Countable tightness is not discretely reflexive in $\sigma$-compact spaces

TL;DR

This paper provides examples of $\sigma$-compact spaces that are not countably tight, yet have the property that closures of discrete subsets are countably tight or even first countable, addressing a question by V. V. Tkachuk.

Contribution

It constructs new examples of $\sigma$-compact spaces illustrating that countable tightness is not discretely reflexive, answering an open question.

Findings

Examples of $\sigma$-compact spaces not countably tight

Closures of discrete subsets can be countably tight or first countable

Some examples are consistent and some are in ZFC

Abstract

Answering a question raised by V. V. Tkachuk, we present several examples of $\sigma$-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.