Box and Nabla Products that are D-Spaces

TL;DR

This paper investigates conditions under which box and nabla products of certain topological spaces are D-spaces, revealing that specific structural properties of the spaces influence their D-space status.

Contribution

It establishes new criteria for when box and nabla products are hereditarily D-spaces, especially for spaces with properties like being scattered, metrizable, or first countable.

Findings

Hereditarily D for scattered, hereditarily paracompact, or finite scattered height spaces

Hereditarily D if X is first countable with w(X) ≤ ω₁

Box product D if X is compact, first countable, or w(X) ≤ ω₁

Abstract

A space $X$ is $D$ if for every assignment, $U$, of an open neighborhood to each point $x$ in $X$ there is a closed discrete $D$ such that $\bigcup \{U(x) : x \in D\}=X$. The box product, $\square X^\omega$, is $X^\omega$ with topology generated by all $\prod_n U_n$, where every $U_n$ is open. The nabla product, $\nabla X^\omega$, is obtained from $\square X^\omega$ by quotienting out mod-finite. The weight of $X$, $w(X)$, is the minimal size of a base, while $\mathfrak{d}=\mathop{cof} \omega^\omega$. It is shown that there are specific compact spaces $X$ such that $\square X^\omega$ and $\nabla X^\omega$ are not $D$, but: (1) $\square X^\omega$ and $\nabla X^\omega$ are hereditarily $D$ if $X$ is scattered and either hereditarily paracompact or of finite scattered height, or if $X$ is metrizable (and $w(X)\le \mathfrak{d}$ for $\square X^\omega$); (2) $\nabla X^\omega$ is hereditarily $D$ if $X$ is first countable and $w(X)\le \omega_1$, or consistently if $X$ is first countable and $|X|\le \mathfrak{c}$, or $w(X)\le \omega_1$; and (3) $\square X^\omega$ is $D$ consistently if $X$ is compact and either first countable or $w(X)\le \omega_1$.