Topologische Affine Ebenen Mit Nichtstetigem Parallelismus

✍ Scribed by Eberhard Eisele

• Publisher: Springer
• Year: 1991
• Tongue: English
• Weight: 1023 KB
• Volume: 40
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00189915

✦ Synopsis

In this paper, we present a general method of constructing topological affine planes having non-continuous parallelism. We prove that a topological affine plane E with point set L k x L k, and with a special 'K-algebraic slope' has a topological affine subplane with noncontinuous parallelism (Satz 4.6). Here, K is a real-closed subfield of a real-closed field L. The crucial tools needed to make our method work are the notion of a slope and the notion of Kalgebraicity, a concept which is introduced and intensively studied here. As an application of our general method, we obtain in Section 5 affine Salzmann planes with lines being bent countably infinitely often admitting a subplane with non-continuous parallelism. This provides a negative answer to a question posed by H. Salzmann 1-13, p. 52].