and Applied Analysis 3 Definition 1.4 see 18, 20–23 . 1 The intersection of all fuzzy α-closed resp., semiclosed, preclosed, γ-closed, semi-preclosed, β-closed sets containing a fuzzy set A is called a fuzzy α-closure resp., semiclosure, preclosure, γ-closure, semi-preclosure, β-closure of A. 2 The union of all fuzzy α-open resp., semiopen, preopen, γ-open, semi-preopen, β-open sets contained i...

Let (X,m, H) be a hereditary m-space. A subset of X is said to θ-H-compact relative if for every cover U by m(θ)-open sets X, there exists finite U0 such that \ ∪ ∈ H. We obtain several properties these sets. Also, we define and investigate two kinds strong forms “θ-H-compact X”.

In this article we originate a new class of Grill Set, namely GSβ-Open which is parallel to the β Open Set in Topological Space (X, θ, G). addition, entitle GSβ-continuous and GSβ-open functions by applying review some its important properties. Many examples are given explain concept lucidly. The properties GSβ open sets investigated studied. theorems based on arbitrary union finite intersectio...

In this note, we study the cut locus of the free, step two Carnot groups Gk with k generators, equipped with their left-invariant Carnot-Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed in [Mya02, Mya06] and [MM16a], by exhibiting sets of cut points Ck ⊂ Gk which, for k > 4, are strictly larger than conjectured ones. While the latter were, re...

The concepts of fuzzy θ-open (θ-closed)sets and fuzzy θ-closure operator are introduced and discussed in intuitionistic fuzzy topological spaces. As applications of these concepts, certain functions are characterized in terms of intuitionistic fuzzy θ-closure operator.

In this paper analogous to [1], we introduce a new class of sets called ωθ˜-µ-open in generalized topological spaces which lies strictly between the θ˜µ-open and ω-µ-open sets. We prove that collection forms topology. Finally, several characterizations properties have been given.