Topologies and incidence structure on \(R^n\)-geometries (Q2785254)

An \(\mathbb R ^n\)-geometry consists of the point space \(\mathbb R ^n\) (with \(n \geq 2\)) and a system \({\mathcal L}\) of subsets, called lines, such that every two points are on a unique line and all lines are topologically closed and homeomorphic to \(\mathbb R\). If, in addition, \({\mathcal L}\) carries a topology such that the map of joining two points by a line is continuous, then \((\mathbb R ^n, {\mathcal L})\) is called a topological \(\mathbb R ^n\)-geometry. According to a result of Skornyakov [see \textit{H. Salzmann} et al.: Compact Projective Planes, de Gruyter (1996; Zbl 0851.51003)], \(\mathbb R ^2\)-geometries (``planes'') are automatically topological with respect to the Hausdorff topology on \({\mathcal L}\). This statement is false in higher dimensions [cf. \textit{D. Betten}, Result. Math. 12, 37-61 (1987; Zbl 0631.51006)]. NEWLINENEWLINENEWLINEIn view of this situation, the author investigates several other topologies on the line space of an \(\mathbb R ^n\)-geometry. For the ``classical'' topologies (i.e. the final topology \(F\), the open meet topology \(OM\) and the Hausdorff topology \(H\)) he proves: The join map is continuous and open with respect to some Hausdorff topology \(\mathcal T\) on \({\mathcal L}\) if, and only if, \({\mathcal T} = F = OM = H\). In this case, \(({\mathcal L}, T)\) is a locally compact second countable space. A sufficient condition for \(F=OM=H\) is also given: Endow the set \(\mathcal H\) of all parametrizations of the elements of \(\mathcal L\) with the compact-open topology. Fix two real numbers \(p,q\) with \(p < q\). If the quotient map \({\mathcal H} \rightarrow ({\mathcal L}, H)\) is continuous and if the map \({\mathcal H} \rightarrow \mathbb R ^n \times \mathbb R ^n\); \(l \mapsto (l(p),l(q))\) is open, then \(F=OM=H\).