RemarkRemark: AI-adaptedProof: Not applicablejudge 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 product, Euclidean-metric and norm topologies on $R^{n}$ agree, and for $n = 1$ they agree with the real-line topology

For $n \geq 1$ , the product topology on $R^{n}$ is the metric topology of each of $d_{1}$ , $d_{2}$ , and $d_{\infty}$ (For $n \geq 1$ the product topology on $n$ copies of the usual topology of $R$ is the metric topology of $d_{\infty}$ on $R^{n}$ , and hence also of $d_{1}$ and $d_{2}$ , so $R^{n}$ as a product and $R^{n}$ as a metric space are one space). Every norm on $R^{n}$ is equivalent to the Euclidean norm, hence induces that same topology (For $n \geq 1$ all norms on $R^{n}$ are equivalent). Thus open, closed, compact, connected, and continuous below have one unambiguous Euclidean meaning.

When $n = 1$ , the Euclidean metric is the usual metric $d_{R} (s, t) = ∣ s - t ∣$ , and the metric and real-line formulations of continuity and compactness agree (Dictionary: for $A \subseteq R$ with the metric $d (x, y) = ∣ x - y ∣$ , continuity and uniform continuity of $f : A \to R$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $R$ is compact in the open-cover sense of $R$ exactly when it is a compact metric subspace). This page works throughout with $n \geq 1$ .

Depends on

Dependency tree · 2 levels

Sources