collectionwise normal

A Hausdorff topological space $X$ is called collectionwise normal if any discrete collection of sets $\{U_{i}\}$ in $X$ can be covered by a pairwise-disjoint collection of open sets $\{V_{j}\}$ such that each $V_{j}$ covers just one $U_{i}$ . This is equivalent to requiring the same property for any discrete collection of closed sets.

A Hausdorff topological space $X$ is called countably collectionwise normal if any countable discrete collection of sets $\{U_{i}\}$ in $X$ can be covered by a pairwise-disjoint collection of open sets $\{V_{j}\}$ such that each $V_{j}$ covers just one $U_{i}$ . This is equivalent to requiring the same property for any countable discrete collection of closed sets.

Any metrizable space is collectionwise normal.

References

1 Steen, Lynn Arthur and Seebach, J. Arthur, Counterexamples in Topology, Dover Books, 1995.

Title	collectionwise normal
Canonical name	CollectionwiseNormal
Date of creation	2013-03-22 14:49:54
Last modified on	2013-03-22 14:49:54
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	5
Author	mathcam (2727)
Entry type	Definition
Classification	msc 54D20
Defines	countably collectionwise normal