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.

The property is intermediate in strength between paracompactness and normality, and occurs in metrisation theorems.

Properties

A collectionwise normal T₁ space is collectionwise Hausdorff.
A collectionwise normal space is normal.
A paracompact space is collectionwise normal.
A metrizable space is collectionwise normal.
An F_σ-set in a collectionwise normal space is also collectionwise normal in the subspace topology. In particular, this holds for closed subsets.
The Moore metrisation theorem states that a collectionwise normal Moore space is metrisable.

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 $F_{i}(i\in I)$ of subsets of X is called a separated family if for every i, we have $F_{i}\cap cl\left(\bigcup _{j\neq i}F_{j}\right)=\emptyset$ , with cl denoting the closure operator in X, in other words if the family of $F_{i}$ is discrete in its union. The following conditions are equivalent:

X is hereditarily collectionwise normal.
Every open subspace of X is collectionwise normal.
For every separated family $F_{i}$ of subsets of X, there exists a pairwise disjoint family of open sets $U_{i}(i\in I)$ , such that $F_{i}\subseteq U_{i}$ .

Examples of hereditarily collectionwise normal spaces

Every linearly ordered topological space (LOTS)
Every generalized ordered space (GO-space)
Every metrizable space
Every monotonically normal space

References

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.