separable metacompact spaces are Lindelöf

Recall that a topological space $X$ is metacompact if every open cover has a point-finite open refinement which covers $X$. Further a space is called Lindelöf if every open cover has a countable sub-cover.

Assuming the law of excluded middle:

Every separable metacompact topological space $X$ is Lindelöf .

If $\{U_i\}_{i\in I}$ is an open cover with no countable subcover, let $\{U_j\}_{j\in J}$ be a point finite refinement. Then $\{U_j\}_{j\in J}$ is uncountable and so some point of the countable dense set is in $U_j$ for uncountably many $j \in J$.

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 12:13:15. See the history of this page for a list of all contributions to it.