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Q-Sets, Δ-Sets, and L-Spaces

Pourya Memarpanahi, Paul Szeptycki

TL;DR

The paper investigates whether Lindelöf $Q$-set spaces or Lindelöf $Δ$-set spaces exist in ZFC, focusing on Moore’s L-space as a crucial test case. It employs the Moore construction, including a $C$-sequence, the oscillation function, and coherence arguments, to show that Moore’s $L$-space is not a $Q$-set space in ZFC. It further shows that, under the assumption that all Aronszajn trees are special, Moore’s $L$-space is not a $Δ$-set space, linking Δ-ness to tree properties. These results clarify the relationship between Lindelöf $Q$-set and $Δ$-set spaces and illuminate how tree-theoretic assumptions influence the possible topological structures of L-spaces, with implications for the existence of such spaces in ZFC and related forcing contexts.

Abstract

The question whether there is a Lindelof Q-set space or Lindelof $Δ$-set space is considered. We show that J. Moore's ZFC $L$-space is not a Q-set space in ZFC and, assuming all Aronszajn trees are special, it is not a $Δ$-set space.

Q-Sets, Δ-Sets, and L-Spaces

TL;DR

The paper investigates whether Lindelöf -set spaces or Lindelöf -set spaces exist in ZFC, focusing on Moore’s L-space as a crucial test case. It employs the Moore construction, including a -sequence, the oscillation function, and coherence arguments, to show that Moore’s -space is not a -set space in ZFC. It further shows that, under the assumption that all Aronszajn trees are special, Moore’s -space is not a -set space, linking Δ-ness to tree properties. These results clarify the relationship between Lindelöf -set and -set spaces and illuminate how tree-theoretic assumptions influence the possible topological structures of L-spaces, with implications for the existence of such spaces in ZFC and related forcing contexts.

Abstract

The question whether there is a Lindelof Q-set space or Lindelof -set space is considered. We show that J. Moore's ZFC -space is not a Q-set space in ZFC and, assuming all Aronszajn trees are special, it is not a -set space.
Paper Structure (2 sections, 9 theorems, 25 equations)

This paper contains 2 sections, 9 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Moore's L-space

Key Result

Proposition 1

B-note If there is an L-space in $\operatorname{ZFC}$, then under $\operatorname{MA}(\omega_1),$ there exists a Lindelöf $Q$-set space which is not hereditarily separable.

Theorems & Definitions (28)

  • Proposition 1
  • proof
  • Definition 1
  • Definition 2
  • Definition 3: Upper Trace Function
  • Definition 4: Lower Trace Function
  • Remark 1
  • Remark 2
  • Definition 5
  • Definition 6: Oscillation Function
  • ...and 18 more