Quasi-Elastic Spaces

Topology originated from the Koenigsbelg Seven Bridge Problem'answer provided by Euler in 1735, he found a necessary and sufficient condition of the connected unicusal network problem. As one of the most impotant branch of toplogy, general topology derived from the foundation of Mathematics analysis and functional analysis. In nearly 100 years after, the compactification and metrization in topology spaces have been deeply studied. Metric spaces are generalized to generalized metric spaces. Stratifiable space which is defined by weakening some condition of the Nagata-Smirnov-Bing metrization theorem is a class of very significant spaces in generalized metric space. Tamano and Vaughan generalized stratifiable space to elastic space and proved that every elastic space is paracompact. From then on, more and more topologists began to study elastic space. For example, Borges proved elastic space is monotonically normal. Gartside and Moody showed that elastic space is well-ordered (F) and proto-metrisable space is elastic space. etc.This thesis contains three chapters. We will generalize elastic space by replacing 'pair base'with'quasi-pair base'or'pair network'and study the generalized space.First chapter is introduction of all the prearrangement knowledge. We introduce the origin of topology firstly and then present a stratifiable space which is a class of very significant spaces in generalized metric spaces. The definition of elastic space required for our article is provided. In conclusion, all conceptions related are made into a diagram in the end of this chapter.Second chapter is about quasi-elastic space which is a generalization of elastic space. Section one is a preliminary including the conception of elasticity and its properties. Section two is main contents containing definition of quasi-pair base and quasi-elastic space. Subspaces of quasi-elastic spaces are also quasi-elastic spaces. Quasi-elastic spaces are well-ordered (F). In section three we generalize elastic spaces to network-elastic spaces and k network-elastic spaces. We obtain a conclusion that network-elastic space is equivalent to T1 space.Third chapter is about point extendable properties of quasi-pair base. Section one is a preliminary including point extension, unitary point extendable pair base, weakly point extendable pair base and theorems. In section two we define unitary point extendable quasi-pair base, quasi proto-metrisable space and prove that space having a unitary point extendable quasi-pair base is a hereditary property, space that having a unitary point extendable quasi-pair base is a quasi-elastic space. A space is quasi proto-metrisable if and only if it has a unitary point extendable quasi-pair base. Weekly point extendable quasi-pair base is first described in section three, spaces that having a weekly point extendable quasi-pair base are quasi-elastic spaces. Final, all conclusions involved in this paper are summed up and a number of questions are presented.