Co-sobriety of Hoare power spaces
Is it true that for every co-sober topological space the Hoare power space is co-sober? In particular, does this hold when is or metrizable?
Equivalent formulations 1Other wordings
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
Hoare-power/Scott-sobriety characterization
For every -space is co-sober is sober
source: A countable metric space whose Hoare power space is not co-sober
References
Primary source
Additional references
- A countable metric space whose Hoare power space is not co-sober — arXiv — Xiaoquan Xu, Wei Ji
Progress summary
An unrefereed preprint claims a countable metric counterexample, settling the preservation question negatively while awaiting verification.
The problem asks whether Hoare power spaces preserve co-sobriety. In October 2026, Xiaoquan Xu and Wei Ji reported a construction showing that this fails even for countable metrizable spaces.
October 2026 counterexample
Xu and Ji use a Ponomarev construction and a characterization via Scott sobriety to transfer a known countable counterexample to a countable metrizable space. The related Scott-sobriety question is also answered negatively: the cited work states that is not sober and uses a retract argument to transfer non-sobriety.
Current status (as of October 2026): A preprint claims the co-sobriety-preservation question is settled negatively, including for countable metrizable spaces, but the result remains unverified.
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