# nLab second-countable spaces are Lindelöf

## Statement

###### Theorem

Using countable choice, then: Every second-countable topological space $X$ is Lindelöf, i.e. any open cover admits a countable subcover.

###### Proof

Let $\{U_n\}$ be a countable base of the topology. Given any open cover $\{V_\lambda\}$ of $X$, we can form the index set $I\subset \mathbb{N}$ of those $U_n$ that are contained in some $V_\lambda$. By assumption $\bigcup_{i\in I} U_{i} = \bigcup_\lambda V_\lambda = X$. The axiom of countable choice provides now a section of $\bigsqcup_{i\in I} \{\lambda \mid U_i \subset V_\lambda\}\to I$.

Axioms: axiom of choice (AC), countable choice (CC).

### Implications

• a metric space has a $\sigma$-locally discrete base

• Nagata-Smirnov metrization theorem

• a second-countable space has a $\sigma$-locally finite base: take the the collection of singeltons of all elements of a countable cover of $X$.

• second-countable spaces are separable: use the axiom of countable choice to choose a point in each set of a countable cover.

• separable spaces satisfy the countable chain condition: given a dense set $D$ and a family $\{U_\alpha : \alpha \in A\}$, the map $D \cap \bigcup_{\alpha \in A} U_\alpha \to A$ assigning $d$ to the unique $\alpha \in A$ with $d \in U_\alpha$ is surjective.

• separable spaces are weakly Lindelöf: given a countable dense subset and an open cover choose for each point of the subset an open from the cover.

• Lindelöf spaces are trivially also weakly Lindelöf.

• a space with a $\sigma$-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a $\sigma$-locally finite base.

• a first-countable space is obviously Fréchet-Urysohn.

• a Fréchet-Uryson space is obviously sequential.

• a sequential space is obviously countably tight.

Last revised on August 20, 2023 at 12:53:18. See the history of this page for a list of all contributions to it.