# 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 $n$ 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 $\bigsqcup_{i\in I} \{\lambda \mid U_i \subset V_\lambda\}\to I$.

propertiesimplications
second-countable: there is a countable base of the topology.A second-countable space has a $\sigma$-locally finite base: take the the collection of singeltons of all elements of countable cover of $X$.
$\sigma$-locally finite base, i.e. $X$ has a countably locally finite base, e.g. a metrisable topological space by Nagata-Smirnov metrization theorem.second-countable spaces are separable: choose a point in each set of countable cover.
separable: there is a countable dense subset.second-countable spaces are Lindelöf
Lindelöf: every open cover has a countable sub-cover.weakly Lindelöf spaces with countably locally finite base are second countable
weakly Lindelöf: every open cover has a countable subcollection the union of which is dense.separable metacompact spaces are Lindelöf
countable choice: the natural numbers is a projective object in Set.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.
metacompact: every open cover has a point-finite open refinement.Lindelöf spaces are trivially also weakly Lindelöf.

Last revised on April 2, 2019 at 12:23:03. See the history of this page for a list of all contributions to it.