# when the intersection of the closure of a decreasing sequence of sets has empty interior?

Hi everyone: Suppose that $(A_{n})_{n}$ is a decreasing sequence of subsets of a ball in $\mathbb{R}^{N}$ $(N\geq2)$. If the intersection of all the $A_{n}$'s is empty, can the intersection of their closure contains an open set? Under which extra condition(s) can we conclude that $\bigcap_{n}\overline{A_{n}}$ has empty interior? Thanks for your help.

Hints. For the first question: Think of a countable dense subset $A_0$ of the given ball. Now write $A_0 = \{a_k \mid k \in \mathbf N\}$, and remove element by element to get $A_n$.
For the second one: A possible condition is that $\operatorname{diam} A_n \to 0$.
• Thanks. How about concluding that $\bigcap_{n\geq0}\partial A_{n}$ (boundaries) has empty interior? – M. Rahmat Nov 18 '15 at 23:03