Statements (26)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:appliesTo |
gptkb:Euclidean_space
|
| gptkbp:concerns |
gptkb:topology
|
| gptkbp:field |
mathematical analysis
|
| gptkbp:firstPublished |
1895
|
| gptkbp:form |
Every open cover of a closed and bounded subset of R^n has a finite subcover.
|
| gptkbp:generalizes |
gptkb:Bolzano–Weierstrass_theorem
|
| https://www.w3.org/2000/01/rdf-schema#label |
Heine–Borel theorem
|
| gptkbp:implies |
every closed and bounded interval in R is compact
|
| gptkbp:namedAfter |
gptkb:Émile_Borel
gptkb:Eduard_Heine |
| gptkbp:relatedTo |
closed set
compactness bounded set open cover finite subcover |
| gptkbp:state |
A subset of Euclidean space is compact if and only if it is closed and bounded.
|
| gptkbp:usedIn |
functional analysis
real analysis metric spaces |
| gptkbp:bfsParent |
gptkb:Tychonoff_theorem
gptkb:庞加莱-伯雷尔定理 gptkb:Eduard_Heine gptkb:Banach–Alaoglu_theorem gptkb:Bolzano–Weierstrass_theorem |
| gptkbp:bfsLayer |
6
|