Pointwise k-Pseudo Metric Space

In this paper, the concept of a k-(quasi) pseudo metric is generalized to the L-fuzzy case, called a pointwise k-(quasi) pseudo metric, which is considered to be a map d:J(LX)×J(LX)⟶[0,∞) satisfying some conditions. What is more, it is proved that the category of pointwise k-pseudo metric spaces is isomorphic to the category of symmetric pointwise k-remote neighborhood ball spaces. Besides, some L-topological structures induced by a pointwise k-quasi-pseudo metric are obtained, including an L-quasi neighborhood system, an L-topology, an L-closure operator, an L-interior operator, and a pointwise quasi-uniformity.

On Strongly b − θ -Continuous Mappings in Fuzzifying Topology

In this article, we will define the new notions (e.g., b − θ -neighborhood system of point, b − θ -closure (interior) of a set, and b − θ -closed (open) set) based on fuzzy logic (i.e., fuzzifying topology). Then, we will explain the interesting properties of the above five notions in detail. Several basic results (for instance, Definition 7, Theorem 3 (iii), (v), and (vi), Theorem 5, Theorem 9, and Theorem 4.6) in classical topology are generalized in fuzzy logic. In addition to, we will show that every fuzzifying b − θ -closed set is fuzzifying γ -closed set (by Theorem 3 (vi)). Further, we will study the notion of fuzzifying b − θ -derived set and fuzzifying b − θ -boundary set and discuss several of their fundamental basic relations and properties. Also, we will present a new type of fuzzifying strongly b − θ -continuous mapping between two fuzzifying topological spaces. Finally, several characterizations of fuzzifying strongly b − θ -continuous mapping, fuzzifying strongly b − θ -irresolute mapping, and fuzzifying weakly b − θ -irresolute mapping along with different conditions for their existence are obtained.

Fuzzy Knowledge Distance with Three-Layer Perspectives in Neighborhood System

Information granule is the basic element in granular computing (GrC), and it can be obtained according to the granulation criterion. In neighborhood rough sets, current uncertainty measures focus on computing the knowledge granulation of single granular space and have two main limitations: (i) neglecting the structural information of boundary regions and (ii) the inability to reflect the difference between neighborhood granular spaces with the same uncertainty for approximating a target concept. Firstly, a fuzziness-based uncertainty measure for neighborhood rough sets is introduced to characterize the structural information of boundary regions. Moreover, from the perspective of distance, based on the idea of density peaks, we present a fuzzy-neighborhood-granule-distance- (FNGD-) based method to discover the relationship between granules in a granular space. Then, to characterize the difference between granular spaces for approximating a target concept, we present the fuzzy neighborhood granular space distance (FNGSD) and fuzzy neighborhood boundary region distance (FNBRD). FNGD, FNGSD, and FNBRD are hierarchically organized from fineness to coarseness according to the semantics of granularity, which provide three-layer perspectives in the neighborhood system.

The Lattice Structures of Approximation Operators Based on L-Fuzzy Generalized Neighborhood Systems

Following the idea of L -fuzzy generalized neighborhood systems as introduced by Zhao et al., we will give the join-complete lattice structures of lower and upper approximation operators based on L -fuzzy generalized neighborhood systems. In particular, as special approximation operators based on L -fuzzy generalized neighborhood systems, we will give the complete lattice structures of lower and upper approximation operators based on L -fuzzy relations. Furthermore, if L satisfies the double negative law, then there exists an order isomorphic mapping between upper and lower approximation operators based on L -fuzzy generalized neighborhood systems; when L -fuzzy generalized neighborhood system is serial, reflexive, and transitive, there still exists an order isomorphic mapping between upper and lower approximation operators, respectively, and both lower and upper approximation operators based on L -fuzzy relations are complete lattice isomorphism.

Ordinary Single Valued Neutrosophic Topological Spaces

We define an ordinary single valued neutrosophic topology and obtain some of its basicproperties. In addition, we introduce the concept of an ordinary single valued neutrosophic subspace.Next, we define the ordinary single valued neutrosophic neighborhood system and we show thatan ordinary single valued neutrosophic neighborhood system has the same properties in a classicalneighborhood system. Finally, we introduce the concepts of an ordinary single valued neutrosophicbase and an ordinary single valued neutrosophic subbase, and obtain two characterizations of anordinary single valued neutrosophic base and one characterization of an ordinary single valuedneutrosophic subbase.

On Tri-α-Open Sets in Fuzzifying Tritopological Spaces

In this paper, we introduced and studied (1,2,3)-α-open set, (1,2,3)-α-neighborhood system, (1,2,3)-α-derived, (1,2,3)-α-closure, (1,2,3)-α-interior, (1,2,3)-α-exterior, (1,2,3)-α-boundary, (1,2,3)-α-convergence of nets, and (1,2,3)-α-convergence of filters in fuzzifying tritopological spaces.