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.
Second countability: an at most countable basis for the topology
Definition
A topological space is second countable when its topology has a basis that is at most countable (Basis and subbasis for a topology, and the topology generated by a family of sets, Finite, countably infinite, countable, uncountable). Thus every open set is a union of members of one at most countable family .
Remarks
The basis is global. This differs from first countability, where the countable family is allowed to depend on the point.
Depends on
Used by
- Under choice, every regular T₁ second-countable space is metrizable Corollary
- Under choice, the five cardinal functions recover first countability, second countability, separability, Lindelöfness, and ccc at the ℵ₀ threshold Corollary
- An uncountable discrete space is metrizable and has a discrete basis, but is not second countable Example
- Assuming choice, the lower-limit plane is first countable, separable, and ccc, but not second countable or Lindelöf 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 first countable space is second countable False statement
- Refuted: every separable space is second countable False statement
- Second countability is hereditary Proposition
- Assuming countable choice, a countable product of second countable spaces is second countable Theorem
- 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 Lindelöf Theorem
- Assuming countable choice, every second countable space is separable Theorem
- Every second countable space is first countable Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 24 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)
- Second-countable space (Wikipedia) (standard reference, not scraped)