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A note on the Ramsey property. (English) Zbl 0953.03058

The author presents an elementary proof of Silver’s theorem [J. Silver, J. Symb. Log. 35, 60-64 (1970; Zbl 0216.01304)] saying that every analytic subset of the space \([\mathbb{N}]\) of infinite subsets of \(\mathbb{N}\) with pointwise convergence topology is completely Ramsey. The main tool used in the reviewed paper is the closure \(\widehat X\) of a set \(X\subseteq [\mathbb{N}]\) in Ellentuck’s topology. Actually \(X\subseteq {\widehat X}\subseteq \overline X\) and, as the author proves, \(\widehat X\) is completely Ramsey for any \(X\subseteq [\mathbb{N}]\).

MSC:

03E05 Other combinatorial set theory
03E15 Descriptive set theory

Citations:

Zbl 0216.01304
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