Embeddings into \(c_ 0\)
\textit{I. Aharoni} [Israel J. Math. 19(1974), 284-291 (1975; Zbl 0303.46012)] proved that each separable metric space can be Lipschitz embedded into \(c_0(\omega_0)\), \textit{P. Assouad} [ibid. 31, 97-100 (1978; Zbl 0387.54003)] simplified his proof, improved constants of Lipschitz embeddings and suggested a generalization of his result. We present the final improvements of constants for Lipschitz embeddings into \(c^+_0(\omega_0)\) and another generalization of Aharoni's result which comes from the theory of uniform spaces. Namely we prove that each metric space \((X,\rho)\) whose metric uniformity has a base consisting of point-finite uniform covers (such a base of uniform covers will be called a point-finite base; a base of uniform covers is a collection of uniform covers such that each uniform cover is refined by some cover that belongs to a base) can be uniformly embedded into \(c^+_0(\kappa)\) where \(\kappa=\text{dens }X\). We also prove that a uniformity of each \(c_0(\kappa)\) has a point-finite base of uniform covers, hence the \(c_0(\kappa)\) are universal spaces for metrizable uniformities with a point-finite base. It implies a well-known fact [\textit{G. Vidossich}, Proc. Am. Math. Soc. 25, 551-553 (1970; Zbl 0181.50903)] that each separable uniformity has a point-finite base. Investigating the Lebesgue number of point-finite uniform refinements we point out the reason why a uniform embedding cannot be generally replaced by a Lipschitz one in the nonseparable case. As a byproduct of this analysis we obtain full information on the Lebesgue number of point-finite uniform refinements in \(c_0(\kappa)\) and an internal characterization of metric spaces which are Lipschitz embeddable into \(c_0(\kappa)\). The difference between \(c_0\) and \(c^+_0\) is mentioned. \((c_0(\kappa)\) denotes a Banach space the underlying set of which is formed by all real-valued mappings \(f:\kappa\to\mathbb{R}\) such that \(\text{card(supp } f)\leq\omega_0\) and \(\lim f=0\), and which is equipped with the sup-norm. \(c^+_0(\kappa)\) denotes its metric subspace consisting of all mappings the values of which are nonnegative.) For a real number \(t\), \((t)^+=\max(t,0)\), \((t)^-= \max(-t,0)\). For a cardinal number \(\alpha\), \((\alpha)^+\) denotes the successor of \(\alpha\).
scientific article; zbMATH DE number 600041