nLab paracompact spaces satisfying the countable chain condition are Lindelöf

Statement

Recall that a topological space $X$ is paracompact if every open cover has a refinement by a locally finite open cover. Further a space is called Lindelöf if every open cover has a countable sub-cover.

Theorem

Assuming the axiom of choice:

Every paracompact space $X$ satisfying the countable chain condition 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 locally finite refinement, which again must not be countable. This is to say that each point possesses an open neighborhood so small that it is only an element of finitely many $U_j$‘s. Inductively (using Zorn's lemma) construct a maximal system $\{V_\lambda\}_{\lambda \in \Lambda}$ of pairwise disjoint opens being contained in at most finitely many $U_j$’s. Due to maximality $\bigcup_{\lambda\in\Lambda} V_\lambda$ is dense. This fact implies by the countable chain condition that $\Lambda$ is countable. Moreover it implies that each $U_j$ intersects at least one $V_\lambda$. But this is to say that there are at most countably many $U_j$’s. This is a contradiction.

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 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.