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

Definitions

Definition

(locally finite cover)

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

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

Definition

(refinement of open covers)

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

Then a refinement of this open cover is a set of open subsets $\{V_j \subset X\}_{j \in J}$ which is still an open cover in itself and such that for each $j \in J$ there exists an $i \in I$ with $V_j \subset U_i$.

Statement

Theorem

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

Corollary

Every metric space $X$ has a $\sigma$-locally discrete base.

Proof

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

Related statements

• metrization theorem

• metric spaces are paracompact

• paracompact Hausdorff spaces are normal

• locally compact and sigma-compact spaces are paracompact

• second-countable regular spaces are paracompact

• regular Lindelöf spaces are paracompact

• CW-complexes are paracompact Hausdorff spaces

• paracompact Hausdorff spaces equivalently admit subordinate partitions of unity

References

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

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