A strictly decreasing nested sequence of non-empty compact subsets of S has a non-empty intersection with empty interior.

Question

S is an Hausdorff topological space. A decreasing nested sequence of non-empty compact subsets of S has a non-empty intersection. In other words, supposing $C_{k}$ is a sequence of non-empty, compact subsets of a topological we know that $\cap_{k\in N}C_{k}\not= \emptyset$

If I assume that the sequence is striclty decreasing $C_{k+1} \subset C_{k}$ can I say the following

$Int(\cap_{k\in N}C_{k})= \emptyset$?

remark: In my opinion $C_{k+1} \subset C_{k}$ does not imply that diameters of $C_{k}$ are strictly decreasing.

Any simple proof? Thanks.

Answer 1

Consider the following intersection \begin{equation} \overset{\infty}{\underset{n=1}{\bigcap}} \left[-1-\frac{1}{n}, 1+\frac{1}{n} \right]=[-1,1]. \end{equation}

Notice that given $k \in \mathbb{N}$ we have $\left[-1-\frac{1}{k}, 1+\frac{1}{k} \right] \supset \left[-1-\frac{1}{k+1}, 1+\frac{1}{k+1} \right]$, but $\text{Int}([-1,1])=(-1,1).$