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Connectedness in the scale of uniform subspaces of R

Published online by Cambridge University Press:  09 April 2009

G. D. Richardson
G. D. Richardson
Affiliation:
Department of MathematicsEast Carolina University, Greenville, North Carolina 27834, U.S.A.
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Kent [4] showed that each uniform space (S, u) could be embedded in a complete, uniform lattice, called the scale of (S, u). The scale was first introduced by Bushaw [3] for studying stability in topological dynamics. In [5], the notions of connectedness and local connectedness were studied. This note is a follow-up of [5]; the purpose being to characterise the uniform subspaces of the reals, R, which have connected (locally connected) scales. The reader is asked to refer to [5] for definitions and notation not given here.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 4 , December 1974 , pp. 461 - 463
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Bourbaki, N., Topologie Generale, Chapt. I, II (Paris 1951).Google Scholar
[2]Bushaw, D., ‘On boundedness in uniform spaces’, Fund. Math. 56 (1965), 295–300.Google Scholar
[3]Bushaw, D., ‘A stability criterion for general systems’, Math. Systems Theory 1 (1967), 79–88.CrossRefGoogle Scholar
[4]Kent, D. C., ‘On the scale of a uniform space’, Inventiones Math. 4 (1967), 159–164.Google Scholar
[5]Lesile, G. S. and Kent, D. C., ‘Connectedness in the scale of a uniform space’, J. Austral. Math. Soc. 13 (1972), 305–312.CrossRefGoogle Scholar
[6]Whyburn, G. T., Analytic Topology, (Amer. Math. Soc. Colloq. Pub.) 28.Google Scholar