weakly Lindelöf spaces with countably locally finite base are second countable

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? spaces 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.

properties | implications |
---|---|

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. |

Created on April 2, 2019 at 12:17:27. See the history of this page for a list of all contributions to it.