,c)<\max (\rho (x,a),\quad \rho (x,b))\}.\) The metric \(\rho\) is scalene whenever \(I_{\rho}(a,b)\neq \emptyset\) for every a,b\(\in X\); the metric \(\rho\) is locally scalene whenever for every point \(x\in X\) there is a neighbourhood U of x such that \(\rho_ U\) (i.e. \(\rho\) \(| (U\times U))\) is scalene. One of the main results concerns absolute retracts and absolute neighbourhood retracts: Theorem (3.10). If a compactum X has a scalene metric, then \(X\in AR\). Theorem (3.11). If a locally compact space has a locally scalene metric, then \(X\in ANR\). Moreover, each point \(x\in X\) has a compact neighbourhood \(U\in AR.\) Theorem (3.10) (Theorem (3.11)) is a step towards a solution of the following problem: to characterize the class AR (ANR) by means of the existence of a metric satisfying some metric conditions. (The author refers to such a characterization as a metric one, though this is evidently a topological characterization.) Another interesting result concerns the selection problem: Theorem (5.1). If Y is any topological space and X is a metric space with metric \(\rho\), then for every continuous function \(\phi\) : \(Y\to 2^ X\), if \(\rho_{\phi (y)}\) is a scalene metric for each \(\phi\) (y), then \(\phi\) has a continuous selection \(s: Y\to X\). Theorem (5.1) is a modification of a Michael theorem concerning a function \(\phi\) : \(Y\to 2^ X\) for X being a Banach space and each \(\phi\) (y) being its closed convex subset.