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.
Every topological group is uniformizable, and assuming dependent choice it is completely regular
Statement
Every topological group is uniformizable. Assuming dependent choice, every topological group is completely regular.
Facts & Assumptions
Given: A topological group.
Its left uniformity induces its given topology (The left and right uniformities of a topological group induce its topology, and inversion interchanges them).
Under dependent choice, uniformizable spaces are completely regular (Assuming dependent choice, a nonempty topological space is uniformizable if and only if it is completely regular, The axiom of dependent choice: a relation in which every element is related to something admits an -indexed chain).
Proof
The left uniformity of [L1] makes the group uniformizable.
Under dependent choice, [L2] applied to step 1.1 makes it completely regular.
Depends on
- The left and right uniformities of a topological group induce its topology, and inversion interchanges them
- Assuming dependent choice, a nonempty topological space is uniformizable if and only if it is completely regular
- The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 50 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- M. Megrelishvili, Lecture Notes in Topological Groups (standard reference, not scraped)
- M. Kunzinger, General Topology (standard reference, not scraped)