On parametric limit superior of a sequence of analytic sets. (English) Zbl 1094.03032

Głąb, Szymon

Real Anal. Exch. 31(2005-2006), No. 1, 285-289 (2006).

Summary: Let \(A_x\) stand for the \(x\)-section of a set \(A\subset 2^\omega\times 2^\omega\). We prove that any sequence \(A^j\subset 2^\omega\times 2^\omega\), \(j\in\omega\), of analytic sets with uncountable \(\limsup_{j\in H}A_x^j\) for all \(x\in 2^\omega\) and \(H\in [\omega]^\omega\) admits a perfect set \(P\subset 2^\omega\) and \(H\in [\omega]^\omega\) with uncountable \(\bigcap_{j\in H}A_x^j\) for all \(x\in P\). This is a parametric version of Komjáth’s theorem [P. Komjáth, Anal. Math. 10, 283–293 (1984; Zbl 0569.03021)].

MSC:

03E15	Descriptive set theory
54H05	Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
28A05	Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets

Keywords:

Borel set; limit superior; parametrized Ellentuck theorem; analytic sets; perfect set

Citations:

Zbl 0569.03021