Abstract
Every sober dcpo P with the Scott topology is U k -admitting, that is, for every Scott open set U and a filtered family F of compact upper sets of P,⋂F⊂U implies K⊂U for some K ⊂ F. In this note, we construct an example of a U k -admitting dcpo which is not sober, thus giving a negative answer to an open problem posed by Heckmann in 1991. Moreover, we prove that for every locally compact dcpo, U k -admitting is equivalent to sober.
Project supported by the National Natural Science Foundation of China.
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Kou, H. (2001). U K -Admitting DCPOS Need not be Sober. In: Keimel, K., Zhang, GQ., Liu, YM., Chen, YX. (eds) Domains and Processes. Semantic Structures in Computation, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0654-5_3
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DOI: https://doi.org/10.1007/978-94-010-0654-5_3
Publisher Name: Springer, Dordrecht
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