Lawson–Mislove Problem 535

Let $P$ be a nonempty continuous dcpo, equipped with its Scott topology. Determine whether, without assuming that $P$ is bicomplete, the following classifications hold: (1) the Isbell and Scott topologies on $C(X,P)$ coincide for every core-compact space $X$ if and only if $P$ is bounded complete; (2) they coincide for every compact core-compact space $X$ if and only if $P$ is conditionally bounded complete; and (3) they coincide for every RW-space $X$ if and only if $P$ is a pointed continuous $L$ -domain.

References

Primary source

arXiv

Additional references

Scott--Isbell Coincidence for Continuous Dcpos beyond Bicompleteness — arXiv — Yuxu Chen, Yunjie Wen, Xiaoyong Xi

domain-theory general-topology

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A new unrefereed preprint claims to solve the problem by removing the extra completeness assumption from all three known classifications.

Lawson–Mislove Problem 535 concerns a longstanding question in function-space topology and the relationship between Scott and Isbell structures.

September 2026 claimed solution

On September 23, 2026, a report identified a preprint by Yuxu Chen, Yunjie Wen, and Xiaoyong Xi claiming that all three prior classifications remain valid without the bicompleteness hypothesis, thereby completing the problem and extending Scott–Isbell coincidence theory. This is a claimed result in a new unrefereed preprint, not an independently verified solution.

Current status (as of September 2026): A preprint claims the problem is solved beyond bicompleteness, but the result is unverified.

Sources

No solutions have been posted yet.