Calibres, compacta and diagonals
Volume 232 / 2016
Fundamenta Mathematicae 232 (2016), 1-19
MSC: Primary 54E35; Secondary 54D30, 54G20.
DOI: 10.4064/fm232-1-1
Abstract
For a space $Z$ let $\mathcal {K}(Z)$ denote the partially ordered set of all compact subspaces of $Z$ under set inclusion. If $X$ is a compact space, $\Delta $ is the diagonal in $X^2$, and $\mathcal {K}(X^2 \setminus \Delta )$ has calibre $(\omega _1,\omega )$, then $X$ is metrizable. There is a compact space $X$ such that $X^2 \setminus \Delta $ has relative calibre $(\omega _1,\omega )$ in $\mathcal {K}(X^2 \setminus \Delta )$, but which is not metrizable. Questions of Cascales et al. (2011) concerning order constraints on $\mathcal {K}(A)$ for every subspace of a space $X$ are answered.
