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On relations between certain intrinsic topologies in partially ordered sets

Published online by Cambridge University Press:  24 October 2008

A. J. Ward
A. J. Ward
Affiliation:
Emmanuel CollegeCambridge
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In the first part of this paper we consider partially ordered sets for which the intrinsic ‘interval topology’ ((1), p. 60) is Hausdorffian. The main result of this section is the following: the interval topology of a lattice is Hausdorffian if and only if convergence with respect to the interval topology is equivalent to strong o*-convergence. This may be regarded as an answer either to Birkhoff's problem 23 ((1), p. 62) or to his problem 25 ((1), p. 64). In the case of a complete lattice, we have an alternative formulation. The interval topology of a complete lattice is Hausdorffian if and only if every filter (or, alternatively, every directed net) in the lattice has an o-convergent refinement. This condition may be regarded as a strong type of compactness.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 51 , Issue 2 , April 1955 , pp. 254 - 261
Copyright
Copyright © Cambridge Philosophical Society 1955

References

REFERENCES

(1)Birkhoff, G.Lattice theory, revised ed. (Colloq. Publ. Amer. math. Soc. no. 25, 1948).Google Scholar
(2)Birkhoff, G. and Frink, O.Representations of lattices by sets. Trans. Amer. math. Soc. 64 (1948), 299–316.CrossRefGoogle Scholar
(3)Bourbaki, N.Topologie générate, chaps. 1 and 2, revised ed. (Actualités sci. industr. no. 1142, Paris, 1951).Google Scholar
(4)Bourbaki, N.Integration, chaps. 1–4 (Actualités sci. industr. no. 1175, Paris, 1952).Google Scholar
(5)Floyd, E. E.Boolean Algebras with pathological order topologies. Pacific J. Math, (to appear).Google Scholar
(6)Frink, O.Ideals in partially ordered sets. Amer. math. Mon. 61 (1954), 223–34.CrossRefGoogle Scholar
(7)McShane, E. J.Order-preserving maps and integration processes (Ann. Math. Stud. no. 31, Princeton, 1953).Google Scholar
(8)Northam, F. S.The interval topology of a lattice. Proc. Amer. math. Soc. 4 (1953), 824–7.CrossRefGoogle Scholar
(9)Rennie, B. C.Lattices. Proc. Lond. math. Soc. (2), 52 (1951), 386–400; see also The theory of lattices (Cambridge, 1951, privately distributed and available in some libraries).Google Scholar