Closedness of certain classes of functions in the topology of uniform convergence

Abstract

In this paper, closedness of certain classes of functions in

Set-open topologies on function spaces

“…It is well-known (e.g., [13], p. 227-229) that, for Y a uniform space, C(X, Y ) is U X -closed in F (X, Y ) but not necessarily U p -closed. Later, some authors also obtained variants of these results for some other classes of functions, not necessarily continuous [11,16,34]. In this section, we establish some results for the class C α (X, Y ) of all "α-continuous" functions from X into Y [21,16].…”

“…In section 2, we study their comparability and also coincidence of such topologies; we also discuss their existence and their relationship with some uniform convergence topologies. In section 3, we establish some results on closedness and completeness of the space C α (X, Y ) of all α-continuous functions, from X into Y [16,21]. Here, we shall need to assume that Y is a regular topological space, which is equivalent to Y being a locally symmetric quasi-uniform space [6,25].…”

Quasi-uniform convergence topologies on function spaces- Revisited

“…Later, some authors obtained variants of these results for spaces of quasi-continuous, somewhat continuous and bounded functions in the case of Y a uniform space [16,22,37,40]. In this section, we examine their U A -closedness and right K-completeness in the setting of Y a locally symmetric quasi-uniform or locally uniform spaces.…”

R-spaces and closedness/completeness of certain function spaces in the topology of uniform convergence

“…In the same vein, Hoyle [10] showed that the set SW(X, Y) of all somewhat continuous functions from a space X into a uniform space Y is closed in Y X in the topology of uniform convergence. Furthermore, Kohli and Aggarwal in [14] proved that the function space SC(X, Y ) of quasicontinuous ( ≡ semicontinuous) functions, C α (X, Y ) of α-continuous functions, and L(X, Y) of cl-supercontinuous functions are closed in Y X in the topology of uniform convergence. In this section we strengthen the results of [14] and show that the set [32] if for each x ∈ A there exists a clopen set H such that…”