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| Topological Representations of Distributive Hypercontinuous Lattices |
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Citation： |
Xiaoquan XU,Jinbo YANG.Topological Representations of Distributive Hypercontinuous Lattices[J].Chinese Annals of Mathematics B,2009,30(2):199~206 |
| Page view： 1943
Net amount： 1320 |
Authors： |
Xiaoquan XU; Jinbo YANG; |
Foundation： |
the National Natural Science Foundation of China (Nos. 10331010, 10861007),
the Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 2007B14), the
Jiangxi Provincial Natural Science Foundation of China (Nos. 0411025, 2007GZS0179), the Foundation of
the Education Department of Jiangxi Province (No. GJJ08162) and the Doctoral Fund of Jiangxi Normal
University. |
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| Abstract： |
The concept of locally strong compactness on domains is generalized to general
topological spaces. It is proved that for each distributive hypercontinuous lattice L, the
space SpecL of nonunit prime elements endowed with the hull-kernel topology is locally
strongly compact, and for each locally strongly compact space X, the complete lattice of all
open sets O(X) is distributive hypercontinuous. For the case of distributive hyperalgebraic
lattices, the similar result is given. For a sober space X, it is shown that there is an order
reversing isomorphism between the set of upper-open filters of the lattice O(X) of open
subsets of X and the set of strongly compact saturated subsets of X, which is analogous
to the well-known Hofmann-Mislove Theorem. |
Keywords： |
Hypercontinuous lattice, Locally strongly compact space, Hull-kernel
topology, Hyperalgebraic lattice, Strongly locally compact space |
Classification： |
06B15, 06B35, 54D45 |
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