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Geometric properties of some totally ordered compact sets

Mohammad Daher, Khalil Saadi

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.

Geometric properties of some totally ordered compact sets

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.

Paper Structure

This paper contains 7 sections, 62 theorems, 92 equations.

Table of Contents

  1. Construction of totally ordered compact sets
  2. Surjectivity of $h$
  3. Continuity of $h$
  4. Geometric properties of the compact sets $K$
  5. Construction of a totally ordered compact set homeomorphic to $K_{\mu }(E)$
  6. General properties of ($C(L),\tau _{p})$ when L is a Rosenthal compact set
  7. The case of a separable connected abelian group verifying the condition (**)

Key Result

Lemma 1

The point $h_{\mu ,E}^{{}}(\theta )$ is unique.

Theorems & Definitions (98)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 1
  • Example 1
  • Example 2
  • Proposition 1
  • Definition 1
  • Corollary 1
  • Corollary 2
  • ...and 88 more