On Borel $σ$-algebras of topologies generated by two-point selections

Abstract

a topology τ f on X by using the relation: x ≤ y if either f ({x, y}) = x or x = y, for every x, y ∈ X. We are mainly concern with the two-point selections on the real line R. In this paper, we study the σ-algebras of Borel, each one denoted by B f (R), of the topologies τ f 's defined by a two-point selection f on R. We prove that the assumption c = 2 <c implies the existence of a family {fν :for distinct µ, ν < 2 c . By assuming that c = 2 <c and c is regular, we also show that there are 2 2 c many σ-algebras on R that contain [R] ≤ω and none of them is the σ-algebra of Borel of τ f for any two-point selection f : [R] 2 → R. Several examples are given to illustrate some properties of these Borel σ-algebras.