The I(L)-inducification Of (IC) L-cotopological Spaces And The KKM Type Theorems In Pretopological Spaces

An induced I-space or a topologically generated I-space was defined by Weiss and Lowen in 1975 and 1976, and has been proved to be a very important class of L-topological spaces for an appropriate lattice L. In 1980, a natural generalization of induced I-spaces, namely the weakly induced I-topological spaces, was given by Martin. (IC) L-cotopological spaces studied in the first part of this paper is a type of L-cotopological spaces which is a generalization of these induced I-spaces described before. The (IC)-interfications, the (IC)-extrafications and the I(L) inducification of L-cotopological spaces are defined, then the relation of them is discussed.In 1909, Brouwer published a series of papers such as "Continuous bijections which map a curved surface to itself" to establish the coincidence theorems. From the 30's in the 20th century, the people started to pay attention to the problems of the coincident points of the set valued fuctions. In 1937, von. Neumann used the definition of the coincident pionts of set valued functions in his research of the basic theorem in game theory(the saddle points theorem). In 1991, Chang Shinsen gave the definition of generalized KKM(that is Knaster-Kuratowski-Mazurkiewicz) type functions. From then, people began to study the generalization and the application in economic of the coincidence theorems. In the second part, the author defines the preparatory finite continuous space(preFC-space), and prooves the KKM type theorems and coincidence theorems in preFC-spaces, furthermore the existence of the equilibrium points for abstract generalized vector equilibrium problems is showed in preFC-spaces.The main content of this paper is as follows:1. Following the definitions of (IC) L-cotopological spaces, some characterizations of (IC) L-cotopological spaces are given, the (IC)-interfications(that is the biggest (IC) L-cotopological space contained in the given one), the (IC)-extrafication(that is the smallest (IC) L-cotopological space containing the given one) and the I(L) inducification of L-cotopological spaces are defined. Then based on the definition of the I(L) inducification of L-cotopological spaces we obtain the conclusion that the order of the (IC)-extrafication and the I(L) inducification of L-cotopological spaces is changebale, but the order of the (IC)-interfications and the I(L) inducification of L-cotopological spaces is unchangebale.2. At first, pretopological space, preFC-space and preFC-KKM classes are defined, then several KKM type theorems and coincidence theorems in preFC-spaces are prooved, furthermore the existence of the equilibrium points for abstract generalized vector equilibrium problems is showed in preFC-spaces. These results have improved the work of the predecessors.