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Completeness of hyperspaces of compact subsets of quasi-metric spaces. (English) Zbl 1274.54046

Summary: We study the hyperspace \(\mathcal K_0(X)\) of non-empty compact subsets of a Smyth-complete quasi-metric space \((X,d)\). We show that \(\mathcal K_0(X)\), equipped with the Hausdorff quasi-seudometric \(H_d\) 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 \((\mathcal K_0(X),H_d)\) is Smyth-complete if \((X,d)\) is Smyth-complete and all compact subsets of \(X\) are \(d^{-1}\)-precompact.

MSC:

54B20 Hyperspaces in general topology
54E99 Topological spaces with richer structures
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