Functionally regular spaces

Approximation of continuous functions on pseudocompact spaces

“…The spaces are precisely the spaces such that the weak topology is compact. The functionally regular closed spaces are precisely the functionally Hausdorff closed spaces that are functionally regular [1]. From the proof of Theorem 1, (a) -» (b) we have the following corollary.…”

“…Definition 2. A space is functionally Hausdorff (regular) if points (points and closed sets) are functionally separated [1].…”

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

“…Any intersection of zero sets is called z-closed set ( [15,30] Definition 2.1. A topological space X is said to be (i) functionally regular ( [3,39]) if for each closed set F in X and each x / ∈ F there exists a continuous real-valued function f defined on X such that f (x) / ∈ f (F ). (ii) ultra Hausdorff [35] if every pair of distinct points in X are contained in disjoint clopen sets.…”

“…We reflect upon interrelations and interconnections that exist among R cl -spaces and separation axioms which already exist in the lore of mathematical literature and lie between zero dimensionality and R 0 -spaces. The class of R cl -spaces properly contains each of the classes of zero dimensional spaces and ultra Hausdorff spaces [35] and is strictly contained in the class of R 0 -spaces ( [20,33]) which in its turn properly contains each of the classes of functionally regular spaces ( [3,39]) and functionally Hausdorff spaces. The organization of the paper is as follows: Section 2 is devoted to preliminaries and basic definitions.…”

On spaces with aG?-basis

“…The main purpose of this paper is to introduce a new class of functions called 'Fsupercontinuous functions', study their basic properties and discuss their place in the hierarchy of strong variants of continuity that already exist in the mathematical literature. The notion of F -supercontinuous functions arise naturally in case either domain or range is a functionally regular space ( [1], [2]). It turns out that the class of F -supercontinuous functions properly includes the class of z-supercontinuous functions [8] and is strictly contained in the class of Rsupercontinuous functions [14] which in turn is properly contained in the class of continuous functions.…”