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Barely alternating real almost chains and extension operators for compact lines

Antonio Avilés, Maciej Korpalski

Abstract

Assume $\text{MA}(κ)$. We show that for every real chain of size $κ$ in the quotient Boolean algebra $P(ω)/fin$ we can find an almost chain of representatives such that every $n\inω$ oscillates at most three times along the almost chain. This is used to show that for every countable discrete extension of a separable compact line $K$ of weight $κ$ there exists an extension operator $E:C(K)\longrightarrow C(L)$ of norm at most three.

Barely alternating real almost chains and extension operators for compact lines

Abstract

Assume . We show that for every real chain of size in the quotient Boolean algebra we can find an almost chain of representatives such that every oscillates at most three times along the almost chain. This is used to show that for every countable discrete extension of a separable compact line of weight there exists an extension operator of norm at most three.
Paper Structure (3 sections, 7 theorems, 10 equations)

This paper contains 3 sections, 7 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Barely alternating almost chains
  3. Countable discrete extensions of compact lines

Key Result

Theorem 1.3

Under $\text{MA}(\kappa)$, for a set $X$ of cardinality $\kappa$, every real almost chain $\{A_x : x \in X\}$ of subsets of $\omega$ has a barely alternating finite adjustment $\{B_x : x \in X\}$.

Theorems & Definitions (13)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 3 more