Collectionwise normal space

In mathematics, a topological space formula_1 is called collectionwise normal if for every discrete family "F""i" ("i" ∈ "I") of closed subsets of formula_1 there exists a pairwise disjoint family of open sets "U""i" ("i" ∈ "I"), such that "F""i" ⊂ "U""i". A family formula_3 of subsets of formula_1 is called "discrete" when every point of formula_1 has a neighbourhood that intersects at most one of the sets from formula_3.
An equivalent definition demands that the above "U""i" ("i" ∈ "I") are themselves a discrete family, which is stronger than pairwise disjoint.
Some authors assume that formula_1 is also a T1 space as part of the definition.
The property is intermediate in strength between paracompactness and normality, and occurs in metrization theorems.
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 formula_8 of subsets of "X" is called a separated family if for every "i", we have formula_9, with cl denoting the closure operator in "X", in other words if the family of formula_10 is discrete in its union. The following conditions are equivalent: