| In this paper, the concept of a WF - fuzzy metric space is introduced. Some fundamental properties of WF - fuzzy metric spaces are researched in a deepgoing way. Trie main indues seven sections:1. The characterization and some important examples of WF - fuzzy metric spaces are given.2. The fuzzy topological structure and the crisp topological structure of a WF -fuzzy metric space are described.The equivalent characterizations of convergence of a sequence of fuzzy points in the two topological structure are discussed. What' s more, the characterizations of opener and closed set and closure are also given.4. The two kinds of density of WF - fuzzy metric spaces and the two kinds of separability corresponding to them are researched.5. The three kinds of completeness of WF - fuzzy metric spaces, that is, the two kinds of completeness about its fuzzy topological structure (completeness and completeness on every layer) and the completeness about its crisp topological structure, are studied. Some of their properties and relations among them are discussed. In addition, the nested closed - ball theorem on WF - fuzzy metric spaces is proved.6. The fixed point theorems for mappings of contractive type and locally contractive type on WF - fuzzy metric spaces, which extend several important fixed point theorems, are established.7.The relations between a WF - fuzzy metric space (X,p) and the crisp metric spaces (X,px)(XG (0,1]) that are generated by the the WF-fuzzy metric space (X, p) are investigated, and the intrinsic connections between them are revealed. |
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