is called collectionwise normal if for every discrete family Fi (i ∈ I) of closed subsets of
An equivalent definition[1] of collectionwise normal demands that the above Ui (i ∈ I) themselves form a discrete family, which is a priori stronger than pairwise disjoint.
is also a T1 space as part of the definition, but no such assumption is made here.
The property is intermediate in strength between paracompactness and normality, and occurs in metrization theorems.
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.
, with cl denoting the closure operator in X, in other words if the family of