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.

单词	CollectionwiseNormal
释义	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.
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