Formula omitted]-selection principles for sequences of functions

Abstract

We generalize three classical selection principles (Arzela–Ascoli theorem, Mazurkiewicz’s theorem and Helly’s theorem) on the ideal convergence. In particular, we show that for every analytic P-ideal I with the BW property (and every Fσ ideal I) the following selection theorems hold: •If 〈fn〉n is a sequence of uniformly bounded equicontinuous functions on [0,1] then there exists A∉I such that 〈fn〉n∈A is uniformly convergent;•if 〈fn〉n is a sequence of uniformly bounded continuous functions then there exists a perfect set P and a set A∉I such that 〈fn↾P〉n∈A is pointwise convergent;•if 〈fn〉n is a sequence of uniformly bounded monotone functions then there exists a set A∉I such that 〈fn〉n∈A is pointwise convergent.