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.
Under choice, for the usual real line, under the raw convention
Example
Specializing is a countable dense subset of , and rational open boxes form a countable basis to , rational-endpoint intervals and the rational points give countable bases and dense sets for the usual real line. The inequalities and then give countable upper bounds for all five functions (Under choice, and ). No finite family can be a basis or a local base, and finite covers or cellular families have arbitrarily large finite witnesses. Thus the five raw functions are all .
Depends on
- Under choice, weight $w(X)$, density $d(X)$, local character $\chi(x,X)$, and character $\chi(X)$ as raw cardinal minima and a supremum
- Under choice, Lindelöf degree $L(X)$ and cellularity $c(X)$ as raw cardinal functions
- Under choice, $c(X)\le d(X)\le w(X)$ and $\chi(X),L(X)\le w(X)$
- Under choice, every metrizable space has $w(X)=d(X)$
- Assuming countable choice, a metrizable space is second countable if and only if it is separable if and only if it is Lindelöf
- $\mathbb{Q}^n$ is a countable dense subset of $\mathbb{R}^n$, and rational open boxes form a countable basis
- The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
49 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
- D. H. Fremlin, Measure Theory, Chapter 5A (standard reference, not scraped)
- Separable space (Wikipedia) (standard reference, not scraped)