Collectionwise normal space

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