Box and Nabla Products that are D-Spaces

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 $\mathfrak{d}=\mathrm{cof}{\omega }^{\omega }$. 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 $w(X)\le {\omega }_1$.