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The order topology in a bicompactly generated lattice

Published online by Cambridge University Press:  09 April 2009

D. C. Kent  and
C. R. Atherton
D. C. Kent
Affiliation:
Washington State UniversityPullman, WashingtonU.S.A.
C. R. Atherton
Affiliation:
Washington State UniversityPullman, WashingtonU.S.A.
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The concept of a compactly generated lattice has been studied extensively in connection with decomposition theory (see [1]). This paper investigates the order topology in a lattice which is, along with its dual, compactly generated (hence, bicompactly generated). We show that order convergence is topological and that the order topology is Hausdorff, totally disconnected, and has an open subbase of ideals and dual ideals in any bicompactly generated lattice; furthermore, with an additional restriction, the lattice operations are continuous in the order topology. Next we consider the order topology in certain special types of compactly generated lattices, namely atomic Boolean algebras and sub-complete lattices of atomic Boolean algebras in the former structures the order topology is uniformizable, in the latter, compact.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 8 , Issue 2 , May 1968 , pp. 345 - 349
Copyright
Copyright © Australian Mathematical Society 1968

References

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