Homeomorphisms of two-point sets

Abstract

Abstract. Given a cardinal κ ≤ c, a subset of the plane is said to be a κ-point set if and only if it meets every line in precisely κ many points. In response to a question of Cobb, we show that for all 2 ≤ κ, λ < c there exists a κ-point set which is homeomorphic to a λ-point set, and further, we also show that it is consistent with ZFC that for all 2 ≤ κ < c, there exists a κ-point set X such that for all 2 ≤ λ < c, X is homeomorphic to a λ-point set. On the other hand, we prove that it is consistent with ZFC that for all 2 ≤ κ, λ < c, there exists a κ-point set, such that for all homeomorphisms f : R 2 → R 2 , if f (X) is a λ-point set, then λ = κ.