Omega_mu-metrizability and supercompleteness, by Giuliano Artico

Topology Atlas Document # zaab-05.htm | Production Editor: R. Flagg

Topology Atlas Invited Contributions, vol. 1, issue 4 (1996), 56.

Omega_mu-metrizability and supercompleteness
by

Giuliano Artico

(Dipartimento di Matematica Pura e Applicata, via Belzoni 7, 35131 Padova, Italy)

A uniform space X is said to be supercomplete provided that the hyperspace H(X) equipped with the Hausdorff uniformity is complete. A complete metric space is supercomplete [3]. In 1969 it has been claimed that a complete omega_mu-metric space is supercomplete [4] (omega_mu is any uncountable regular cardinal). Counterexamples prove that this result does not hold [1]. Supercompleteness in omega_mu-metric spaces involves set-theoretic questions. The situation has been clarified by the following recent results (here k=omega_mu; k^k and 2^k are equipped with the <k-box uniformity):

a) K^K is not supercomplete.

b) 2^K is supercomplete iff 2^K is K-compact iff K is weakly compact.

c) A k-metric space is k-compact iff it is supercomplete and k-totally bounded.

d) Every complete k-totally bounded k-metric space is k-compact iff there are no k-Aronszajn trees.

Rerefences

[1] G.Artico and U.Marconi, "A Complete omega_mu-Metric Space is not Necessarily Supercomplete", Bol. Un. Mat. It. (7) 9--A (1995), 633-637. Postscript file is available by anonymous ftp.

[2] G.Artico and U.Marconi and J.Pelant, "On Supercomplete omega_mu-Metric Spaces", to appear.

[3] J.R.Isbell, "Uniform spaces", Mathematical Surveys nr 12, AMS, Providence, Rhod Island, (1964).

[4] F.W.Stevenson and W.J.Thron, "Results on omega_mu-metric spaces", Fund. Math., 65 (1969), 317-324.

Received by the editors: January 12, 1996.