Ideal convergence of continuous functions

For a given ideal I ⊆ P ( ω ) , IC ( I ) 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.