Quasi-uniform hyperspaces of compact subsets

Let (X, u) be a quasi-uniform space, K(X) be the family of all nonempty compact subsets of (X, u) . In this paper, the notion of compact symmetry for (X, u) is introduced, and relationships between the Bourbaki quasi-uniformity and the Vietoris topology on K( X) are examined. Furthermore we establis...

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Bibliographic Details
Published in:Topology and its applications 1998, Vol.87 (2), p.117-126
Main Authors: Cao, Jiling, Künzi, H.P.A., Reilly, I.L., Romaguera, S.
Format: Article
Language:English
Subjects:
Bourbaki quasi-uniformity
Compactly symmetric
Complete
Small-set symmetric
Vietoris topology
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Summary:Let (X, u) be a quasi-uniform space, K(X) be the family of all nonempty compact subsets of (X, u) . In this paper, the notion of compact symmetry for (X, u) is introduced, and relationships between the Bourbaki quasi-uniformity and the Vietoris topology on K( X) are examined. Furthermore we establish that for a compactly symmetric quasi-uniform space (X, u) the Bourbaki quasi-uniformity u ∗ on K(X) is complete if and only if u is complete. This theorem generalizes the well-known Zenor-Morita theorem for uniformisable spaces to the quasi-uniform setting.
ISSN:0166-8641
1879-3207
DOI:10.1016/S0166-8641(97)00133-8