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.

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.…”