Scott--Isbell Coincidence for Continuous Dcpos beyond Bicompleteness

@misc{chen2026scottisbell,
  title = {{Scott--Isbell Coincidence for Continuous Dcpos beyond Bicompleteness}},
  author = {Yuxu Chen and Yunjie Wen and Xiaoyong Xi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28123},
  eprint = {2609.28123},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28123},
  note = {CC BY-NC-SA 4.0},
}

TY  - UNPB
TI  - Scott--Isbell Coincidence for Continuous Dcpos beyond Bicompleteness
AU  - Yuxu Chen
AU  - Yunjie Wen
AU  - Xiaoyong Xi
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28123
UR  - https://arxiv.org/abs/2609.28123
AB  - Lawson and Mislove posed the following problem in 1990 as Problem~535 in \emph{Open Problems in Topology}: for a core-compact space X and a dcpo P equipped with its Scott topology, under what conditions on P do the Isbell and Scott topologies on C(X,P) agree?It was proved that, for a nonempty bicomplete continuous dcpo P, the Isbell and Scott topologies on C(X,P) coincide for every core-compact space X if and only if P is bounded complete; for every compact core-compact space X if and only if P is conditionally bounded complete; and for every RW-space X if and only if P is a pointed continuous L-domain.We remove the bicompleteness assumption from all three classifications by combining the forbidden-retract theorem of Jia, Jung and Li with a separation theorem for powers of downward well-ordered chains and suitable Alexandrov test spaces.
ER  -