On the concept of topological well-ordered space

Since the general definition of a topology is based on the characteristics of the standard Euclidean topology, the relationships between the ordering on real numbers and its topology have been generalized over time and studied from many aspects. The compatibility of partially ordered sets with the topology on it has been studied by many researchers. Wellorderness, on the other hand, is an important concept of set theory. However, there is no direct relationship between topology and the concept of well-orderness. In this study, we define the concept of topological well-orderness, which we regard as a topological generalization of well-orderness in set theory, and investigate its basic properties. In addition, the relationship between graphs and topological well-ordered spaces is examined.