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On supercomplete $$\omega_\mu$$-metric spaces. (English) Zbl 0892.54020
Consider a regular cardinal $$\omega_\mu$$. An $$\omega_\mu$$-metric on a set $$X$$ is a symmetric function $$d:X\times X\to\omega_\mu+1$$ that satisfies (1) $$x=y$$ iff $$d(x,y)=\omega_\mu$$ and (2) $$d(x,z)\leq \max\{d(x,y),d(y,z)\}$$ – where $$\omega_\mu$$ is assumed to be ordered upside down: $$\alpha<\beta$$ iff $$\beta\in\alpha$$. Many concepts from metric theory carry over to $$\omega_\mu$$-metrics: uniformities, topologies, completeness, Hausdorff metric, etc.
In [Fundam. Math. 65, 317-324 (1969; Zbl 0187.20002)] F. W. Stevenson and W. J. Thron proved that the hyperspace of a complete $$\omega_\mu$$-metric space is again complete (for metric spaces this is an old results of Hahn’s). Their proof contains a gap and in the first paper [Boll. Unione Mat. Ital., VII. Ser., A 9, No. 3, 633-637 (1995), see above)] the authors show that, for every nonzero $$\mu$$, the space $$\omega_\mu^{\omega_\mu}$$ of all functions of $$\omega_\mu$$ to itself, with the $$\omega_\mu$$-metric of ‘first difference’, is a counterexample. For limit $$\mu$$ the proof needs $$\lozenge$$ on $$\omega_\mu$$.
In the second paper the $$\lozenge$$ assumption is removed and spaces with a complete hyperspace (supercomplete spaces) are studied in more detail. For instance the familiar theorem that ‘compact equals completeness plus total boundedness’ is false for $$\omega_\mu$$-metric spaces when ‘finite’ is replaced with ‘cardinality less than $$\omega_\mu$$’ – the correct form substitutes supercompleteness for completeness.
The paper concludes with an investigation of the supercompleteness of the subspace $$2^{\omega_\mu}$$ of $$\omega_\mu^{\omega_\mu}$$. If $$\omega_\mu$$ is not strongly inaccessible then both spaces are uniformly homeomorphic, so the authors consider strongly inaccessible $$\omega_\mu$$ only. For such $$\omega_\mu$$ supercompleteness of $$2^{\omega_\mu}$$ is equivalent to its ‘compactness’ which in turn is equivalent to weak compactness of the cardinal $$\omega_\mu$$ – the last equivalence is due to D. Monk and D. Scott [Fundam. Math. 53, 335-343 (1964; Zbl 0173.00803)]. The authors provide an alternative proof by way of uniform embeddings of $$\omega_\mu$$-trees, endowed with a natural $$\omega_\mu$$-metric, into $$2^{\omega_\mu}$$; these trees are never supercomplete, but a tree is complete iff it is $$\omega_\mu$$-Aronszajn. Also, if there are no $$\omega_\mu$$-Aronszajn trees then compact equals complete and totally bounded in the class of $$\omega_\mu$$-metric spaces.
Reviewer: K.P.Hart (Delft)

##### MSC:
 54E99 Topological spaces with richer structures 54B20 Hyperspaces in general topology 54E15 Uniform structures and generalizations 03E55 Large cardinals