Let $\{A_n\}$ be a sequence of perfect sets in a metric space. Then, is $\bigcap_{n\in\omega} A_n$ perfect?
Recently, i have studied some Cantor-like sets and i got this very natural question.
Let $C_1$ be a closed connected set in a metric space $X$ and $\alpha\in \mathbb{N}^\mathbb{N}$
Define $A(n)$ be a union of disjoint $n$ closed connected subsets of $A$. (With Axiom of choice, it can be well-defined)
Define recursively, $C_{n+1}=C_n (\alpha_n)$.
I believe $\bigcap_{n\in\mathbb{N}}C_n$ is perfect.
Is it true or is there any generalization similar to this? Sorry in advance that i'm not really good at describing things..
- For any closed connected metric space $A$, there exists a perfect subset $B$ such that $Int(B)=\emptyset$? Or possibly, "Every perfect set in a metric space contains a perfect subset with empty interior"?