I was wondering if someone can please help me with the following problems:
- Show that $\mathbb{Q}$ is not locally compact.
- Prove that if $X$ is Lindelöf and $Y$ is compact then $X \times Y$ is Lindelöf.
I think I got 1, here's my work:
Suppose $\mathbb{Q}$ is locally compact and let $x \in \mathbb{Q}$. Then by definition of local compactness (for Hausdorff spaces) we can find an open set $U \subset \mathbb{Q}$ such that $\overline{U}$ is compact. Since the open intervals form a basis for the usual topology then we can find an open interval $(a,b)$ such that $x \in (a,b) \cap \mathbb{Q} \subset U$. Then observe $[a,b] \cap \mathbb{Q} \subseteq \overline{U}$. But $[a,b] \cap \mathbb{Q}$ is closed so we have a closed subset of the compact set $\overline{U}$ , thus $[a,b] \cap \mathbb{Q}$ is compact. Hence it suffices to show this set is not compact. Now pick an irrational number $z \in (a,b)$ then we can find a sequence $\{x_{n}\}$ of consisting of rational numbers in $(a,b)$ such that $x_{n} \rightarrow z$. But $[a,b] \cap \mathbb{Q}$ is compact so sequentially compact. Therefore the sequence $\{x_{n}\}$ must have a subsequence converging to a point in $(a,b) \cap \mathbb{Q}$, which is impossible because every subsequence converges to $p$ and $p$ is irrational. Is this OK?
2) Stuck in this one for a while. How to prove this?