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Dow–Tkachuk Question 4.8: pseudocomplete countable closures in $C_p(X)$ force $X$ to be a $P$ -space

Yes, in ZFC. The paper proves a stronger statement: if a Tychonoff space $X$ is not a $P$ -space, then $C_p(X)$ contains a closed, separable, metrizable linear subspace that is not pseudocomplete. A closed separable subspace is the closure of a countable subset, so if all such closures are pseudocomplete, $X$ must be a $P$ -space. Dow and Tkachuk had shown this under the extra assumption that $X$ is functionally countable (their Theorem 3.10); the result removes that assumption. The companion Question 4.9 (whether $X$ must also be functionally countable) is not addressed.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Oct 12, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-assisted. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Dow–Tkachuk Question 4.8: pseudocomplete countable closures in $C_p(X)$ force $X$ to be a $P$-space · Dow–Tkachuk Q4.8 ($C_p(X)$ and $P$-spaces)

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

ChatGPT Astra (proof)
model · ai model contributor · OpenAI

Claude Opus 5.5 (Lean formalization)
model · ai model contributor · Anthropic

Lucas Alves de Almeida
human · human collaborator

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Dow–Tkachuk Question 4.8: pseudocomplete countable closures in $C_p(X)$ force $X$ to be a $P$ -space

specialist registry · passed

No independent mathematician has checked this yet. Checked by this site on 15 October 2026, not rebuilt. Question 4.8 was read in the open-access Dow-Tkachuk paper (Eur. J. Math. 11 (2025), Art. 5) and the statement matches; their paper assumes all spaces Tychonoff and defines pseudocompleteness by Oxtoby's sequences of pi-bases, with no extra regularity clause. The Lean headline question_4_8 in Question48.lean (commit 8b4c7d0, SHA-256 matching the repository's certificate) was read with its definitions: Cp X is the continuous functions as a subspace of the product X -> R (pointwise, not Mathlib's compact-open C(X, R)), closure is taken in Cp X, Pseudocomplete is exactly the paper's definition, and IsPSpace says countable intersections of open sets are open, equivalent to every G-delta being open. The hypothesis is CompletelyRegularSpace, weaker than Tychonoff, so the formal statement is if anything stronger. A grep of the file found no sorry, admit, axiom, native_decide, unsafe, opaque or implemented_by; the axiom list (propext, Classical.choice, Quot.sound) is the author's build output, not re-run here. The paper's short proof was also read and no gap was found; that is not an expert review. The formal statement was written in the same workflow as the proof and is not independently anchored.

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Dow–Tkachuk Question 4.8: pseudocomplete countable closures in $C_p(X)$ force $X$ to be a $P$ -space

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean 4 formalization (Question48.lean)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Paper (PDF)

paper · pending

paper link supplied by the VibeMathed registry.

Source ↗

Dow and Tkachuk, Reflecting topological properties in closed separable subspaces (Question 4.8)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean 4 formalization (Question48.lean)

lean artifact · pending

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Compute record

No linked compute attempts recorded.

Lineage and corrections

This event attributed to ChatGPT Astra (proof)

Paper (PDF) evidence for this event

Dow–Tkachuk Question 4.8: pseudocomplete countable closures in $C_p(X)$ force $X$ to be a $P$ -space parent of this event

This event attributed to Lucas Alves de Almeida

Dow–Tkachuk Question 4.8: pseudocomplete countable closures in $C_p(X)$ force $X$ to be a $P$ -space evidence for this event

Lean 4 formalization (Question48.lean) evidence for this event

VibeMathed record: Dow–Tkachuk Question 4.8: pseudocomplete countable closures in $C_p(X)$ force $X$ to be a $P$ -space evidence for this event

This event attributed to Claude Opus 5.5 (Lean formalization)

Dow and Tkachuk, Reflecting topological properties in closed separable subspaces (Question 4.8) evidence for this event

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