In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

  is called collectionwise normal if for every discrete family F_i (i ∈ I) of closed subsets of there exists a pairwise disjoint family of open sets U_i (i ∈ I), such that F_i ⊂ U_i. A family of subsets 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 demands that the above U_i (i ∈ I) are themselves a discrete family, which is stronger than pairwise disjoint.

Many authors assume that is also a T₁ space as part of the definition.

The property is intermediate in strength between paracompactness

Paracompact space

In mathematics, a paracompact space is a topological space in which every open cover admits a locally finite open refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by ....

 and normality

Normal space

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...

, and occurs in metrisation theorems

Metrization theorem

In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metricd\colon X \times X \to [0,\infty)...

.

Properties

• A collectionwise normal T₁ space is collectionwise Hausdorff
  Collectionwise Hausdorff space
  In mathematics, in the field of topology, a topological space is said to be collectionwise Hausdorff if given any closed discrete collection of points in the topological space, there are pairwise disjoint open sets containing the points...
  
  .
• A collectionwise normal space is normal
  Normal space
  In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
  
  .
• A paracompact space
  Paracompact space
  In mathematics, a paracompact space is a topological space in which every open cover admits a locally finite open refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by ....
  
   is collectionwise normal.
• An F_σ-set in a collectionwise normal space is also collectionwise normal in the subspace topology
  Subspace topology
  In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
  
  . In particular, this holds for closed subsets.
• The Moore metrisation theorem states that a collectionwise normal Moore space is metrisable.