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 $\mathcal{F}$ 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, 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 F_σ-set in a collectionwise normal space is also collectionwise normal in the subspace topology. In particular, this holds for closed subsets.

References

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