5
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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 $\text{Int}([-1,1])=(-1,1).$

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.