TOPOLOGIES AND INCIDENCE STRUCTURE ON Rn-GEOMETRIES
Jang-Hwan Im · Journal of the Korean Mathematical Society · 2002
An R $^{n}$ -geometry (P $^{n}$ , L) is a generalization of the Euclidean geometry on R $^{n}$ (see Def. 1.1). We can consider some topologies (see Def. 2.2) on the line set L such that the join operation V : P $^{n}$ $\times$ P $^{n}$ \ $\Delta$ longrightarrow L is continuous. It is a notable fact that in the case n = 2 the introduced topologies on L are same and the join operation V : P $^2$ $\times$ P $^2$ \ $\Delta$ longrightarrow L is continuous and open [10, 11]. It is a fundamental topological property of plane geometry, but in the cases n $\geq$ 3, it is no longer true. There are counter examples [2]. Hence, it is a fundamental problem to find suitable topologies on the line set L in an R $^{n}$ -geometry (P $^{n}$ , L) such that these topologies are compatible with the incidence structure of (P $^{n}$ , L). Therefore, we need to study the topologies of the line set L in an R $^{n}$ -geometry (P $^{n}$ , L). In this paper, the relations of such topologies on the line set L are studied.