The line geometry of a class of linear spaces Original Research Article

In this paper we study the properties of the Grassmannian of lines of a linear space. In order to obtain a well-defined notion of a Grassmannian, we consider just a class of linear spaces; more precisely, the ones having rank greater than three in which a weaker version of the exchange axiom holds. Such spaces will be called normal linear spaces. We state six properties of the Grassmannian which turn out to be characteristic. Furthermore, by requiring that some geometric mappings are continuous, we obtain a characterization in the topological case too.