Let $K(X)$ the space of all non-empty compact subsets of $X$ equipped with the topology from the Hausdorff metric.
if $X$ is metrizable and $K_n\in K(X)$, $K_1 \supseteq K_2 \supseteq \ldots$. Then $\lim_n K_n=\bigcap K_n$. In particular if $K_n$ is the union of the $2^{n}$ many closed intervals ocuurring in the n th step of the construccction of the Cantor ser $E_{\frac{1}{3}}$, $K_n \to E_{\frac{1}{3}}$.?
a suggestion for this exercise ? please.
The definition of Hausdorff metric I will use, starting from a metric space $(X,d)$, is $d_H(K_1, K_2) = \max(\sup_{x \in K_1} d(x, K_2), \sup_{x \in K_2} d(x, K_1))$.
So suppose $K = \cap_n K_n$, where $K_{n+1} \subseteq K_n$ for all $n$, and all $K_n$ are non-empty compact (so $K$ is too). For $x \in K$, $x \in K_n$ as well, so $d(x, K_n) = 0$, so $\sup_{x \in K} d(x, K_n) = 0$ for all $n$. This takes care of one term in the formula for $d_H(K, K_n)$. We see that in this case $d_H(K_n, K) = \sup_{x \in K_n} d(x,K)$.
Also, as $K_{n+1} \subseteq K_n$ for all $n$, we have that $d_H(K_{n+1},K) = \sup_{x \in K_{n+1}} d(x,K) \le \sup_{x \in K_n} d(x,K) = d_H(K_n, K)$, as the sup of a larger set can only be larger, so the sequence of distances $d_H(K_n, K)$ is mononotonically decreasing, and positive, so assume $\lambda > 0$ would be its infimum; if $\lambda = 0$ we would be done, as in any metric space, $K_n \rightarrow K$ iff $d_H(K_n, K) \rightarrow 0$.
So pick $x_n \in K_n$ such that $d(x_n, K) \ge \frac{\lambda}{2} > 0$. Then as all $x_n$ are in the compact $K_0$, some subsequence $(x_{n_k})$ of $(x_n)$ converges to some $x \in X$, and by the decreasingness of the $K_n$ one easily sees that $x \in K_n$ (as all terms of a tail of the subsequence are in $K_n$, which is closed), for all $n$, so $x \in K$ by definition. But as $y \rightarrow d(y,K)$ is continuous, this contradicts the fact that all $d(x_{n_k}, K) \ge {\lambda \over 2}$, while their limit has distance $0$ to $K$.
So indeed $d_H(K_n, K) \rightarrow 0$ and we are done with the statement.
So we have that if we have decreasing family $K_n$ of non-empty compact in the hyperspace (in the Vietoris topology or Hausdorff metric), the intersection is their topological limit.
The remark about the Cantor set is then an immediate consequence of this and the Cantor set's definition.