PropositionStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31‡ rests on unproved material (inherited)

How 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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Under choice, for $Y\subseteq X$ , $w(Y)\le w(X)$ and $\chi(y,Y)\le\chi(y,X)$

Statement

Assuming choice, $Y\subseteq X$ implies $w(Y)\le w(X)$ and $\chi(y,Y)\le\chi(y,X)$ for $y\in Y$ .

Facts & Assumptions

Given: A subspace $Y\subseteq X$ , a basis $\mathcal B$ of $X$ of cardinality $w(X)$ , and a local base $\mathcal N_y$ at $y\in Y$ of cardinality $\chi(y,X)$ .

[L1]

In the subspace topology, the open subsets of $Y$ are the traces $U\cap Y$ of open subsets $U$ of $X$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

[L2]

Weight is the least cardinality of a basis and local character is the least cardinality of a neighbourhood base at a point (Under choice, weight $w(X)$ , density $d(X)$ , local character $\chi(x,X)$ , and character $\chi(X)$ as raw cardinal minima and a supremum).

Proof

technique · direct

1.1

The family $\{B\cap Y:B\in\mathcal B\}$ is a basis of $Y$ by [L1] and has cardinality at most $|\mathcal B|=w(X)$ .

1.2

The family $\{N\cap Y:N\in\mathcal N_y\}$ is a local base at $y$ in $Y$ by [L1] and has cardinality at most $|\mathcal N_y|=\chi(y,X)$ .

2.1

Applying the two minima in [L2] to the families of steps 1.1 and 1.2 yields $w(Y)\le w(X)$ and $\chi(y,Y)\le\chi(y,X)$ .

step 1.1step 1.2L2∎

Depends on

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: 33 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

D. H. Fremlin, Measure Theory, Chapter 5A (standard reference, not scraped)

Under choice, for Y⊆XY\subseteq XY⊆X, w(Y)≤w(X)w(Y)\le w(X)w(Y)≤w(X) and χ(y,Y)≤χ(y,X)\chi(y,Y)\le\chi(y,X)χ(y,Y)≤χ(y,X)