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, a continuous surjection does not increase density or Lindelöf degree

Statement

Assume the Axiom of Choice. If $f:X\to Y$ is continuous and onto, then $d(Y)\le d(X)$ and $L(Y)\le L(X)$ .

Facts & Assumptions

Given: The Axiom of Choice and a continuous surjection $f:X\to Y$ (The Axiom of Choice).

[L1]

A continuous map has open preimages of open sets (Continuity of a map of topological spaces at a point and globally, For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f(\overline{A}) \subseteq \overline{f(A)}$ ).

[L2]

The least dense-set cardinality $d(Z)$ and the least cardinal bounding subcovers $L(Z)$ exist for every topological space $Z$ (Under choice, $d(X)$ is a well-defined cardinal, Under choice, $L(X)$ is a well-defined cardinal).

Proof

technique · direct

1.1

If $D\subseteq X$ is dense, then $f[D]$ is dense in $Y$ : a nonempty open $V\subseteq Y$ has nonempty open preimage by surjectivity and [L1], so that preimage meets $D$ and $V$ meets $f[D]$ .

1.2

For an open cover $\mathcal U$ of $Y$ , the family $\{f^{-1}[U]:U\in\mathcal U\}$ is an open cover of $X$ ; a subfamily indexed by at most $L(X)$ members covers $X$ , and the corresponding members of $\mathcal U$ cover $Y$ by surjectivity.

L1L2

2.1

Taking $D$ with $|D|=d(X)$ , step 1.1 gives a dense subset of $Y$ of cardinality at most $d(X)$ ; hence $d(Y)\le d(X)$ by [L2].

step 1.1L2

3.1

Step 1.2 gives $L(Y)\le L(X)$ by [L2], and together with step 2.1 this proves both inequalities.

step 2.1step 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: 52 results over 14 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)