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 $j \in J$.

Related statements and properties

Properties

• second-countable: there is a countable base of the topology.

• metrisable: the topology is induced by a metric.

• $\sigma$-locally discrete base: the topology of $X$ is generated by a $\sigma$-locally discrete base.

• $\sigma$-locally finite base: the topology of $X$ is generated by a countably locally finite base.

• separable: there is a countable dense subset.

• Lindelöf: every open cover has a countable sub-cover.

• weakly Lindelöf: every open cover has a countable subcollection the union of which is dense.

• countable choice: the natural numbers are a projective object in Set.

• metacompact: every open cover has a point-finite open refinement.

• countable chain condition: A family of pairwise disjoint open subsets is at most countable.

• first-countable: every point has a countable neighborhood base

• Frechet-Uryson space: the closure of a set $A$ consists precisely of all limit points of sequences in $A$

• sequential topological space: a set $A$ is closed if it contains all limit points of sequences in $A$

• countably tight: for each subset $A$ and each point $x\in \overline A$ there is a countable subset $D\subseteq A$ such that $x\in \overline D$.

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.

• second-countable spaces are Lindelöf.

• weakly Lindelöf spaces with countably locally finite base are second countable.

• separable metacompact spaces are Lindelöf.

• 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.

• paracompact spaces satisfying the countable chain condition are Lindelöf.