Collectionwise normal space

Collectionwise normal space

In mathematics, a topological space X is called collectionwise normal if for every discrete family Fi (iI) of closed subsets of X there exists a pairwise disjoint family of open sets Ui (iI), such that FiUi. A family of subsets \mathcal{F} of subsets of X is called discrete when every point of X has a neighbourhood that intersects at most one of the sets from \mathcal{F}. An equivalent definition demands that the above Ui (iI) are themselves a discrete family, which is stronger than pairwise disjoint.

Many authors assume that X is also a T1 space as part of the definition.

The property is intermediate in strength between paracompactness and normality, and occurs in metrisation theorems.

Properties

References

  • Engelking, Ryszard, General Topology, Heldermann Verlag Berlin, 1989. ISBN 3-88538-006-4

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