Alphabeta MathHow 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.
The long line is locally Euclidean and Hausdorff but not a manifold under the library convention
Statement refuted
Every Hausdorff locally Euclidean space is a manifold.
Facts & Assumptions
Given: The long line and the Axiom of Countable Choice .
The A-page refutation already proves that is Hausdorff and locally Euclidean but not second countable, hence not a manifold under the library convention (Hausdorff and locally Euclidean do not by themselves make a manifold).
The long-ray construction and its order topology are those of The closed long ray under the lexicographic order, and the long line, with the order topology, and its order-theoretic connectedness properties are recorded in The long ray is a linear continuum, hence connected; every one of its at most countable subsets is bounded above, assuming countable choice.
Counterexample
By [L1], the long line is Hausdorff and locally Euclidean.
The same cited refutation shows that fails second countability, so it is not a manifold under the library convention. The structural details of [F1] identify the witness but do not change that conclusion.
Thus is the required counterexample.
Depends on
- Hausdorff and locally Euclidean do not by themselves make a manifold
- The closed long ray $\omega_1 \times [0,1)$ under the lexicographic order, and the long line, with the order topology
- The long ray is a linear continuum, hence connected; every one of its at most countable subsets is bounded above, assuming countable choice
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
27 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
- Long line (topology) (Wikipedia) (standard reference, not scraped)
- MIT OpenCourseWare, The Long Line (standard reference, not scraped)