Low-Distortion Embeddings of Infinite Metric Spaces into the Real Line
We present a proof of a Ramsey-type theorem for infinite metric spaces due to Matoušek. Then we show that for every K>1 every uncountable Polish space has a perfect subset that K-bi-Lipschitz embeds into the real line. Finally we study decompositions of infinite separable metric spaces into subsets that, for some K>1, K-bi-Lipschitz embed into the real line.
Geschke, Stefan. (2009). "Low-Distortion Embeddings of Infinite Metric Spaces into the Real Line". Annals of Pure and Applied Logic, 157(2-3), 148-160. http://dx.doi.org/10.1016/j.apal.2008.09.014