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.
is disconnected, whereas is polygonally connected
Example
The invertible real matrices are the nonzero real numbers, so . This set is disconnected: it contains and but not the intermediate point , so it is not order-convex and cannot be connected by The connected subspaces of with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in ". In contrast, For , the punctured space is polygonally connected gives polygonal connectedness of . Connectedness is understood through Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets.
Depends on
- For $n\ge2$, the punctured space $\mathbb{R}^n\setminus\{0\}$ is polygonally connected
- The connected subspaces of $\mathbb{R}$ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in $\mathbb{R}$"
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
26 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
- General linear group (standard reference, not scraped)
- Connected space (Wikipedia) (standard reference, not scraped)