This is an announcement for the paper "The Kolmogorov-Riesz compactness theorem" by Harald Hanche-Olsen and Helge Holden. Abstract: We show that the Arzela-Ascoli theorem and Kolmogorov compactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly's theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem. Archive classification: math.CA math.FA Mathematics Subject Classification: 46E30, 46E35; 46N20 The source file(s), kolmogorov.tex: 36412 bytes, is(are) stored in gzipped form as 0906.4883.gz with size 12kb. The corresponding postcript file has gzipped size 213kb. Submitted from: holden@math.ntnu.no The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0906.4883 or http://arXiv.org/abs/0906.4883 or by email in unzipped form by transmitting an empty message with subject line uget 0906.4883 or in gzipped form by using subject line get 0906.4883 to: math@arXiv.org.