# every $\sigma $-compact set is Lindelöf

###### Theorem 1.

Every $\sigma $-compact (http://planetmath.org/SigmaCompact) set is Lindelöf (every open cover has a
countable^{} subcover).

###### Proof.

Let $X$ be a $\sigma $-compact. Let $\mathcal{A}$ be an open cover of $X$ . Since $X$ is $\sigma $-compact, it is the union of countable many compact sets,

$$X=\bigcup _{i=0}^{\mathrm{\infty}}{X}_{i}$$ |

with ${X}_{i}$ compact. Consider the cover ${\mathcal{A}}_{i}=\{A\in \mathcal{A}:{X}_{i}\cap A\ne \mathrm{\varnothing}\}$ of the set ${X}_{i}$. This cover is well defined, it is not empty and covers ${X}_{i}$: for each $x\in {X}_{i}$ there is at least one of the open sets $A\in \mathcal{A}$ such that $x\in A$.

Since ${X}_{i}$ is compact, the cover ${\mathcal{A}}_{i}$ has a finite subcover. Then

$${X}_{i}\subseteq \bigcup _{j=0}^{{N}_{j}}{A}_{i}^{j}$$ |

and thus

$$X\subseteq \bigcup _{i=0}^{\mathrm{\infty}}\left(\bigcup _{j=0}^{{N}_{j}}{A}_{i}^{j}\right).$$ |

That is, the set $\left\{{A}_{i}^{j}\right\}$ is a countable subcover of $\mathcal{A}$ that covers $X$. ∎

Title | every $\sigma $-compact set is Lindelöf |
---|---|

Canonical name | EverysigmacompactSetIsLindelof |

Date of creation | 2013-03-22 17:34:07 |

Last modified on | 2013-03-22 17:34:07 |

Owner | joen235 (18354) |

Last modified by | joen235 (18354) |

Numerical id | 14 |

Author | joen235 (18354) |

Entry type | Theorem |

Classification | msc 54D45 |