Sur quelques points de la théorie des fonctions¶
Borel, É. (1895). Sur quelques points de la théorie des fonctions. Annales scientifiques de l'École normale supérieure, 12, 9-55.
Cited by¶
3 citations across 3 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Boundedness
- In finite-dimensional Euclidean space, the Heine-Borel theorem asserts that a subset is compact if and only if it is closed and bounded — so for $\mathbb{R}^n$ with the Euclidean metric, compactness and (closed-and-bounded) coincide.
This sourcethe structural bridge linking boundedness (magnitude) to compactness (topology) in finite dimensions.
- In finite-dimensional Euclidean space, the Heine-Borel theorem asserts that a subset is compact if and only if it is closed and bounded — so for $\mathbb{R}^n$ with the Euclidean metric, compactness and (closed-and-bounded) coincide.
- Completeness
- … completeness/incompleteness contrast is sharpened by the infinite expressive power of the systems involved), `boundedness` (the Heine-Borel theorem links bounded-and-closed in $\mathbb{R}^n$ to compactness, which under metric completeness becomes the conjunction of total-boundedness and completeness;
This sourceThe standard attribution for the covering-compactness result underlying the Heine-Borel theorem (every closed and bounded subset of R^n is compact), with prior contributions from Eduard Heine, Pierre Cousin, and Henri Lebesgue.
- … completeness/incompleteness contrast is sharpened by the infinite expressive power of the systems involved), `boundedness` (the Heine-Borel theorem links bounded-and-closed in $\mathbb{R}^n$ to compactness, which under metric completeness becomes the conjunction of total-boundedness and completeness;
- Topology
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