These operations depend on special properties of the incidence structures in question. An affine plane is called a topological affine plane if P and L carry topologies such that these operations are continuous. We require that P and L are compact spaces, and that joining and intersecting are continuous. In addition, we stipulate that D be open in the space of pairs of lines. Again, we ignore certain degenerate cases.
Classical examples are the real projective, affine, and hyperbolic planes.
Hilbert , Direct limits and maximality of stable planes . The length of such a chain is defined as the number of elements minus 1. A generalized n -gon has the following properties. The diameter is n , that is, any two elements of V are connected by a chain of length at most n . Moreover, there exists at most one chain of length less than n between any two elements of V .