For a space Z let 𝒦(Z) denote the partially ordered set of all compact subspaces of Z under set inclusion. If X is a compact space, Δ is the diagonal in X², and 𝒦(X²∖Δ) has calibre (ω₁,ω), then X is metrizable. There is a compact space X such that X²∖Δ has relative calibre (ω₁,ω) in 𝒦(X²∖Δ), but which is not metrizable. Questions of Cascales et al. (2011) concerning order constraints on 𝒦(A) for every subspace of a space X are answered.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-1-1,
author = {Paul Gartside and Jeremiah Morgan},
title = {Calibres, compacta and diagonals},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {1-19},
zbl = {1339.54021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-1-1}
}
Paul Gartside; Jeremiah Morgan. Calibres, compacta and diagonals. Fundamenta Mathematicae, Tome 233 (2016) pp. 1-19. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-1-1/