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A short proof of Ellentuck’s theorem. (English) Zbl 0962.03042

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)

MSC:

03E05 Other combinatorial set theory
03E15 Descriptive set theory
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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
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