The Two Points Theorem of Mazurkiewicz

✍ Scribed by Péter Komjáth; James H. Schmerl

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 96 KB
• Volume: 99
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.2002.3278

✦ Synopsis

It is proved that Mazurkiewicz's construction of a subset of the plane meeting each line at exactly two points can be generalized to any vector space over an infinite field. # 2002 Elsevier Science (USA) An old theorem of Mazurkiewicz [2] asserts that the plane R 2 has a subset which meets each line at exactly two points. The statement of this theorem does not involve any of the structure of R 2 beyond its being a vector space over a field. How much does this result actually depend upon the field being R? And how much does it depend upon what the dimension of the vector space is? The purpose of this note is to point out that the answer to both of these questions is: very little.

Theorem. Let V be a vector space over an infinite field F. Then there is a subset M V such that whenever ' V is a line, then jM \ 'j ¼ 2: Let k ¼ jFj5@ 0 : The usual inductive proof of the Mazurkiewicz theorem, for example, as in the survey article [1], can be adapted here as long as