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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:
 300000 Other combinatorial set theory 3e+15 Descriptive set theory
##### Keywords:
Ramsey property; analytic set; Ellentuck topology
Zbl 0216.01304
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