Study On Some Properties Of L-closure Spaces

In 1965, Professor L. A. Zadeh established the theory of fuzzy set and extended the crisp set. Then, in 1968, C. L. Chang introduced the concept of fuzzy topological space, and some concepts in general topology were extended to fuzzy topological spaces. During the development of the theory of fuzzy topological spaces, the study on fuzzy closure spaces is proceeding. Although fuzzy closure space is similar to fuzzy topological space to some extent, there are some differences between them, some of which are interesting. A. S. Mashhour and M. H. Ghanim introduced the concept of fuzzy closure space in "Fuzzy closure space", and R. Srivastava defined fuzzy closure space also. Based on the latter, Rekha Srivastava and Manjari Srivastava studied subspace of a fuzzy closure space, the sum fuzzy closure space of a family of pairwise disjoint fuzzy closure spaces and product space of a family of fuzzy closure spaces. However, all the above works are in the fixed-basis setting [0,1]. Professor Wu-Neng Zhou generalized the concept of fuzzy closure space and introduced the concept of L-closure space, where L is a completely distributive lattice with an order-reversing involution, and studied many important properties for L-closure spaces, such as category property, especially.Based on the definition of L-closure operator, the author defines L-closure space, L-open set, L-closed set, L-continuous mapping and L-homeomorphism between L-closure spaces, connected L-subset and N-compact L-subset in L-closure space and so on, and discusses some properties of L-closure spaces, such as Urysohn lemma in L-closure spaces and union of a family of connected L-subsets which satisfies some conditions is connected. At last, the relation between density and cocellularity of an L-topological space is discussed and two ralated theorems are extended. Dependent set of an L-subset in a product L-topological space is defined and a result on dependent set of union L-subsets is given. In this paper, the author defines L according to different contents in order to make the conclusions as wide as possible. The main content of this paper is as follows:1. Following the definition of L-closure operator, L-closure space, L-open set, L-closed set are defined, and several other methods to define L-closure space are given. L-continuous mapping, L-open mapping, L-closed mapping and L- homeo-morphism are defined and Urysohn lemma in L-closure spaces is proved.