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Euclidean representations of topologically ordered spaces

Part of: Special properties

Published online by Cambridge University Press:  26 February 2010

H. B. Griffiths
H. B. Griffiths
Affiliation:
Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton, SO9 5NH
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Extract

A topological ordered space (or pospace) is a poset (X, <) with a topology on X for which the relation < is closed in the product X × X. The topology of X is then necessarily Hausdorff. The basic theory of pospaces was developed by Nachbin in his book [5]; and others have extended it, but the resulting body of knowledge is not very geometrical. There are few concrete examples, other than the unit interval I with its natural order, and Euclidean spaces (Rn, ≤), the Hilbert cube (H, ≤) (each with the vector order), and some function spaces.

MSC classification

Secondary: 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
Type
Research Article
Information
Mathematika , Volume 39 , Issue 1 , June 1992 , pp. 104 - 135
Copyright
Copyright © University College London 1992

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References

1.Bonnesen, T. and Fenchel, W.. Theorie der Convexen Körper (Chelsea, New York, 1948).Google Scholar
2.Griffiths, H. B.. Convergence of convex sets and the Fourier-Motzkin process. To appear.Google Scholar
3.Hausdorff, F.. Mengenlehre, Third Edition (Chelsea).Google Scholar
4.Munkres, J.. Topology (A First Course) (Prentice Hall, N.J., 1975).Google Scholar
5.Nachbin, L.. Topology and Order (Van Nostrand, New York, 1965).Google Scholar
6.Nada, S. I. M.. Studies on Topological Ordered Spaces, Ph.D. Thesis (Southampton, 1986).Google Scholar
7.Schrijver, A.. Theory of Linear and Integer Programming (Wiley, 1986).Google Scholar
8.Spanier, E. H.. Algebraic Topology (McGraw Hill, 1966).Google Scholar
9.Steen, L. A. and Seebach, J. A.. Counter-examples in Topology (Holt Rinehard Winston, 1968).Google Scholar
10.Ward, A. J.. The topological characterisation of an open interval. Proc. Lond. Math. Soc., 41 (1936), 191–198.Google Scholar
11.Ward, L. E.. Partially ordered topological spaces. Proc. Amer. Math. Soc., 5 (1954), 144–161.Google Scholar
12.Wilder, R. L.. Topology of Manifolds, Amer. Math. Soc. Colloquium Publications, XXXII (1949).CrossRefGoogle Scholar