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.
Separability: the existence of an at most countable dense subset
Definition
A topological space is separable if some at most countable subset is dense in (Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets, Finite, countably infinite, countable, uncountable). Equivalently, every nonempty open subset of meets .
Depends on
Used by
- Under choice, the five cardinal functions recover first countability, second countability, separability, Lindelöfness, and ccc at the ℵ₀ threshold Corollary
- Assuming countable choice, ω₁ is first countable and countably compact but is not separable or Lindelöf Example
- The one-point compactification of the discrete real line is compact and Lindelöf but is neither first countable nor separable Example
- Under choice, the lower-limit line is regular and separable but not second countable and therefore not metrizable Example
- FALSE: every regular space is metrizable False statement
- Refuted: every separable space is second countable False statement
- Refuted: separability is hereditary False statement
- Under choice, if |I|>2^ℵ₀, then the Cantor cube 2^I is not separable Lemma
- Every separable space satisfies the countable chain condition Proposition
- Assuming countable choice, a metrizable space is second countable if and only if it is separable if and only if it is Lindelöf Theorem
- Assuming countable choice, every second countable space is separable Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 27 results over 10 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
- UCR General Topology Notes (standard reference, not scraped)
- Separable space (Wikipedia) (standard reference, not scraped)