Alexandrov compactification

E173177

compactification functorial construction in topology topological construction

The Alexandrov compactification is a topological construction that adds a single “point at infinity” to a non-compact space to make it compact.

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All labels observed (2)

Label	Occurrences
Alexandroff compactification	1
Alexandrov compactification canonical	1

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 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Pavel Alexandrov → notableFor → Alexandrov compactification ⓘ

Alexandrov compactification → alsoKnownAs → Alexandrov compactification ⓘ

this entity surface form: Alexandroff compactification