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On the order scale of a uniform space

Part of: Special properties Spaces with richer structures

Published online by Cambridge University Press:  09 April 2009

D. C. Kent
D. C. Kent
Affiliation:
Department of Mathematics Washington State UniversityPullman, Washington 99164, U.S.A.
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Abstract

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The order topology is compact and T2 in both the scale and retracted scale of any uniform space (S, U). if (S, U) is T2 and totally bounded, the Samuel compactification associated with (S, U) can be obtained by uniformly embedding (S, U) in its order retracted scale (that is, the retracted scale with its order topology). This implies that every compact T2 space is both a closed subspace of a complete, infinitely distributive lattice in its order topology, and also a continuous, closed image of a closed subspace of a complete atomic Boolean algebra in its order topology.

MSC classification

Secondary: 54E15: Uniform structures and generalizations 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 34 , Issue 2 , April 1983 , pp. 248 - 257
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Birithoff, G., Lattice theory, (Amer. Math. Soc. Colloq. Publ. 25 (third edition) New York, 1967).Google Scholar
[2]Bushaw, D., ‘A stability criterion for general systems’, Math. Systems Theory 1 (1967), 79–88.CrossRefGoogle Scholar
[3]Bushaw, D., ‘The scale of a uniform space’, Proc. of the International Symp. on Topology and its Applications, Hercegnovi, Yugoslavia, 08 25–31, 1968.Google Scholar
[4]Dobbertin, H., Erné, M. and Kent, D. C., ‘A note on order convergence in complete lattices, to appear.Google Scholar
[5]Erné, M. and Week, S., ‘Order convergence in lattices’, Rocky Mountain J. Math. 10 (1980). 805–818.CrossRefGoogle Scholar
[6]Kent, D. C., ‘On the scale of a uniform space’, Inventiones Math. 4 (1967), 159–164.Google Scholar
[7]Kent, D. C. and Atherton, C. R., ‘The order topology in a bicompactly generated lattice’, J. Austral. Math. Soc. 8 (1968), 345–349.Google Scholar
[8]Leslie, G. C. and Kent, D. C., ‘Connectedness in the scale of a uniform space’, J. Austral. Math. Soc. 13 (1972), 305–312.Google Scholar
[9]Ramsey, O. C. Jr, Some properties of the scale of a uniform space, (Ph. D. Dissertation, Washington State University, 1967).Google Scholar
[10]Richardson, G. D., ‘Connectedness in the scale of uniform subspaces of R’, J. Austral. Math. Soc. 18 (1974), 461–463.Google Scholar
[11]Richardson, G. D. and Wolf, E. M., ‘Topological properties of the scale of a uniform space’, J. Austral. Math. Soc., to appear.Google Scholar