# 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 CollectionwiseNormal 2013-03-22 14:49:54 2013-03-22 14:49:54 mathcam (2727) mathcam (2727) 5 mathcam (2727) Definition msc 54D20 countably collectionwise normal