Alexandrov compactification
E173177
The Alexandrov compactification is a topological construction that adds a single “point at infinity” to a non-compact space to make it compact.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Alexandroff compactification | 1 |
| Alexandrov compactification canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1509443 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Alexandrov compactification Context triple: [Pavel Alexandrov, notableFor, Alexandrov compactification]
-
A.
Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
-
B.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
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C.
Noetherian space
A Noetherian space is a topological space in which every descending chain of closed subsets stabilizes, mirroring the finiteness conditions of Noetherian rings in algebra.
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D.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
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E.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Alexandrov compactification Target entity description: The Alexandrov compactification is a topological construction that adds a single “point at infinity” to a non-compact space to make it compact.
-
A.
Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
-
B.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
-
C.
Noetherian space
A Noetherian space is a topological space in which every descending chain of closed subsets stabilizes, mirroring the finiteness conditions of Noetherian rings in algebra.
-
D.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
-
E.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
compactification
ⓘ
functorial construction in topology ⓘ topological construction ⓘ |
| adds | single point at infinity ⓘ |
| alsoKnownAs |
Alexandrov compactification
ⓘ
surface form:
Alexandroff compactification
one-point compactification ⓘ |
| appliesTo | locally compact non-compact Hausdorff spaces ⓘ |
| categoryTheoreticView | defines a functor from locally compact Hausdorff spaces to compact Hausdorff spaces ⓘ |
| codomain | compact topological space ⓘ |
| constructionStep |
add a new point often denoted infinity
ⓘ
define neighborhoods of infinity as complements of compact subsets of the original space ⓘ |
| contrastWith | Stone–Čech compactification which is maximal and often adds many points ⓘ |
| domain | non-compact topological space ⓘ |
| example |
one-point compactification of R is homeomorphic to S^1 for R^1
ⓘ
one-point compactification of R^n is homeomorphic to S^n ⓘ one-point compactification of a discrete countable space is homeomorphic to a convergent sequence with its limit point ⓘ one-point compactification of an open interval (0,1) is homeomorphic to S^1 ⓘ |
| failsToBeHausdorffIf | the original space is not locally compact ⓘ |
| field | general topology ⓘ |
| goal | to obtain a compact space from a non-compact space ⓘ |
| historicalPeriod | 20th century topology ⓘ |
| namedAfter | Pavel Alexandrov ⓘ |
| notation |
X ∪ {∞} for the compactified space of X
ⓘ
X^* for the compactified space of X ⓘ |
| outputProperty |
Hausdorff if the original space is locally compact Hausdorff
ⓘ
compactness ⓘ |
| preserves |
connectedness
ⓘ
local compactness away from the added point ⓘ local connectedness away from the added point ⓘ |
| property | unique up to homeomorphism for a given locally compact Hausdorff space ⓘ |
| relatedConcept |
Freudenthal compactification
ⓘ
Stone–Čech compactification ⓘ compactification of a topological space ⓘ |
| requires |
Hausdorff space for uniqueness up to homeomorphism
ⓘ
original space to be non-compact to be nontrivial ⓘ |
| specialCaseOf | compactification by adjunction of boundary points ⓘ |
| topologyDefinedBy | open sets of original space plus sets whose complement is compact ⓘ |
| universalProperty |
every continuous map from the original space to a compact space sending points escaping to infinity to a single point factors uniquely through it
ⓘ
minimal compactification adding only one point ⓘ |
| usedFor |
defining reduced cohomology via compact spaces
ⓘ
studying behavior of functions at infinity ⓘ treating non-compact spaces as compact by adding a point at infinity ⓘ |
| usedIn |
algebraic topology
ⓘ
dynamical systems ⓘ functional analysis ⓘ homotopy theory ⓘ potential theory ⓘ |
| yields | a compact Hausdorff space when applied to a locally compact Hausdorff space ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Alexandrov compactification Description of subject: The Alexandrov compactification is a topological construction that adds a single “point at infinity” to a non-compact space to make it compact.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Alexandroff compactification