nLab Lindelöf topological space

Contents

Definition

A topological space is a Lindelöf space if every open cover has a countable sub-cover.

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 $x\in \overline D$ .

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.

On product spaces with Lindelöf spaces:

Michael Barr, John Kennison, Robert Raphael, On productively Lindelöf spaces, Scient. Math. Japon. 65 (2007) 23–36 [pdf, pdf]