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Semi-proximal spaces and normality

Khulod Almontashery, Paul Szeptycki

Abstract

We consider the relationship between normality and semi-proximality. We give a consistent example of a first countable locally compact Dowker space that is not semi-proximal, and two ZFC examples of semi-proximal non-normal spaces. This answers a question of Nyikos. One of the examples is a subspace of $(ω+1) \times ω_1$. In contrast, we show that every normal subspace of a finite power of $ω_1$ is semi-proximal.

Semi-proximal spaces and normality

Abstract

We consider the relationship between normality and semi-proximality. We give a consistent example of a first countable locally compact Dowker space that is not semi-proximal, and two ZFC examples of semi-proximal non-normal spaces. This answers a question of Nyikos. One of the examples is a subspace of . In contrast, we show that every normal subspace of a finite power of is semi-proximal.
Paper Structure (4 sections, 14 theorems, 25 equations)

This paper contains 4 sections, 14 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. A not semi-proximal Dowker space
  3. A semi-proximal non-normal $\Psi$-space
  4. Semi-proximal and normal subspaces of finite powers of $\omega_1$

Key Result

Lemma 1

If $A\subset (\omega_1\times\{k\})$ is uncountable, then $\overline{A}(n)$ contains a club for all $n>k$

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 23 more