A new dcpo whose Scott topology is well-filtered but not sober. (English) Zbl 1507.06010
Summary: A topological space is sober if each of its nonempty closed irreducible sets is the closure of a unique singleton. A weaker topological property is the well-filteredness first considered by Heckmann. Johnstone constructed the first directed complete poset whose Scott topology is not sober. Heckmann asked whether the Scott topology of a directed complete poset is sober if it is assumed to be well-filtered. Kou constructed the first counterexample to give a negative answer. In this short note, based on our recent work on the dcpo models of topological space, we give another simpler example of directed complete poset whose Scott topology is well-filtered but not sober.
MSC:
| 06B30 | Topological lattices |
| 06B35 | Continuous lattices and posets, applications |
| 54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |
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