Set-theoretic constructions of two-point sets

Abstract

Abstract. A two-point set is a subset of the plane which meets every line in exactly two points. By working in models of set theory other than ZFC, we demonstrate two new constructions of two-point sets. Our first construction shows that in ZFC + CH there exist two-point sets which are contained within the union of a countable collection of concentric circles. Our second construction shows that in certain models of ZF, we can show the existence of two-point sets without explicitly invoking the Axiom of Choice.1. Introduction. Given a cardinal κ, a subset of the plane is said to be a κ-point set if and only if it meets every line in exactly κ many points and is said to be a partial κ-point set if and only if it meets every line in at most κ many points. We are particularly interested in the case that κ = 2, and we refer to such sets as two-point sets.The existence of two-point sets was first shown by Mazurkiewicz [8]. (A French translation is given in [9].) A two-point set is classically constructed via transfinite recursion, making use of an arbitrary well-ordering of the collection of all lines in the plane. We will demonstrate two constructions of two-point sets which result by varying the use of this well-ordering.The first construction shows that working in ZFC+CH, we can construct a two-point set which is contained in the union of a countable collection of concentric circles of unbounded radius. In general, classical constructions of two-point sets can be thought of as starting with a c-point set and then refining it to a two-point set, and so this construction is of interest because we obtain a two-point set by refining an ℵ 0 -point set. We achieve this development by being careful about the well-ordering of the collection of all lines that we choose to help us in our construction.The second construction shows that in ZF, if we assume that the real line satisfies a certain number of properties, we can construct a two-point set without explicitly invoking the Axiom of Choice. It is a question of Erdős,

Homeomorphisms of two-point sets

“…However, to demonstrate the existence of n-point sets for 2 ≤ n < ℵ 0 , it seems apparent that we must resort to transfinite techniques. The standard approach, which we take in this paper, is essentially due to Mazurkiewicz 2 [13] and is based on the existence of a wellordering of the real line, but we note that Chad et al [3] describe an alternative construction of two-point sets which is consistent with ZF and requires only that some suitable fragment of the real line be well-ordered.…”

“…Mauldin gave three problems concerning two-point sets, and to this day, the only remaining problem is to determine if a two-point set can be chosen to be a Borel subset of the plane. 3 This problem is apparently very deep, and it is likely that if we are to make any progress on it, then we will need to further our knowledge about the structure of two-point sets.…”

Some 2-point sets

“…Lemma 1 is Lemma 4.1 of [1]. As stated in [1], it says that there are at most 23 possible r; this does not seem to be the optimal number.…”

“…There have been other constructions of 2-point sets such as in [3] where it was shown that they exist in arbitrary vector spaces over arbitrary infinite fields. Recently, Chad, Knight & Suabedissen [1] proved that the Continuum Hypothesis implies that there is a 2-point set included in the union of countably many concentric circles. Their conclusion is even stronger, being precisely the statement of the Theorem below.…”

“…Their conclusion is even stronger, being precisely the statement of the Theorem below. Subsequent to [1], Miller [6] constructed models of ZFC in which the continuum is arbitrarily large and there is a 2-point set that is included in the union of ω 1 circles.…”