<p>Let X and Y be topological spaces, F(X,Y) the set of all functions from X into Y and C(X,Y) the set of all continuous functions in F(X,Y). We study various set-open topologies tλ (λ ⊆ P(X)) on F(X,Y) and consider their existence, comparison and coincidence in the setting of Y a general topological space as well as for Y = R. Further, we consider the parallel notion of quasi-uniform convergence topologies Uλ (λ ⊆ P(X)) on F(X,Y) to discuss Uλ-closedness and right Uλ-K-completeness properties of a certain subspace of F(X,Y) in the case of Y a locally symmetric quasi-uniform space. We include some counter-examples to justify our comments.</p>

The main purpose of this paper is to introduce and study new topological properties called C-almost normality and L-almost normality. A space X is called a C-almost normal (resp. L-almost normal) space if there exist an almost normal space Y and a bijective function f : X → Y such that the restriction function f|A : A → f(A) is a homeomorphism for each compact (resp. Lindelöf) subspace A ⊆ X. We investigate these properties and present some examples to illustrate the relationships among them with other kinds of topological properties.

This work studies a new version of normality called epi-quasi normality, which lies between epi-normality and epi-mild normality. In this paper, we investigate this property and present some examples that illustrate the relationships between epi-quasi normality and other kinds of both normality and regularity.

Following the notion of so-called C-normality - a weaker version of normality in topological spaces as proposed by A. V. Arhangel’skii, further weaker version called CC-normality is studied by Kalantan et al [14]. In this paper, we investigate various type of properties such as CC-complete regularity, CC-almost complete regularity, CC-regularity, CC-almost regularity, CCT3 and CC-Tychonoffness. A space (X, T ) is called a CC-completely regular (resp. CC-almost completely regular, CC-regular, CC-almost regular, CCT3, CC-Tychonoff) space if there exist a completely regular (resp. almost completely regular, regular, almost regular, T3, Tychonoff) space Y and a bijective function f : X → Y such that the restriction function f|A : A → f(A) is a homeomorphism for each countably compact subspace A ⊆ X. We study these properties and present some examples to illustrate the relationships among them with other forms of topological properties.