Type: Article

On Borel ?-algebras of topologies generated by two-point selections

García-Ferreira, S.;

Abstract:

A two-point selection on a set X is a function f :[X]2 ? X such that f(F) ? F for every F ? [X]2. It is known that every two-point selection f : [X]2 ? X induced a (Tychonoff) topology ?f on X by using the left and right open intervals generated by the relation: x ? y if either f({x,y})=x or x = y, for every x,y ? X. In this paper, we are mainly concern with the two-point selections on the real line R. We study the ?-algebras of Borel, each one denoted by Bf(R), of the topologies ?f’s defined by a two-point selection f on R. By using also the intervals induce by a twopoint selection f it is possible to define an outer measure on R which generalize the Lebesgue outer measure, denoting by Mf its corresponding measurable sets. It is proved that if C ? R satisfies that |C| = |R\C| = c, then there is f ? Sel2(R) such that C ?Mf \Bf(R). We construct a family {f? : ?<2c} of two-point selections on R such that Bf? (R)= Bf? (R) for distinct ?,? < 2c and they are metrizable. Besides, we construct a set {f? : ?<2c} of two-point selections such that each ?f? is second countable and Bf? (R)= Bf? (R) for distinct ordinals ?,? < 2c. By assuming that c =2<c and c is regular, we show the existence of 22c many ?-algebras on R that contain [R]?? and none of them is the ?-algebra of Borel of ?f for any two-point selection f on R. Several examples are given to illustrate some properties of these Borel ?-algebras.