Recall that a topological space is weakly Lindelöf if every open cover has a countable subcollection the union of which is dense.
Every weakly Lindelöf space with $\sigma$-locally finite base is second countable.
Let $\mathcal{V}$ be a countably locally finite base. For each $x \in X$, there is a neighborhood $N_x$ meeting countably many members of $\mathcal{V}$. If $X$ is weakly Lindelöf, there is a countable $\{N_n\}_n$ which covers a dense subset of $X$. Then $\mathcal{U} = \{V\in \mathcal{V} \mid N_n \cap V \neq \emptyset\}$ is a countable basis for X.
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.
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 $x\in \overline D$.
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.
weakly Lindelöf spaces with countably locally finite base are second countable.
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.
Last revised on April 3, 2020 at 15:51:36. See the history of this page for a list of all contributions to it.