Strongly Connected Spaces And Locally Strongly Connected Spaces

Abstract Connectedness is an important property in topology. Based on the definition of connectedness, in this paper, the strongly connectedness is defined, the sub-continuous mapping and continuous mapping are introduced and the prop-erties of strongly connected space and locally strongly connected space are discussed. Meanwhile, the topological properties of two categories are studied.The key points and the main contents of this paper are as follows:The first chapter studies on strongly connected space. Some relevant concepts and important conclusions about strongly connected space are introduced in this chapter.The second chapter concentrates on locally strongly connected space. By using the method of topology, the definition of locally strongly connected space and its second definition are introduced, and the properties of sum, quotient and product are discussed.The third chapter focuses on a theorem of category about locally strongly con-nected space, that is, the category constituted by locally strongly connected space and continuous mapping is a topological construct.The fourth chapter, based on the studies in previous three chapters, further raises many questions that are worth researching. Take the following three questions for example. First, using the way of the second definition of connected space, the 2d space reflected by connected space is changed into Z-space reflected by strongly connected space. Under this method, is there any new connected space if Z-space is changed? Second, is the set constructed by locally strongly connected space and sub-continuous mapping a category? If yes, is it a topological construct? Third, can category theorem be generalized to L-topology? Among those questions, some still remains unsolved, and more research should be done in the future.