On the ideal convergence of sequences of quasi-continuous functions (Q2787127)

The classes of pointwise ideal limits, i.e. limits with respect to dual filters to various ideals on \(\omega\), of sequences of quasi-continuous real functions on Baire metric spaces \(X\) are studied. One of the main results characterizes those Borel ideals \(\mathcal I\) on \(\omega\) for which the family of pointwise \(\mathcal I\)-limits of sequences of quasi-continuous functions coincides with the family of all pointwise limits of sequences of quasi-continuous functions. The latter family coincides with the set of all pointwise discontinuous (cliquish) functions by results of \textit{Z. Grande} [Fundam. Math. 129, No. 3, 167--172 (1988; Zbl 0657.26003)] and \textit{C. Richter} and \textit{I. Stephani} [Real Anal. Exch. 29, No. 1, 299--322 (2004; Zbl 1068.54015)]. The other Borel ideals lead to the set of all functions with the Baire property.NEWLINENEWLINEThe methods of proof recall those of \textit{M. Laczkovich} and \textit{I. Recław} [Fundam. Math. 203, No. 1, 39--46 (2009; Zbl 1172.03025)], and \textit{G. Debs} and \textit{J. Saint Raymond} [ibid. 204, No. 3, 189--213 (2009; Zbl 1179.03046)] who independently characterized the Borel ideals \(\mathcal I\) for which the \(\mathcal I\)-limits of continuous functions lead to the first Baire class. The dual filters are in both cases, for limits of continuous and for limits of quasi-continuous functions, characterized by combinatorial properties which are equivalent to the existence of a winning strategy of the respective game studied by \textit{C. Laflamme} in the same paper [Contemp. Math. 192, 51--67 (1996; Zbl 0854.04004)].