# nLab separable metacompact spaces are Lindelöf

## Statement

Recall that a topological space $X$ is metacompact if every open cover has a point-finite open refinement which covers $X$. Further a space is called Lindelöf if every open cover has a countable sub-cover.

###### Theorem

Assuming the law of excluded middle:

Every separable metacompact topological space $X$ is Lindelöf .

###### Proof

The proof goes by contradiction: Assume there is an open cover $\{U_i\}_{i\in I}$ with no countable subcover. Let $\{U_j\}_{j\in J}$ be a point finite refinement, which again must not be countable. Let $D\subset$ be a countable dense set. Then $\{U_j\}_{j\in J}$ is uncountable and so some point $x \in D$ is in $U_j$ for infinetly (even uncountably) many $j \in J$.

### 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 countable cover of $X$.

• second-countable spaces are separable: use the axiom of countable choice to choose a point in each set of 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 April 3, 2020 at 17:43:25. See the history of this page for a list of all contributions to it.