Geometric properties of some totally ordered compact sets

TL;DR

This paper constructs examples of totally ordered compact sets with complex topological properties, demonstrating non-measurability and isomorphic subspaces within certain function spaces, advancing understanding of their geometric and measure-theoretic features.

Contribution

It introduces new totally ordered non-metrizable compact sets and explores their function spaces' topological and measure-theoretic properties, including non-universal measurability.

Findings

Existence of totally ordered compact sets with non-measurable function spaces

Construction of isomorphic subspaces within these function spaces

Demonstration of differences in Borel sigma-algebras for these spaces

Abstract

In this paper, we show that there are a totally ordered compact K separable (K is Rosenthal compact set), a Hausdorff topology T' on C(K) and two closed subspaces Y1, Y2 of (C(K); Tp) such that (C(K);T') is not universally measurable, (C(K),Tp) = (Y1,Tp) + (Y2,Tp);(Y1,Tp) is isomorphic to (Y2,Tp), (Yj ,Tp) = (Yj,T'), j=1,2; and Bor((C(K)XC(K),T'XT')) is not equal to Bor(C(K)),T'))XC(K)),T')) this is the main result of this work. We start this work to construct totally ordered non metrisable compact sets K(E) from a reference set E which is totally ordered, and from a positive Borel measure on E satisfying some reasonable assumptions.