Separation by ambivalent sets. (English) Zbl 1065.26006

Prus-Wiśniowski, Franciszek

Real Anal. Exch. 30(2004-2005), No. 1, 289-294 (2005).

A characterization of when two sets in \([0,1]\) can be separated by ambivalent sets is shown. Two applications to some proofs of known theorems are presented. In particular it is proved that for disjoint subsets \(A\), \(B\) the following conditions are equivalent: (i) \(A\) and \(B\) can be separated by ambivalent sets, (ii) \(A\) and \(B\) can be separated by a Baire 1 function, (iii) there is no perfect set \(K\) such that both \(A\) and \(B\) are dense in \(K\).

Reviewer: Zbigniew Grande (Bydgoszcz)

MSC:

26A21

Classification of real functions; Baire classification of sets and functions

Keywords:

\(G_{\delta }\)-set; \(F_{\sigma }\)-set; ambivalent set; Baire 1 function