Separation axioms | Properties of topological spaces

Collectionwise normal space

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 of collectionwise normal demands that the above Ui (i ∈ I) are themselves a discrete family, which is stronger than pairwise disjoint. Some authors assume that is also a T1 space as part of the definition. The property is intermediate in strength between paracompactness and normality, and occurs in metrization theorems. (Wikipedia).

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Code samples derived from work by Joey de Vries, @joeydevries, author of https://learnopengl.com/ All code samples, unless explicitly stated otherwise, are licensed under the terms of the CC BY-NC 4.0 license as published by Creative Commons, either version 4 of the License, or (at your o

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Related pages

Neighbourhood (mathematics) | Paracompact space | Compact space | Topological space | Separated sets | Countably compact space | Closure (topology) | Subspace topology | Metrizable space | Moore space (topology) | Normal space | Order topology | Monotonically normal space | Hausdorff space | Hereditarily normal space