M-Fuzzifying K-Pseudo Metric Space And Its Induced M-Fuzzigying Structures

Metric space plays an important role in the theoretical research and practical application of mathematics.Fuzzy mathematics is a mathematical theory and method for studying and dealing with fuzzy phenomena.With the development of fuzzy set theory,classical metrics were generalized to fuzzy cases.Fuzzy metrics are not only widely used in fuzzy control,fuzzy image processing,fuzzy neural networks,fuzzy data analysis and other fields,but also its theoretical research has attracted much attention,so this paper mainly introduces the M-fuzzifying k-pseudo metric based on monotonically decreasing fuzzy non-negative real numbers and its induced M-fuzzifying topology and M-fuzzifying convex structure,enriching and improving the research content of fuzzy metrics.Firstly,the definition of M-fuzzifying k-pseudo metric and the example of " k " are introduced,and the definition of M-fuzzifying topologies and M-fuzzifying convex structure are reviewed,where M-fuzzifying topologies include an M-fuzzifying neighborhood system,an M-fuzzifying topology,an M-fuzzifying closure operator and an M-fuzzifying interior operator;M-fuzzifying convex structures include an M-fuzzifying interval operator,an M-fuzzifying convex structure and an M-fuzzifying hull operator.Secondly,the construction methods of some M-fuzzifying topologies and M-fuzzifying convex structures induced by an M-fuzzifying k-pseudo metric are given,and the rationality of the construction method of the M-fuzzifying structures given in this paper is proved from the perspective of theoretical demonstration.In addition,it is proved that there is a one-to-one correspondence between M-fuzzifying k-pseudo metric and crisp nests of k-pseudo metric in this paper,that is,M-fuzzifying k-pseudo metric and crisp nests of k-pseudo metric can induce each other.Finally,because crisp nests of k-pseudo metric can induce the classical topology,and crisp nests of topology can generate M-fuzzifying topology,at the same time,M-fuzzifying k-pseudo metric can also induce M-fuzzifying topology.In this paper,it is proved that the M-fuzzifying topology induced by an M-fuzzifying k-pseudo metric is consistent with the M-fuzzifying topology induced by crisp nests of k-pseudo metric.