Semiaffine stable planes

TL;DR

This paper characterizes semiaffine stable planes as being either affine, projective, or punctured projective planes, extending the understanding of their structure in topological and linear space contexts.

Contribution

It introduces the concept of semiaffine stable planes and classifies their structure, bridging topological and linear space theories.

Findings

Semiaffine stable planes are classified as affine, projective, or punctured projective planes.

The main result holds in topological stable planes but not in general linear spaces.

Comparison with finite and infinite linear spaces highlights differences in structural properties.

Abstract

A locally compact stable plane of positive topological dimension will be called semiaffine if for every line $L$ and every point $p$ not in $L$ there is at most one line passing through $p$ and disjoint from $L$. We show that then the plane is either an affine or projective plane or a punctured projective plane (i.e., a projective plane with one point deleted). We also compare this with the situation in general linear spaces (without topology), where P. Dembowski showed that the analogue of our main result is true for finite spaces but fails in general.