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- Collectionwise_normal_space abstract "In mathematics, a topological space is called collectionwise normal if for every discrete family Fi (i ∈ I) of closed subsets of there exists a pairwise disjoint family of open sets Ui (i ∈ I), such that Fi ⊂ Ui. A family of subsets of is called discrete when every point of has a neighbourhood that intersects at most one of the sets from .An equivalent definition demands that the above Ui (i ∈ I) are themselves a discrete family, which is stronger than pairwise disjoint. Many authors assume that is also a T1 space as part of the definition, i. e., for every pair of distinct points, each has an open neighborhood not containing the other. A collectionwise normal T1 space is a collectionwise Hausdorff space.Every collectionwise normal space is normal (i. e., any two disjoint closed sets can be separated by neighbourhoods), and every paracompact space (i. e., every topological space in which every open cover admits a locally finite open refinement) is collectionwise normal. The property is therefore intermediate in strength between paracompactness and normality.Every metrizable space (i. e., every topological space that is homeomorphic to a metric space) is collectionwise normal. The Moore metrisation theorem states that every collectionwise normal Moore space is metrizable.An Fσ-set in a collectionwise normal space is also collectionwise normal in the subspace topology. In particular, this holds for closed subsets.".
- Collectionwise_normal_space wikiPageID "9715483".
- Collectionwise_normal_space wikiPageRevisionID "475357169".
- Collectionwise_normal_space hasPhotoCollection Collectionwise_normal_space.
- Collectionwise_normal_space subject Category:Properties_of_topological_spaces.
- Collectionwise_normal_space subject Category:Separation_axioms.
- Collectionwise_normal_space type Abstraction100002137.
- Collectionwise_normal_space type AuditoryCommunication107109019.
- Collectionwise_normal_space type Communication100033020.
- Collectionwise_normal_space type Maxim107152948.
- Collectionwise_normal_space type Possession100032613.
- Collectionwise_normal_space type PropertiesOfTopologicalSpaces.
- Collectionwise_normal_space type Property113244109.
- Collectionwise_normal_space type Relation100031921.
- Collectionwise_normal_space type Saying107151380.
- Collectionwise_normal_space type SeparationAxioms.
- Collectionwise_normal_space type Speech107109196.
- Collectionwise_normal_space comment "In mathematics, a topological space is called collectionwise normal if for every discrete family Fi (i ∈ I) of closed subsets of there exists a pairwise disjoint family of open sets Ui (i ∈ I), such that Fi ⊂ Ui. A family of subsets of is called discrete when every point of has a neighbourhood that intersects at most one of the sets from .An equivalent definition demands that the above Ui (i ∈ I) are themselves a discrete family, which is stronger than pairwise disjoint.".
- Collectionwise_normal_space label "Collectionwise normal space".
- Collectionwise_normal_space label "Espace collectivement normal".
- Collectionwise_normal_space sameAs Espace_collectivement_normal.
- Collectionwise_normal_space sameAs m.02pp_k4.
- Collectionwise_normal_space sameAs Q5146123.
- Collectionwise_normal_space sameAs Q5146123.
- Collectionwise_normal_space sameAs Collectionwise_normal_space.
- Collectionwise_normal_space wasDerivedFrom Collectionwise_normal_space?oldid=475357169.
- Collectionwise_normal_space isPrimaryTopicOf Collectionwise_normal_space.