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

Abstract

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.