Planar-topologische Fastkörper

Abstract

Topological planar nearfields (F,T) are suitable for coordinatizing topological affine and projective planes only if the solutions of equations of the type\( ax - bx = c, a \neq b \), depend continuously on\( a, b, c \in F \). In this case we call T a p-topology and deal with the problem, under which conditions a nearfield topology even is a p-topology. Satisfactory answers are given in each of the following three situations:¶¶1. T is induced by a valuation.¶2. T is locally compact.¶3. T is derived from a coupling \( \kappa \) on a topological skewfield with an open kernel and with continuous images \( \kappa(a) \).¶¶Furthermore, it is shown that the interval topology of an ordered nearfield always is a p‐topology.