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 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.
