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 |