Selections and suborderability

Giuliano Artico, et al. "Selections and suborderability." Fundamenta Mathematicae 175.1 (2002): 1-33. <http://eudml.org/doc/282933>.

@article{GiulianoArtico2002,
abstract = {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.},
author = {Giuliano Artico, Umberto Marconi, Jan Pelant, Luca Rotter, Mikhail Tkachenko},
journal = {Fundamenta Mathematicae},
keywords = {(weak) selection; countably compact; pseudocompact; (sub)orderable; topological group; scattered space; Vietoris topology; Fell topology; ordinal space},
language = {eng},
number = {1},
pages = {1-33},
title = {Selections and suborderability},
url = {http://eudml.org/doc/282933},
volume = {175},
year = {2002},
}

TY - JOUR
AU - Giuliano Artico
AU - Umberto Marconi
AU - Jan Pelant
AU - Luca Rotter
AU - Mikhail Tkachenko
TI - Selections and suborderability
JO - Fundamenta Mathematicae
PY - 2002
VL - 175
IS - 1
SP - 1
EP - 33
AB - 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.
LA - eng
KW - (weak) selection; countably compact; pseudocompact; (sub)orderable; topological group; scattered space; Vietoris topology; Fell topology; ordinal space
UR - http://eudml.org/doc/282933
ER -