Calibres, compacta and diagonals

Paul Gartside¹ ; Jeremiah Morgan¹

¹ Department of Mathematics University of Pittsburgh Pittsburgh, PA 15260, U.S.A.

Fundamenta Mathematicae, Tome 232 (2016) no. 1, pp. 1-19

Résumé

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.

Détail

DOI : 10.4064/fm232-1-1

Mots-clés : space mathcal denote partially ordered set compact subspaces under set inclusion compact space delta diagonal mathcal setminus delta has calibre omega omega metrizable there compact space setminus delta has relative calibre omega omega mathcal setminus delta which metrizable questions cascales concerning order constraints mathcal every subspace space answered