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.
An uncountable discrete space is metrizable and has a discrete basis, but is not second countable
Example
The real set with the discrete topology is metrizable and has a discrete basis, but is not second countable.
Facts & Assumptions
Given: The set with its discrete topology.
The real line is uncountable ( is uncountable (Cantor's nested intervals, 1874)).
Verification
The zero-one function for and otherwise is a metric inducing the discrete topology; singleton sets form a discrete basis.
Any basis must contain for every : applying the basis condition to the open set produces a basis member containing and contained in . Thus a countable basis would inject the uncountable set into a countable set, contradicting [L1].
This gives the claimed profile.
Depends on
- Discrete families and $\sigma$-locally-finite and $\sigma$-discrete bases
- Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not
- Second countability: an at most countable basis for the topology
- The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- $\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874)
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
51 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
- Encyclopedia of Mathematics, Metrizable space (standard reference, not scraped)