Universal Metric Spaces According to W. Holsztynski

In this note we show, following W. Holsztynski, that there is a continuous metric d on R such that any finite metric space is isometrically embeddable into (R, d). Let M be a family of metric spaces. A metric space U is said to be a universal space for M if any space from M is (isometrically) embeddable in U . Fréchet [2] proved that l (the space of all bounded sequences of real numbers endowed with the sup norm) is a universal space for the family M of all separable metric spaces. Later, Uryson [5] constructed an example of a separable universal space for this M (l is not separable). In this note we establish the following result which is Theorem 5 in [3]. Theorem 1. There exists a metric d in R, inducing the usual topology, such that every finite metric space embeds in (R, d). Our proof essentially follows the original Holsztynski’s approach [4]. We say that a metric space (X, d) is ε–dispersed if d(x, y) ≥ ε for all x 6= y in X (ε > 0). Clearly, any ε– metric space is also ε–dispersed for any positive ε < ε. The following proposition will be used to construct universal spaces for particular families M. Proposition 1. Let f : X → Y be a continuous surjection from a metric space (X, d) onto a metric space (Y,D). Then (X, dε) where dε(x, y) = max{min{d(x, y), ε}, D(f(x), f(y))} (1) is a universal space for the family of ε–dispersed subspaces of (Y,D) and metrics d and dε are equivalent on X.