Let $A_0 \subseteq A_1 \subseteq...$ be a nested sequence of closed subsets of $[0,1]$ such that $\bigcup\limits_{n = 0}^\infty A_n = [0,1]$. What interesting properties can be said about the sets in this sequence?
For example, each set is measurable, so there exists $A_n$ with measure arbitrarily close to 1. However, not every point in $[0,1]$ is necessarily a limit point of some $A_n$. Consider for example $A_n = [0, \frac 12 - \frac 1n] \cup \{\frac 12\} \cup [\frac 12 + \frac 1n, 1]$.
Can anything interesting be said about points in $[0,1]$ that are not limit points of any $A_n$? (e.g. countable, nowhere dense, etc.)