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3 Fundamenta Mathematicae

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C(K) spaces which cannot be uniformly embedded into c₀(Γ)

100%

Pelant J. , Holický P. , Kalenda O.

nr 3

245-254

We give two examples of scattered compact spaces K such that C(K) is not uniformly homeomorphic to any subset of c₀(Γ) for any set Γ. The first one is [0,ω₁] and hence it has the smallest possible cardinality, the other one has the smallest possible height ω₀ + 1.

Selections and suborderability

88%

Artico G. , Marconi U. , Pelant J. , Rotter L. , Tkachenko M.

tom 175

nr 1

1-33

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.

On complexity of metric spaces

63%

Hohti A. , Pelant J.

nr 2

133-142