Generalized Metric Spaces And Coverning Approximation Spaces

This paper consists of two parts and is divided into five chapters. The first four chapters centre on the assumption of Alexandrof, and give characterizations of some generalized metric spaces by means of images of metric spaces. In the last chapter, we discuss the applications of general topology in rough-set theory.In Chapter 1, we investigate so-metrizable spaces. This class of spaces is an im-portant class of generalized metric spaces between metric spaces and sn-metric spaces. We give some characterizations of so-mctrizable spaces and we discuss properties for invariancc and inverse invariance under mappings. Our results enrich and perfect the known theory on so-metrizable spaces.In Chapter 2. we investigate strongly g-developable spaces introduced by Y. Tanaka and Y. Ge for investigating quotient images of metric spaces. We prove that a space is a strongly g-developable space iff it is a sequence-covering, quotient, compact, mssc-imagc of a metric space. This result answers a question about strongly g-developable spaces posed by Y. Tanaka and Y. Ge.In Chapter 3, weak Cauchy sn-symmctric spaces and sn-symmetric spaces in-troduced by S. Lin and Y. Ge are investigated. We characterize weak Cauchy sn-symmetric spaces in terms of images of metric spaces and we give some mapping theorems on weak Cauchy sn-symmetric spaces. As applications of these results, we give some mapping theorems on weak Cauchy symmetric spaces and weak Cauchy semi-metric spaces, which improves correspondent results given by Y. Tanaka.In Chapter 4, we investigate relations between mappings and networks in Ponomarev-systems. We give an example to show that, in Ponomarev-system (f,M, X,g), point-finite property of g does not imply compactness of f. As further investigations for Ponomarev-systems, we give sufficient and necessary conditions for f being compact and g being point-finite in Ponomarev-systems (f,M,X,g) respectively. One of these results corrects a known conclusion on Ponomarev-systems. In addition, we es-tablish some relations between mappings and networks in Ponomarev-systems.At the beginning of Chapter 5, we give a brief introduction on the background of rough set theory and its relationships with point-set topology. Then, by using a proof technique proposed by Z. Balogh et al. for constructing a class of typical symmetrie neighborhood assignments, we answer an open problem raised by W. Zhu on axiomatic issue of a type of covering upper approximation operators, and present a topological property of this covering upper approximation operator. We also give a topological characterization of unary coverings which is a basic concept in covering approximation space. Finally, under the conditions that two important covering upper approximation operators are closure operators, we give topological descriptions for covering approximation space.