Topological planes

On the existence of topological ovals in flat projective planes

“…The real Desarguesian projective plane is the classical example of a fiat projective plane, that is, a projective plane which shares the point set with the real Desarguesian projective plane and whose lines are homeomorphic to the circle § 1 . For non-isomorphic examples the reader is referred to [3) and [4,Chapter 3). Using transfinite induction ( cf.…”

“…It is clear from axioms (Pl) and (P2) and from what we just said that the dual of a ( topological) projective plane is a ( topological) projective plane. In particular, the dual of a flat projective plane is topological and if we regard a point as the set of all lines it is contained in, this dual can be seen to be a flat projective plane, too.· For details the reader is referred to [3] and [4].…”

The inner and outer space of 2-dimensional Laguerre planes

“…From what we just said it is clear that D Y<L /y is an incidence geometry with the real projective plane as point set and sets homeomorphic to S 1 as circles, or better, lines. By [Sa,Theorem 2.5] it now suffices to show that given two distinct points in this geometry there is a uniquely determined line connecting both points. This translates back to showing that the pencil of circles through an arbitrary point p £ § 2 and y(p) is a foliation of S 2 , that is, every point in S 2 \ [p, y(p)} is contained in exactly one such circle.…”

Semi-classical projective planes over half-ordered fields

“…The resulting projective planes are topological projective planes in the sense that the point set and the set of lines carry Hausdorff topologies such that the geometric operations of joining two distinct points by a line and intersecting two distinct lines in a point are continuous; cf. [9]. We call the planes Ph,g semi-classical projective planes because the geometries and topologies on A+ = R + x R and A-= R -x R are the same as on the corresponding subsets of the (topological) real Desarguesian projective plane.…”