Collectionwise Hausdorff space
In mathematics, in the field of topology, a topological space is said to be collectionwise Hausdorff if given any closed discrete subset of, there is a pairwise disjoint family of open sets with each point of the discrete subset contained in exactly one of the open sets.
Here a subset being discrete has the usual meaning of being a discrete space with the subspace topology.
Properties
- Every T1 space that is collectionwise Hausdorff is also Hausdorff.
- Every collectionwise normal space is collectionwise Hausdorff.
- Metrizable spaces are collectionwise normal and hence collectionwise Hausdorff.