Selections and suborderability

Giuliano Artico ¹   ; Umberto Marconi ¹   ; Jan Pelant ²   ; Luca Rotter ¹   ; Mikhail Tkachenko ³

¹ Dipartimento di Matematica Pura e Applicata via Belzoni 7 I-35131 Padova, Italy
² Mathematical Institute Academy of Sciences of Czech Republic Žitná 25 115 67 Praha 1, Czech Republic
³ Departamento de Matemáticas Universidad Autónoma Metropolitana Av. San Rafael Atlixco #186, Col. Vicentina C.P. 09340 Iztapalapa México, D.F., México

Fundamenta Mathematicae, Tome 175 (2002) no. 1, pp. 1-33

Résumé

We extend van Mill–Wattel's results and show that each countably compact completely regular space with a continuous selection on couples is suborderable. The result extends also to pseudocompact spaces if they are either scattered, first countable, or connected. An infinite pseudocompact topological group with such a continuous selection is homeomorphic to the Cantor set. A zero-selection is a selection on the hyperspace of closed sets which chooses always an isolated point of a set. Extending Fujii–Nogura results, we show that an almost compact space with a continuous zero-selection is homeomorphic to some ordinal space, and a (locally compact) pseudocompact space with a continuous zero-selection is an (open) subspace of some space of ordinals. Under the Diamond Principle, we construct several counterexamples, e.g. a locally compact locally countable monotonically normal space with a continuous zero-selection which is not suborderable.

Détail

DOI : 10.4064/fm175-1-1

Keywords: extend van mill wattels results each countably compact completely regular space continuous selection couples suborderable result extends pseudocompact spaces either scattered first countable connected infinite pseudocompact topological group continuous selection homeomorphic cantor set zero selection selection hyperspace closed sets which chooses always isolated point set extending fujii nogura results almost compact space continuous zero selection homeomorphic ordinal space locally compact pseudocompact space continuous zero selection subspace space ordinals under diamond principle construct several counterexamples locally compact locally countable monotonically normal space continuous zero selection which suborderable