Collectionwise normal

In mathematics, a topological space $X$ is called collectionwise normal if for every discrete family "F"_"i" ("i" ∈ "I") of closed subsets of $X$ there exists a pairwise disjoint family of open sets "U"_"i" ("i" ∈ "I"), such that "F"_"i" ⊂ "U"_"i". 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 "U"_"i" ("i" ∈ "I") 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

*A collectionwise normal T₁ space is collectionwise Hausdorff.
*A collectionwise normal space is normal.
*A paracompact space is collectionwise normal.
*An F_σ-set in a collectionwise normal space is also collectionwise normal in the subspace topology. In particular, this holds for closed subsets.
*The "Moore metrisation theorem" states that a collectionwise normal Moore space is metrisable.

References

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