# 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,

 $X=\bigcup_{i=0}^{\infty}X_{i}$

with $X_{i}$ compact. Consider the cover $\mathcal{A}_{i}=\left\{A\in\mathcal{A}:X_{i}\cap A\neq\emptyset\right\}$ 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

 $X\subseteq\bigcup_{i=0}^{\infty}\left(\bigcup_{j=0}^{N_{j}}A_{i}^{j}\right).$

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 EverysigmacompactSetIsLindelof 2013-03-22 17:34:07 2013-03-22 17:34:07 joen235 (18354) joen235 (18354) 14 joen235 (18354) Theorem msc 54D45