Abstract
We consider the poset of all nonempty compact saturated subsets of the Scott space of a poset , equipped with the reverse inclusion order. We also introduce the notion of -lattices, that is, a complete lattice is called a -lattice if for any , . We prove that if and are quasicontinuous domains or -lattices, is order isomorphic to if and only if is order isomorphic to . We also prove that for a compact space , if the Scott space of the open set lattice is sober, then is -compact. Using this result and the Isbell example of a non-sober complete lattice, we present a compact sober space which is not -compact. This gives a positive answer to an open problem posed by Zhao and Ho.
Citation
Zhengmao He. Kaiyun Wang. "A COMPACT SPACE IS NOT ALWAYS -COMPACT." Rocky Mountain J. Math. 52 (6) 2041 - 2051, December 2022. https://doi.org/10.1216/rmj.2022.52.2041
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