Is every sigma-compact space is lindelof space?

Question

Is every sigma-compact topological space is Lindelof space ? I know about $\sigma$-compact and Lindelof space.Say about converse is true or not.

Answer 1

Suppose that $X= \bigcup_{n \in \Bbb N} K_n$ where each $K_n$ is compact.

Let $\mathcal{U}$ be any open cover of $X$.

We have for each $n$ that $K_n \subseteq \bigcup \mathcal{U}$ so there is a finite subcover $\mathcal{U}_n \subseteq \mathcal{U}$ such that $K_n \subseteq \bigcup \mathcal{U}_n$ as $K_n$ is compact.

Then $\bigcup_{n \in \Bbb N} \mathcal{U}_n$ is a countable (a countable union of finite sets is countable) subcover of $\mathcal{U}$.

As $\mathcal{U}$ was arbitrary, $X$ is Lindelöf.