Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The class $C(ω_1)$ and countable net weight

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy

Abstract

Hart and Kunen, and independently in the recent preprint arXiv:2304.13113, Ríos-Herrejón defined and studied the class $C(ω_1)$ of topological spaces $X$ having the property that for every neighborhood assignment $\{U(y) : y \in Y\}$ with $Y \in [X]^{ω_1}$ there is $Z \in [Y]^{ω_1}$ such that $$Z \subset \bigcap \{U(z) : z \in Z\}.$$ It is obvious that spaces of countable net weight, i.e. having a countable network, belong to this class. In this paper we present several independence results concerning the relationships of these and several other classes that are sandwiched between them. These clarify some of the main problems that were raised in the above preprint. In particular, we prove that the continuum hypothesis, in fact a weaker combinatorial principle called super stick, implies that every regular space in $C(ω_1)$ has countable net weight, answering a question that was raised by Hart and Kunen.

The class $C(ω_1)$ and countable net weight

Abstract

Hart and Kunen, and independently in the recent preprint arXiv:2304.13113, Ríos-Herrejón defined and studied the class of topological spaces having the property that for every neighborhood assignment with there is such that It is obvious that spaces of countable net weight, i.e. having a countable network, belong to this class. In this paper we present several independence results concerning the relationships of these and several other classes that are sandwiched between them. These clarify some of the main problems that were raised in the above preprint. In particular, we prove that the continuum hypothesis, in fact a weaker combinatorial principle called super stick, implies that every regular space in has countable net weight, answering a question that was raised by Hart and Kunen.
Paper Structure (4 sections, 20 theorems, 82 equations, 1 figure)

This paper contains 4 sections, 20 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conditions implying $\mathop{\mathrm{nw}}\nolimits(X) < \kappa$ for $X \in C(\kappa)$
  3. Cohen forcing and $C({\omega}_1) \cap T_3 \subset K(\omega_1)$
  4. Some separation results

Key Result

Theorem 1.5

The combinatorial principle ${} \mathord{\hbox{$|$} } }_{\hbox{\tiny SUPER}}$, an obvious consequence of CH, implies $\mathop{\mathrm{nw}}\nolimits(X)={\omega}$ for any $X\in C({\omega}_1) \cap T_3$.

Figures (1)

  • Figure 1: Results concerning regular spaces

Theorems & Definitions (45)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 2.1
  • proof
  • ...and 35 more