Ideal convergence of continuous functions

摘要

For a given ideal I is contained in P(ω), IC(1) denotes the class of separable metric spaces X such that whenever f_n : X → R is a sequence of continuous functions convergent to zero with respect to the ideal I then there exists a set of integers {m_0 < m_1 < ···} from the dual filter F(I) such that lim_(i→∞) f_(m_i) (x) = 0 for all x ∈ X. We prove that for the most interesting ideals I, IC(I) contains only singular spaces. For example, if I = I_d is the asymptotic density zero ideal, all IC(I_d) spaces are perfectly meager while if I = I_b is the bounded ideal then IC(I_b) spaces are σ-sets.