nLab
open covers of metric spaces have open countably locally discrete refinements

Definitions

Definition

(locally finite cover)

Let (X,τ)(X,\tau) be a topological space.

An open cover {U iX} iI\{U_i \subset X\}_{i \in I} of XX is called locally finite if for each point xXx \in X, there exists a neighbourhood U x{x}U_x \supset \{x\} such that it intersects only finitely many elements of the cover, hence such that U xU iU_x \cap U_i \neq \emptyset for only a finite number of iIi \in I.

Definition

(refinement of open covers)

Let (X,τ)(X,\tau) be a topological space, and let {U iX} iI\{U_i \subset X\}_{i \in I} be a open cover.

Then a refinement of this open cover is a set of open subsets {V jX} jJ\{V_j \subset X\}_{j \in J} which is still an open cover in itself and such that for each jJj \in J there exists an iIi \in I with V jU iV_j \subset U_i.

Statement

Theorem

Every open cover of a metric space has a σ \sigma -locally discrete refinement.

Corollary

Every metric space XX has a σ\sigma-locally discrete base.

Proof

Set 𝒰 n{open ball of radius 1/n around xxX}\mathcal{U}_n \coloneqq \{ \text{open ball of radius }\; 1/n \;\text{ around }\;\, x \mid x \in X \}. For each nn let 𝒱 n\mathcal{V}_n be a σ\sigma-locally discrete refinement of 𝒰 n\mathcal{U}_n. By a diagonal argument the family 𝒱 n𝒱 n\mathcal{V} \coloneqq \bigcup_n \mathcal{V}_n is also σ\sigma-locally discrete. Moreover 𝒱\mathcal{V} is a base since for each point xx the balls of radius 1/n1/n form a neighborhood base.

References

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

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

Last revised on April 30, 2019 at 09:57:03. See the history of this page for a list of all contributions to it.