$\Omega$ is open in a topological space $X$ and $\{K_n\}$ are countable open sets such that $\bigcup K_n = \Omega$ and $K_n \subset \text{int} K_{n+1}$.
Suppose there is another compact set $K \subset \Omega$. Then, is there a $K_N \in \{K_n\}$ such that $K \subset K_N$?
If it doesn't hold generally, then does it hold when $X$ is a Euclidean space $\mathbb{R}^n$? Any help would be appreciated!
Best Answer
Always true. The interiors if the sets $K_n$ form an open cover if $K$ and there is a finite subcover. But any finite Union is contained on one of the sets $K_n$.