LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Every uncountable Moore subspace is nonseparable

Statement

If $X \subseteq ω_{1}$ is uncountable, then $(X, τ [X])$ is not separable.

Facts & Assumptions

Given: An uncountable $X \subseteq ω_{1}$ .

[F1]

Moore's clopen-generated topology says that $W_{β} \cap X$ is clopen, contains $β$ , and contains no ordinal below $β$ .

[F2]

Separability: the existence of an at most countable dense subset says that a space is separable when it has a countable dense subset.

Proof

technique · direct

1.1

Let $D \subseteq X$ be countable. Its supremum is a countable ordinal, so uncountability of $X$ gives $β \in X$ with $\sup D < β$ .

given

2.1

By [F1], $W_{β} \cap X$ is an open neighborhood of $β$ and every one of its points is at least $β$ . Hence it is disjoint from $D$ , so $D$ is not dense.

F1step 1.1

3.1

Since every countable $D \subseteq X$ fails to be dense, [F2] proves that $(X, τ [X])$ is nonseparable. The assertion deliberately excludes countable $X$ ; for $X = \emptyset$ or a singleton the displayed argument has no $β$ above $D$ and no nonseparability conclusion is claimed.

F2step 2.1∎

Depends on

Used by

A ZFC L-space Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Moore, A solution to the L space problem, Section 7, printed p. 22 (standard reference, not scraped)