Lindelöf space
E518483
A Lindelöf space is a topological space in which every open cover has a countable subcover, generalizing a key compactness property.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
topological property ⓘ |
| characterizedInMetricSpacesBy | every open cover has a countable subcover if and only if the space is separable and second countable ⓘ |
| closedSubspaceOf | Lindelöf space is Lindelöf ⓘ |
| continuousImageOf | Lindelöf space is Lindelöf NERFINISHED ⓘ |
| countableUnionOf | Lindelöf subspaces need not be Lindelöf ⓘ |
| definition | a topological space in which every open cover has a countable subcover ⓘ |
| doesNotImply |
compactness
ⓘ
paracompactness in general ⓘ σ-compactness ⓘ |
| field | topology ⓘ |
| generalizes | compact space ⓘ |
| hasCardinalityRestriction | no restriction on underlying set cardinality ⓘ |
| hasExample |
every compact space
ⓘ
every second countable space ⓘ the real line with the usual topology ⓘ |
| hasNonExample |
an uncountable discrete space
ⓘ
the product of uncountably many copies of the unit interval with the product topology ⓘ |
| hasOpenCoverCondition | every open cover has a countable subcover ⓘ |
| hasProperty | every open cover admits a countable subcover ⓘ |
| implies |
Lindelöf property
ⓘ
every family of pairwise disjoint nonempty open sets is countable ⓘ every open cover has a countable dense subfamily ⓘ |
| impliesInRegularSpaces | normality under additional set-theoretic assumptions ⓘ |
| isEquivalentInMetricSpacesTo |
Lindelöfness is equivalent to separability and second countability
ⓘ
separability plus completeness does not characterize Lindelöfness ⓘ |
| isHereditaryWithRespectTo | closed subspaces ⓘ |
| isImpliedBy |
compact space
ⓘ
second countable space ⓘ σ-compact space ⓘ |
| isNotPreservedBy |
arbitrary products
ⓘ
uncountable disjoint unions ⓘ |
| isPreservedBy |
closed subspaces
ⓘ
quotient maps ⓘ |
| isStrongerThan | second countability in general ⓘ |
| isWeakerThan | compactness ⓘ |
| namedAfter | Ernst Leonard Lindelöf NERFINISHED ⓘ |
| productWith |
Lindelöf space need not be Lindelöf in general
ⓘ
compact space is Lindelöf ⓘ |
| relatedConcept |
compact space
ⓘ
countably compact space ⓘ paracompact space ⓘ second countable space ⓘ separable space ⓘ σ-compact space ⓘ |
| usedIn |
general topology
ⓘ
set-theoretic topology ⓘ |
| usedToStudy | covering properties of topological spaces ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.