Determinination Of M-closure Systems And Their Products, Sums, And Quotients

The determining of a co-topology (the whole closed sets of a topo-logical space) is an interesting question. From papers [17,18,23,25] we can see that pre-interior operators, pre-exterior operators, pre-boundary operators, pre-derived operators, pre-difference derived operators, pre-neighborhood operators and pre-remote neighborhood operators can be used to determine the pre-co-topology (a generalization of the co-topology). Closure systems (a generalization of the pre-co-topology) is a structure involved by many aspects in mathematics and computer science (see [7,13]). In this paper, the closure system is generalized to the M-closure systems, and it is proved that the M-closure systems, M-weak closure operators, M-weak interior operators, M-weak exterior operators, M-weak boundary operators, M-weak derived operators, M-weak difference derived operators, M-weak neighbor-hood operators, M-weak remote neighborhood operators and M-weak neighborhood operators can be determined reciprocally. In addition, it is proved that the category M-CS of all M-closure spaces and continuous mappings between them is a topologi-cal construct, but not cartesian closed (where M is any nonempty index set). Based on the category theory, the notions of product M-closure spaces, sum M-closure spaces, and quotient M-closure spaces are defined.The main points of this thesis are as follows:In the first chapter, some concepts of M-closure systems and category are in-troduced.In the second chapter, the partial order relations are defined on M-WCL(X) (the set of all M-weak closure operators of X), M-WIN(X) (the set of all M-weak interior operators of X), M-WOU(X) (the set of all M-weak exterior operators of X), M-WB(X) (the set of all M-weak boundary operators of X), M-WD(X) (the set of all M-weak derived operators of X), M-WD*(X) (the set of all M-weak difference derived operators of X), M-WR(X) (the set of all M-weak remote neighborhood operators of X), M-WN(X) (the set of all M-weak neighborhood operators of X), and it is proved tant M-CS(X), M-WCL(X),M-WIN(X), M-WOU(X),M-WB(X), M-WD(X),M-WD*(X), M-WR(X) and M-WN(X) can be determined reciprocally. In the third chapter, it is proved that the category M-CS of all M-closure spaces and continuous mappings between them is a topological construct. Based on the category theory, the notions of product M-closure spaces, sum M-closure spaces, and quotient M-closure spaces are defined. Finally,we proved M-closure system category is not cartesian closed (where M is any nonempty index set).