Strong zero-dimensionality of hyperspaces |
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Authors: | Nobuyuki Kemoto Jun Terasawa |
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Institution: | a Department of Mathematics, Faculty of Education, Oita University, Dannoharu, Oita, 870-1192, Japan b Department of Mathematics, The National Defense Academy, Yokosuka 239-8686, Japan |
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Abstract: | For a space X, X2 denotes the collection of all non-empty closed sets of X with the Vietoris topology, and K(X) denotes the collection of all non-empty compact sets of X with the subspace topology of X2. The following are known:- •
- ω2 is not normal, where ω denotes the discrete space of countably infinite cardinality.
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- For every non-zero ordinal γ with the usual order topology, K(γ) is normal iff whenever cf γ is uncountable.
In this paper, we will prove:- (1)
- ω2 is strongly zero-dimensional.
- (2)
- K(γ) is strongly zero-dimensional, for every non-zero ordinal γ.
In (2), we use the technique of elementary submodels. |
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Keywords: | Strongly zero-dimensional Normal Hyperspace Ordinal Elementary submodel |
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