An R

-geometry (P

, L) is a generalization of the Euclidean geometry on R

(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

P

\

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

P

\

longrightarrow L is continuous and open [10, 11]. It is a fundamental topological property of plane geometry, but in the cases n

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

-geometry (P

, L) such that these topologies are compatible with the incidence structure of (P

, L). Therefore, we need to study the topologies of the line set L in an R

-geometry (P

, L). In this paper, the relations of such topologies on the line set L are studied.