Context:
I am currently doing some exercises on the middle $\lambda$ Cantor set $C_\lambda$ (construction is similar to the usual Cantor set, but we remove the middle $\lambda$ proportion of the intervals). One involves showing that the Hausdorff dimension function mapping the set of compact subsets of $\mathbb{R}$ to $[0,1]$, namely, $\dim_H(.):\mathcal{K}\to[0,1]$ is not continuous. I was thinking of construction a sequence $k_n:=C_{\frac{1}{n}}$. I still need to show this converges to $C_0$ which I'm going to define as the Cantor set constructed by removing the middle point.
Question:
Anyways, I believe that sketch of a proof should work, but I'm curious as to the properties of $C_0$. Would its dimension be $1$ Since the Hausdorff dimension is invariant of countable points? What is its measure? What type of numbers are left in the limiting set of this particular construction? Is there anything 'interesting' about this set compared to the usual $C_\lambda$?
EDIT (construction):
I assume the $C_\lambda$ construction isn't the issue, so I will omit that.
$$(C_{0})_0=[0,1]$$ $$(C_{0})_1=[0,1/2)\cup{}(1/2,1]$$ $$(C_{0})_2=[0,1/4)\cup{}(1/4,1/2)\cup{}(1/2,3/4)\cup{}(3/4,1]$$ $$C_0:=\bigcap_{n=1}^{\infty}{(C_{0})_n}$$
I am asking about the properties of $C_0$ as defined above. However, as a side question, Is this the limiting set of $k_n$, or would that just be [0,1]?