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.
The product, Euclidean-metric and norm topologies on agree, and for they agree with the real-line topology
For , the product topology on is the metric topology of each of , , and (For the product topology on copies of the usual topology of is the metric topology of on , and hence also of and , so as a product and as a metric space are one space). Every norm on is equivalent to the Euclidean norm, hence induces that same topology (For all norms on are equivalent). Thus open, closed, compact, connected, and continuous below have one unambiguous Euclidean meaning.
When , the Euclidean metric is the usual metric , and the metric and real-line formulations of continuity and compactness agree (Dictionary: for with the metric , continuity and uniform continuity of agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of is compact in the open-cover sense of exactly when it is a compact metric subspace). This page works throughout with .
Depends on
- For $n \ge 1$ the product topology on $n$ copies of the usual topology of $\mathbb{R}$ is the metric topology of $d_\infty$ on $\mathbb{R}^n$, and hence also of $d_1$ and $d_2$, so $\mathbb{R}^n$ as a product and $\mathbb{R}^n$ as a metric space are one space
- Dictionary: for $A \subseteq \mathbb{R}$ with the metric $d(x,y) = |x-y|$, continuity and uniform continuity of $f : A \to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace
- For $n \ge 1$ all norms on $\mathbb{R}^n$ are equivalent
Used by
Dependency tree · two levels
55 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
- J. R. Munkres, Topology, 2nd ed., §§19, 20 (standard reference, not scraped)
- Product topology (Wikipedia) (standard reference, not scraped)
- Norm (mathematics) (Wikipedia) (standard reference, not scraped)