Locally Discrete Collection - Properties and Examples

Properties and Examples

1. Locally discrete collections are always locally finite. See the page on local finiteness.

2. If a collection of subsets of a topological space X is locally discrete, it must satisfy the property that each point of the space belongs to at most one element of the collection. This means that only collections of pairwise disjoint sets can be locally discrete.

3. A Hausdorff space cannot have a locally discrete basis unless it is itself discrete. The same property holds for a T₁ space.

4. The following is known as Bing's metrization theorem:

A space X is metrizable iff it is regular and has a basis that is countably locally discrete.

5. A countable collection of sets is necessarily countably locally discrete. Therefore, if X is a metrizable space with a countable basis, one implication of Bing's metrization theorem holds. In fact, Bing's metrization theorem is almost a corollary of the Nagata-Smirnov theorem.