L-pre-topological Space Locally Connected And Imitation Of Compactness

Based on L.A.Zadeh's fuzzy set theory,the notion of fuzzy topological space was proposed by C.L.Chang and some elementary concepts in the general topology were extended to fuzzy topological spaces by him in 1968.As a generalization of the theory of general topology,fuzzy topology was much more complicated.Afterwards,A.S.Mashhour and M.H.Ghanim introduced the concept of fuzzy closure space,based on which,the subspaces,the sum spaces,the product spaces,etc of fuzzy closure spaces were studied by R.Srivastava and M.Srivastava.However,all the above works are in the fixed-basis setting[0,1].In 1995,professor Wu-Neng Zhou generalized the concept of fuzzy closure space to the case of F lattice and introduced the concept of L-closure space,where L is a completly distributive lattice with an order-reversing involution.He also studied many important properties of L-closure spaces.The L-pretopological space and L-closure space in the present thesis have been proved to be equivalent,and they are both extension of the theory of L-topological spaces.Juan Lu has introduced the concepts such as conntedness and N-compactness of L-fuzzy topology spaces to L-closure spaces and discussed their properties.As a further research work of Juan Lu,the present thesis mainly studies the local connectedness and paracompactness in L-pretopological spaces.The present thesis is organized as follows:Chapter 1.Preliminaries.In this chapter,we make a depiction for basic knowledge and basic conclusions of fuzzy sets,lattice theory and category theory which would be used in the following chapters.Chapter 2.Firstly,the equivalent characterizations as well as some properties of L-pretopogical spaces are investigated.Then the local connectedness in L-pretopological space is defined and some equivalent descriptions for L-pretopological spaces of local connectedness are given.Moreover,some properties such as toplogy invariance;the sum property;the product property;the quotient property of L-pretopological spaces are investigated.Chapter 3.The paracompactness in L-pretopological spaces is studied.Similar to the I-strong paracompactness in L-topological sapces,the notion of paracompactness is also introduced in L-pretopological spaces.Some properties such as the product property, the closed hereditary,L-extension,etc are maily investigated.