The notions of closedness and D-connectedness in quantale-valued approach spaces

Abstract

In this paper, we characterize local T0 and T1 quantale-valued gauge spaces, show how these concepts are related to each other and apply them to L-approach distance spaces and L-approach system spaces. Furthermore, we give the characterization of a closed point and D-connectedness in quantale-valued gauge spaces. Finally, we compare all these concepts to each other.

On the topological category of neutrosophic crisp sets

“…(b) By Theorems 3.6 and 3.9 of [27], T 1 at p =⇒ T 0 at p, and if an L-gauge space (X, G) is T 0 (or T 1 ), then (X, G) is T 0 at p (or T 1 at p) [24,27]. (c) There is no relation between D-connectedness and the notion of closedness or T 1 at p [27]. (5) In pqsMet, the category of extended pseudo-quasi-semi metric spaces and contraction maps, (a)…”

“…Moreover, an L-gauge space (X, G) is T 2 , then (X, G) is both N T 2 and P reT 2 , and in the realm of Pre-Hausdorff quantale-valued gauge spaces, T 0 , T 1 and T 2 are equivalent [24]. (b) By Theorems 3.6 and 3.9 of [27], T 1 at p =⇒ T 0 at p, and if an L-gauge space (X, G) is T 0 (or T 1 ), then (X, G) is T 0 at p (or T 1 at p) [24,27]. (c) There is no relation between D-connectedness and the notion of closedness or T 1 at p [27].…”

A Note on Quotient Reflective Subcategories of O-REL

“…Classical separation axioms at some point p (locally) were generalized and have been inspected in [14], where the purpose was to describe the notion of strongly closed sets (resp., closed) in arbitrary set based topological categories [19]. Moreover, the notions of compactness [20], Hausdorffness [14], regular and normal objects [21], perfectness [20], and soberness [22] have been generalized by using the closed and strongly closed sets in some well-defined topological categories over sets [20,[23][24][25][26]. Furthermore, the notion of closedness is suitable for the formation of closure operators [27] in several well-known topological categories [28][29][30].…”

Local $T_0$ and $T_1$ quantale-valued preordered spaces

“…With the advancement of lattice theory, distinct mathematical frameworks have been studied with lattice structures including lattice-valued topology [15], quantalevalued approach space [23,24,28], quantale-valued metric space [25], lattice-valued convergence space [22] and lattice-valued preordered space [15]. This motivates us to study local T 0 and T 1 separation axioms in quantale-valued preordered spaces.…”