On dynamics of convergent sequences of functions on metric spaces

In this paper, we continue the study of the families F_u(X), F_eq(X), and F_p.w.(X) on X=[0,1]. We prove that F_u(X) is a closed nowhere dense subset of F_eq(X), and that F_eq(X) is a closed nowhere dense subset of F_p.w.(X). We also introduce the notion of a δ -ideal-decomposition of a metric space (X,ρ), where δ:X→R⁺, defined as a sequence of pairwise disjoint closed sets covering X and satisfying natural δ -separation conditions. Using this framework, we show that if a sequence {f_n}⊂F_eq(X) converges pointwise to f and does so relative to a δ -ideal-decomposition, then it belongs to F_u(X). Furthermore, we prove that a typical function f∈B₁(X,X) has a Baire–one set-valued map ω_f:x↦ω(x,f), and that ω_f admits a Baire–one selection. Finally, we establish that equi-limits of equi-Baire one families remain equi-Baire one.