Journal of the
Korean Mathematical Society JKMS

An $R^n$-geometry $({\mathcal P}^n, {\mathcal L})$ 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 ${\mathcal L}$ such that the join operation $\vee : {\mathcal P}^n\times {\mathcal P}^n\setminus \triangle \longrightarrow {\mathcal L}$ is continuous. It is a notable fact that in the case $n=2$ the introduced topologies on ${\mathcal L}$ are same and the join operation $\vee : {\mathcal P}^2\times {\mathcal P}^2\setminus \triangle \longrightarrow {\mathcal L}$ 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 ${\mathcal L}$ in an $R^n$-geometry $({\mathcal P}^n, {\mathcal L})$ such that these topologies are compatible with the incidence structure of $({\mathcal P}^n ,{\mathcal L})$. Therefore, we need to study the topologies of the line set ${\mathcal L}$ in an $R^n$-geometry $({\mathcal P}^n, {\mathcal L}).$ In this paper, the relations of such topologies on the line set ${\mathcal L}$ are studied.

Keywords: topological geometry, $R^n$-geometry, continuous and open maps

MSC numbers: 51H10, 54C05, 54C08