Example about Hausdorff distance

Question

As we know Hausdorff distance for two compact sets is defined like:

$$d_H (A,B)=\max\{\sup_{a\in A}(a,B), \sup_{b\in B}(b,A)\}$$

And compact set sequence $C_i$ converge to $C$ iff $ \lim_{i\to\infty}d_H(C_i,C)=0$

Can anyone find $C_i \ C\subset R^2$ are compact and connected such that:

$$\lim\sup_{x\in C_i}(x,C)=0 \ \text{but} \lim\sup_{x\in C}(x,C_i)\neq 0$$

Any suggestions are welcome! Thanks!

Answer 1

Sure - take $C_i=\{0\}$ and $C=\{0, 1\}$. Then $d(x, C)=0$ for all $x\in C_i$ for all $i$, but $\sup_{x\in C} d(x, C_i)=1$ for all $i$.

Answer 2

Take $C=[-1,0]$, and $C_i = [0,1/i]$, then $$ \lim\sup_{x\in C_i}(x,C)=0 \ \text{but} \lim\sup_{x\in C}(x,C_i) = 1. $$