nLab weakly Lindelöf spaces with countably locally finite base are second countable

Statement

Recall that a topological space is weakly Lindelöf if every open cover has a countable subcollection the union of which is dense.

Theorem

Every weakly Lindelöf space with $\sigma$-locally finite base is second countable.

Proof

Let $\mathcal{V}$ be a locally finite collection of nonempty open sets. For each $x \in X$, there is a neighborhood $N_x$ meeting finitely many members of $\mathcal{V}$. As $X$ is weakly Lindelöf, there are $x_n$ such that $\{N_{x_n}\}_n$ covers a dense subset of $X$. Then each member of $\mathcal{V}$ meets some $N_{x_n}$, showing $\mathcal{V}$ is countable.

Therefore any countably locally finite base in a weakly Lindelöf space is countable, showing the space is second countable.

Related statements and properties

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

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.

• 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 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.

• 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.

References

• “Is a $\sigma$-locally finite collection of open sets locally countable?” [MO:a/4878507]