Jeremiah Morgan | Search arXiv

arXiv2024

Products of Directed Sets with Calibre $(ω_1, ω)$

A directed set $P$ is calibre $(ω_1, ω)$ if every uncountable subset of $P$ contains an infinite bounded subset. $P$ is productively calibre $(ω_1, ω)$ if $P \times Q$ is calibre $(ω_1, ω)$ for every directed set $Q$ with calibre $(ω_1, ω)$, and $P$ is powerfully calibre $(ω_1, ω)$ if the countable power of $P$ is calibre $(ω_1, ω)$. It is shown that (1) uncountable products are calibre $(ω_1, ω)$ only in highly restrictive circumstances, (2) many but not all $\sum$-products of calibre $(ω_1, ω)$ directed sets are calibre $(ω_1, ω)$, (3) there are directed sets which are calibre $(ω_1, ω)$ but neither productively nor powerfully calibre $(ω_1, ω)$, and (4) there are directed sets which are powerfully but not productively calibre $(ω_1, ω)$. As an application, the position is established of $\sum ω^{ω_1}$ in the Tukey order among Isbell's classical 10 directed sets.

math.LO↗

arXiv2015

$P$-Paracompact and $P$-Metrizable Spaces

Let $P$ be a directed set and $X$ a space. A collection $\mathcal{C}$ of subsets of $X$ is \emph{$P$-locally finite} if $\mathcal{C}=\bigcup \{ \mathcal{C}_p : p \in P\}$ where (i) if $p \le p'$ then $\mathcal{C}_p \subseteq \mathcal{C}_{p'}$ and (ii) each $\mathcal{C}_p$ is locally finite. Then $X$ is \emph{$P$-paracompact} if every open cover has a $P$-locally finite open refinement. Further, $X$ is \emph{$P$-metrizable} if it has a $(P \times \mathbb{N})$-locally finite base. This work provides the first detailed study of $P$-paracompact and $P$-metrizable spaces, particularly in the case when $P$ is a $\mathcal{K}(M)$, the set of all compact subsets of a separable metrizable space $M$ ordered by set inclusion.

math.GN↗