Selections and suborderability

Tom 175 / 2002

Giuliano Artico, Umberto Marconi, Jan Pelant, Luca Rotter, Mikhail Tkachenko Fundamenta Mathematicae 175 (2002), 1-33 MSC: 54C65, 54B20, 54F05, 54H11, 54G12. DOI: 10.4064/fm175-1-1

Streszczenie

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.

Autorzy

  • Giuliano ArticoDipartimento di Matematica Pura e Applicata
    via Belzoni 7
    I-35131 Padova, Italy
    e-mail
  • Umberto MarconiDipartimento di Matematica Pura e Applicata
    via Belzoni 7
    I-35131 Padova, Italy
    e-mail
  • Jan PelantMathematical Institute
    Academy of Sciences of Czech Republic
    Žitná 25
    115 67 Praha 1, Czech Republic
    e-mail
  • Luca RotterDipartimento di Matematica Pura e Applicata
    via Belzoni 7
    I-35131 Padova, Italy
    e-mail
  • Mikhail TkachenkoDepartamento de Matemáticas
    Universidad Autónoma Metropolitana
    Av. San Rafael Atlixco #186, Col. Vicentina
    C.P. 09340 Iztapalapa
    México, D.F., México
    e-mail

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