On the ideal convergence of sequences of quasi-continuous functions

Ideal equal Baire classes

“…Recently, Natkaniec and Szuca (see [18] and [19]) obtained similar results in the case of quasi-continuous functions instead of continuous functions. Namely, they characterized Baire systems generated by the family of quasi-continuous functions in the case of ideal convergence and ideal discrete convergence for all Borel ideals and metric Baire spaces.…”

Sets of ideal convergence of sequences of quasi-continuous functions

“…The last statement can be proved exactly in the same way as claim (3) in [16,Proposition 9]. We copy here that proof for a convenience of a reader.…”

“…Note that the family of F σ ideals which are ω-+-diagonalizable is a proper subclass of all F σ ideals. (See [16]. )…”

“…(See e.g. [16,Proposition 5].) Thus if I is a Borel ideal which is not ω-+-diagonalizable then player I has a winning strategy in the game G(I), thus there exists an I -tree with each branch in I.…”

“…Let J = {A ⊂ ω: i∈A 1 i < ∞}. Note that J is F σ and such that player I has a winning strategy in the game G(J ) [16], hence there exists a sequence g = (g n ) n ⊂ QC(R) such that …”