Konstruktion topologischer affiner Ebenen mit nichtstetigem Parallelismus durch Knicken von Geraden

Abstract

We prove the existence of a class of topological affine planes having non-continuous parallelism by using [2, Satz 5.2]. For this, we introduce a new method of constructing affine Salzmann-planes with a ‘monotonically increasing’ slope (see 2.1) by bending lines on two special curves, which are not necessary lines. Furthermore, the limit inferior of a sequence of topological planes with fixed point space is defined. As application of our new method, we construct a sequence of affine Salzmann-planes such that the limit inferior of this sequence is again an affine Salzmann-plane and fulfils the assumptions of [2, Satz 5.2]. Applying this theorem repeatedly, we get a sequence of non-isomorphic topological affine subplanes with non-continuous parallelism.