When a topological Hausdorff space X is locally compact and second-countable (has countable weight), can we find a chain of compact sets $\{K_i: n \in \mathbb{N}\}$, where
- $K_0 = \emptyset$ and
- $K_n \subseteq \operatorname{int} K_{n+1}$
- $\bigcup_n K_n = X$ ?
My intuition is that yes, but I am interested in a formal proof. Also, I am not sure, if we really need the secon-countability. I would say that compact neighborhood of any point is enough to create such infinite sequence of sets.
Also, if the space isn´t locally compact, could we still find such chain?
Thank you.