Freudenthal compactification
E679314
The Freudenthal compactification is a topological construction that extends a non-compact, locally compact space by adding a boundary of “ends” to obtain a compact space that more finely captures its asymptotic structure than the one-point (Alexandrov) compactification.
Statements (39)
| Predicate | Object |
|---|---|
| instanceOf |
compactification
ⓘ
construction in topology ⓘ topological construction ⓘ |
| adds |
boundary of ends
ⓘ
ends ⓘ |
| appliesTo |
locally compact space
ⓘ
non-compact space ⓘ |
| assumes |
Hausdorff property of the original space
ⓘ
local compactness of the original space ⓘ |
| captures |
asymptotic behavior of spaces
ⓘ
structure of ends of a space ⓘ |
| characterizedBy | boundary points corresponding to equivalence classes of rays going to infinity ⓘ |
| comparedTo | Alexandrov compactification NERFINISHED ⓘ |
| constructionType | end-compactification ⓘ |
| domain | locally compact, σ-compact spaces ⓘ |
| field |
algebraic topology
ⓘ
geometric topology ⓘ topology ⓘ |
| focusesOn | behavior of sequences and rays escaping to infinity ⓘ |
| generalizes | end compactification of graphs ⓘ |
| introducedBy | Hans Freudenthal NERFINISHED ⓘ |
| namedAfter | Hans Freudenthal NERFINISHED ⓘ |
| produces | compact space ⓘ |
| property |
extends the original space as a dense subset
ⓘ
resulting space is compact and Hausdorff when the original space is locally compact and Hausdorff ⓘ |
| purpose |
to capture asymptotic structure of a space
ⓘ
to compactify non-compact locally compact spaces ⓘ |
| refines | one-point compactification ⓘ |
| relatedConcept |
Alexandrov compactification
NERFINISHED
ⓘ
Stone–Čech compactification NERFINISHED ⓘ end of a topological space ⓘ space of ends ⓘ |
| usedFor |
defining boundaries of non-compact spaces
ⓘ
studying ends of groups via Cayley graphs ⓘ |
| usedIn |
geometric group theory
ⓘ
study of infinite graphs ⓘ study of non-compact manifolds ⓘ topology of manifolds ⓘ |
| yields | a compactification finer than the one-point compactification ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.