Matet, Pierre A short proof of Ellentuck’s theorem. (English) Zbl 0962.03042 Proc. Am. Math. Soc. 129, No. 4, 1195-1197 (2001). The author offers a short proof of Ellentuck’s theorem that characterizes the completely Ramsey subsets of \([\omega]^\omega\) as those having the Baire property with respect to what is now known as the Ellentuck topology. The key step is a technical lemma that readily implies the following two facts: Ellentuck open sets are completely Ramsey and the family of Ellentuck nowhere dense sets is closed under countable unions. From this the main theorem follows at once. Reviewer: K.P.Hart (Delft) Cited in 1 Document MSC: 03E05 Other combinatorial set theory 03E15 Descriptive set theory Keywords:Ellentuck topology; completely Ramsey set; Baire property PDFBibTeX XMLCite \textit{P. Matet}, Proc. Am. Math. Soc. 129, No. 4, 1195--1197 (2001; Zbl 0962.03042) Full Text: DOI References: [1] Fred Galvin and Karel Prikry, Borel sets and Ramsey’s theorem, J. Symbolic Logic 38 (1973), 193 – 198. · Zbl 0276.04003 [2] Erik Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163 – 165. · Zbl 0292.02054 [3] A. R. D. Mathias, Happy families, Ann. Math. Logic 12 (1977), no. 1, 59 – 111. · Zbl 0369.02041 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.