On the Convergence of Sequences of Functions which are Discontinuous on Countable Sets

Let (X, TX ) be a topological space and let (Y, dY ) be a metric space. For a function ƒ : X → Y denote by C(ƒ) the set of all continuity points of ƒ and by D(ƒ) = X\C(ƒ) the set of all discontinuity points of ƒ. Let C(X, Y) = {ƒ : X → Y; ƒ is continuous}, H(X, Y) = {ƒ : X → Y; D(ƒ) is countable}, H 1( X, Y) = {ƒ : X → Y; ∃ h∈C(X, Y) {x; ƒ(x) ≠ h(x)} is countable}, and H 2(X, Y) = H(X, Y)∩H 1(X, Y). In this article we investigate some convergences (pointwise, uniform, quasiuniform, discrete and transfinite) of sequences of functions from H(X, Y), H 1(X, Y) and H 2(X, Y).