On spaces with $\sigma$-closed-discrete dense sets

TL;DR

This paper investigates $e$-separable spaces, which have dense sets as unions of countably many closed discrete sets, focusing on their behavior under products and related cardinal invariants, linking their properties to large cardinal axioms.

Contribution

It establishes the equiconsistency between the existence of a product of many $e$-separable spaces failing to be $e$-separable and the existence of a weakly compact cardinal.

Findings

Product of up to continuum many $e$-separable spaces can fail to be $e$-separable

The behavior of $e$-separable spaces under products is linked to large cardinal axioms

Cardinal invariants related to $e$-separable spaces are studied in depth

Abstract

The main purpose of this paper is to study \emph{$e$-separable spaces}, originally introduced by Kurepa as $K_0'$ spaces; we call a space $X$ $e$-separable iff $X$ has a dense set which is the union of countably many closed discrete sets. We primarily focus on the behaviour of $e$-separable spaces under products and the cardinal invariants that are naturally related to $e$-separable spaces. Our main results show that the statement "there is a product of at most $\mathfrak c$ many $e$-separable spaces that fails to be $e$-separable'" is equiconsistent with the existence of a weakly compact cardinal.