LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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.

The image of a universal net under any map is universal, and a continuous map preserves its limits

Statement

If $x$ is a universal net in $X$ and $f : X \to Y$ is any map, then $f (x)$ is universal. If $f$ is continuous and $x \to p$ , then $f (x) \to f (p)$ .

Facts & Assumptions

Given: A universal net $x : D \to X$ and a map $f : X \to Y$ .

[A1]

$x$ is universal precisely when it eventually enters every subset or its complement (Universal net: eventually in every subset or eventually in its complement).

[L1]

A continuous map preserves every convergent net (A map of topological spaces is continuous at a point if and only if it preserves every net converging to that point).

Proof

technique · direct

1.1

Let $S \subseteq Y$ . By [A1], $x$ is eventually in $f^{- 1} [S]$ or in its complement $f^{- 1} [Y ∖ S]$ ; respectively, $f (x)$ is eventually in $S$ or in $Y ∖ S$ .

2.1

Thus $f (x)$ is universal.

step 1.1A1

3.1

If $f$ is continuous and $x \to p$ , the second assertion is [L1].

L1∎

Depends on

Used by

Assuming the ultrafilter lemma, an arbitrary product of compact Hausdorff spaces is compact Theorem

Dependency tree · two levels

8 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

WVU Math 581 Topology I (standard reference, not scraped)
Net (mathematics) (Wikipedia) (standard reference, not scraped)