A characterization of Sigma(1)(2) sets

作者：Pawlikowski Janusz^*

来源：Fundamenta Mathematicae, 2017, 236(1): 45-49.

DOI：10.4064/fm61-4-2016

• Polish space
• Ellentuck topology
• meager sets
• Mathias forcing

We show that a subset X of a given Polish space X is Sigma(1)(2) iff there is an open set O subset of X x [omega](omega) such that
X = { x is an element of X : there exists r is an element of [omega](omega) {x} x [r](omega) subset of O}.
This implies that if a set U subset of omega(omega) (chi x [omega](omega)) is universal for G(delta) subsets of chi x [omega](omega), then the set of all (v, x) is an element of omega(omega) x chi such that the section U-vx has nonempty interior in the Ellentuck topology is universal for Sigma(1)(2) subsets of chi. It follows that the sigma-ideal of meager sets in the Ellentuck topology is not Sigma(1)(2) on G(delta), a fact established recently by Sabok (2012) with the help of Kleene's Recursion Theorem.

• 出版日期2017