A note on zero-dimensional spaces with the star-finite property

Reichel, Hans-Christian

doi:10.1090/S0002-9939-1974-0328863-3

A note on zero-dimensional spaces with the star-finite property
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Proc. Amer. Math. Soc. 42 (1974), 307-311 Request permission

Abstract:

The paper provides necessary and sufficient conditions for a weakly zero-dimensional metrizable space to be strongly paracompact, i.e., to have the star-finite property. The characterizations use special basis properties of uniformities which induce the topology of X, and yield further characteristics of the class of all metric spaces with ind $X = 0$ and Ind $X > 0$.

References

Bernhard Banaschewski, Über nulldimensionale Räume, Math. Nachr. 13 (1955), 129–140 (German). MR 86287, DOI 10.1002/mana.19550130302
Ben Fitzpatrick Jr. and Ralph M. Ford, On the equivalence of small and large inductive dimension in certain metric spaces, Duke Math. J. 34 (1967), 33–37. MR 205226
J. de Groot, Non-archimedean metrics in topology, Proc. Amer. Math. Soc. 7 (1956), 948–953. MR 80905, DOI 10.1090/S0002-9939-1956-0080905-8
A. F. Monna, Remarques sur les métriques non-archi-médiennes. I, Nederl. Akad. Wetensch., Proc. 53 (1950), 470–481 = Indagationes Math. 12, 122–133 (1950) (French). MR 35982
Jun-iti Nagata, Modern dimension theory, Bibliotheca Mathematica, Vol. VI, Interscience Publishers John Wiley & Sons, Inc., New York, 1965. Edited with the cooperation of the “Mathematisch Centrum” and the “Wiskundig Genootschap” at Amsterdam. MR 0208571

N-compact spaces

On the structure of zero-dimensional spaces

H. C. Reichel, On the metrization of the natural topology on the space $A^{B}$, J. Math. Anal. Appl. 34 (1971), 644–647. MR 275359, DOI 10.1016/0022-247X(71)90103-X

Über uniforme Räume

48

A. C. M. van Rooij, Non-Archimedean uniformities, Kyungpook Math. J. 10 (1970), 21–30. MR 264598