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Countable tightness is not discretely reflexive in $σ$-compact spaces

István Juhász, Jan van Mill

Abstract

Answering a question raised by V. V. Tkachuk, we present several examples of $σ$-compact spaces, some only consistent and some in ZFC, that are not countably tight but in which the closure of any discrete subset is countably tight. In fact, in some of our examples the closures of all discrete subsets are even first countable.

Countable tightness is not discretely reflexive in $σ$-compact spaces

Abstract

Answering a question raised by V. V. Tkachuk, we present several examples of -compact spaces, some only consistent and some in ZFC, that are not countably tight but in which the closure of any discrete subset is countably tight. In fact, in some of our examples the closures of all discrete subsets are even first countable.

Paper Structure

This paper contains 3 sections, 7 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Consistent examples
  3. ZFC examples

Key Result

Lemma 2.2

The family $\{W_\alpha : \alpha < \omega_1\}$ of clopen subsets in $\omega \times X$ is centered and has empty intersection.

Theorems & Definitions (15)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 5 more