Let $M$ be a topological manifold. Is that true that there always exist compact sets $K_n$ with the property $\bigcup_n K_n = M$ and $K_n \subset K ^{\circ} _{n+1}$.
The problem is that this is true is $M$ is a CW complex or has countable base. But does it hold also generally?
The usual definition of manifold is such that every manifold is second countable. Then, has you wrote, there is such a sequence of compact sets. On the other hand, if you don't assume that, such a sequence doesn't have to exist. Imagine, say, an infinite uncountable union of copies of $\mathbb R$.