Remarks on transfinite sequences of functions that preserve convergence

Abstract

In the paper [6] the author investigates sequences of functions (f_n)_n=1^∞  preserving the convergence of sequences which are alternatively called conservative sequences of functions. In this note we extend this concept to the transfinite sequences of functions and prove that in this case the property of being conservative is equivalent to the property of being pointwise convergent. Then we introduce the concept of p-locally convergence for transfinite sequences of functions and show that if X is a locally Lindelöf first-countable T₁-space and a transfinite sequence {f_ξ : X → R}_ξ<Ω converges p-locally uniformly to a function f : X → R, then C(f) = Lim C(f_ξ) where C(f) denotes the set of all continuity points of f.