On supercomplete \(\omega_\mu\)-metric spaces.

*(English)*Zbl 0892.54020Consider 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.

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)