Topological geometry
Topological geometry deals with incidence structures consisting of a point set and a family of subsets of called lines or circles etc. such that both and carry a topology and all geometric operations like joining points by a line or intersecting lines are continuous. As in the case of topological groups, many deeper results require the point space to be (locally) compact and connected. This generalizes the observation that the line joining two distinct points in the Euclidean plane depends continuously on the pair of points and the intersection point of two lines is a continuous function of these lines. Linear geometries Linear geometries are incidence structures in which any two distinct points and are joined by a unique line . Such geometries are called topological if depends continuously on the pair with respect to given topologies on the point set and the line set. The dual of a linear geometry is obtained by interchanging the roles of points and lines. A survey of linear topological geometries is given in Chapter 23 of the Handbook of incidence geometry. The most extensively investigated topological linear geometries are those which are also dual topological linear geometries. Such geometries are known as topological