Property of topological spaces stronger than normality
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. Here 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[1] of collectionwise normal demands that the above Ui (i ∈ I) themselves form a discrete family, which is stronger than pairwise disjoint.
Some authors assume that is also a
T1 space as part of the definition, but no such assumption is made here.
A
Hausdorffparacompact space is collectionwise normal.[2] In particular, every
metrizable space is collectionwise normal. Note: The Hausdorff condition is necessary here, since for example an infinite set with the
cofinite topology is
compact, hence paracompact, and T1, but is not even normal.
Every normal
countably compact space (hence every normal compact space) is collectionwise normal. Proof: Use the fact that in a countably compact space any discrete family of nonempty subsets is finite.
An
Fσ-set in a collectionwise normal space is also collectionwise normal in the
subspace topology. In particular, this holds for closed subsets.
The Moore metrization theorem states that a collectionwise normal
Moore space is
metrizable.
Hereditarily collectionwise normal space
A topological space X is called hereditarily collectionwise normal if every subspace of X with the subspace topology is collectionwise normal.
In the same way that
hereditarily normal spaces can be characterized in terms of
separated sets, there is an equivalent characterization for hereditarily collectionwise normal spaces. A family of subsets of X is called a separated family if for every i, we have , with cl denoting the
closure operator in X, in other words if the family of is discrete in its union. The following conditions are equivalent:[3]
X is hereditarily collectionwise normal.
Every open subspace of X is collectionwise normal.
For every separated family of subsets of X, there exists a pairwise disjoint family of open sets , such that .
Examples of hereditarily collectionwise normal spaces
^Engelking, Theorem 5.1.17, shows the equivalence between the two definitions (under the assumption of T1, but the proof does not use the T1 property).