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Completeness of hyperspaces of compact subsets of quasi-metric spaces

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Abstract

We study the hyperspace K 0(X) of non-empty compact subsets of a Smyth-complete quasi-metric space (X, d). We show that K 0(X), equipped with the Hausdorff quasi-pseudometric H d forms a (sequentially) Yoneda-complete space. Moreover, if d is a T 1 quasi-metric, then the hyperspace is algebraic, and the set of all finite subsets forms a base for it. Finally, we prove that K 0(X), H d ) is Smyth-complete if (X, d) is Smyth-complete and all compact subsets of X are d −1-precompact.

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Authors and Affiliations

  1. Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

    M. Ali-Akbari

  2. School of Mathematics and Computer Science, Amirkabir University of Technology and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

    M. Pourmahdian

Authors
  1. M. Ali-Akbari
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  2. M. Pourmahdian
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Correspondence to M. Pourmahdian.

Additional information

Partially supported by the Institute for Research in Fundamental Sciences, Tehran, Iran, grant No. 88030111.

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Ali-Akbari, M., Pourmahdian, M. Completeness of hyperspaces of compact subsets of quasi-metric spaces. Acta Math Hung 127, 260–272 (2010). https://doi.org/10.1007/s10474-010-9132-8

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  • DOI: https://doi.org/10.1007/s10474-010-9132-8

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