Let rm UC_X denote the unique closed choice operation on a represented Hausdorff space X, and let rm C_X denote closed choice on X. For Baire space obreak ^ obreak , the unique-choice conjecture. rm UC_N^Nmathopequiv_Wrm C_N^N. Unique choice is always Weihrauch-reducible to full choice, and the corresponding equivalence is proved for N. The conjecture asks whether the same phenomenon holds for Baire space.

Unique choice is Weihrauch-equivalent to choice on Baire space

About 16 years old · traced to 2010

Let ${\rm UC}_X$ denote the unique closed choice operation on a represented Hausdorff space $X$ , and let ${\rm C}_X$ denote closed choice on $X$ . For Baire space ${ obreak\begin{\mathbb{N}}^{ obreak\begin{\mathbb{N}}}}$ , the unique-choice conjecture.

{\rm UC}_{{\mathbb{N}}^{\mathbb{N}}}\mathop{\equiv_{\mathrm{W}}}{\rm C}_{{\mathbb{N}}^{\mathbb{N}}}.

Unique choice is always Weihrauch-reducible to full choice, and the corresponding equivalence is proved for ${\mathbb{N}}$ . The conjecture asks whether the same phenomenon holds for Baire space.

References

Primary source

Vasco Brattka, Matthew de Brecht and Arno Pauly, “Closed Choice and a Uniform Low Basis Theorem”, arXiv:1002.2800 (2010).

Posted 1 month ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A reader points to a 2018 paper as a possible proof, but the Baire-space case remains unverified.

The conjecture asks whether unique closed choice and full closed choice have the same Weihrauch degree on Baire space. Earlier literature records the analogous equivalence for natural numbers but presents the Baire-space case as open.

Known results

The equivalence is proved for $\mathbb{N}$ .
$\lim \leq_{\mathrm{W}} \mathrm{UC}_{\mathbb{N}^{\mathbb{N}}}$ .
$\mathrm{UC}_{\mathbb{N}^{\mathbb{N}}}$ is not confined to any finite Borel level.

Community submission (unverified), September 19, 2026

A submitted argument points to Corollary 3.7 of a 2018 paper and argues that it establishes $\mathrm{UC}_{\mathbb{N}^{\mathbb{N}}} \equiv_{\mathrm{W}} \mathrm{C}_{\mathbb{N}^{\mathbb{N}}}$ . This proposed resolution has not been independently verified.

Current status (as of September 2026): The equivalence is settled for $\mathbb{N}$ , while the Baire-space case remains open apart from an unverified community claim.

Sources

This was resolved negatively by Kihara, Marcone, and Pauly. Unique choice is strictly weaker than closed choice. See Corollary 3.7 of https://arxiv.org/abs/1812.01549v2 (HTML version: https://arxiv.org/html/1812.01549v2). This paper is a verified publication in the Journal of Symbolic Logic (https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/open-and-clopen-ramsey-theorems-in-the-weihrauch-lattice/2BFBEEB9ABE915BF447E729BFFA28237).See full solutionHide full solution

See Corollary 3.7 of https://arxiv.org/abs/1812.01549v2 (HTML version: https://arxiv.org/html/1812.01549v2). This paper is a verified publication in the Journal of Symbolic Logic (https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/open-and-clopen-ramsey-theorems-in-the-weihrauch-lattice/2BFBEEB9ABE915BF447E729BFFA28237).