nLab countably locally discrete set of subsets

Definition

Given a topological space $X$, then a set of subsets $\{S_i \subset X\}_{i \in I}$ is countably locally discrete or $\sigma$-locally discrete if it is a countable union of locally discrete sets of subsets.

Note that open covers of metric spaces have open countably locally discrete refinements.

Example

• Every metrisable topological space admits a base which is $\sigma$-locally discrete, see open covers of metric spaces have open countably locally discrete refinements.

Related concepts

• locally finite cover
• locally finite set of subsets
• locally discrete set of subsets
• countably locally finite set of subsets

References

• Ryszard Engelking, General Topology, Heldermann Verlag Berlin, 1989.

• K. Kuratowski, Topology, 2014, vol. 1.