Clearifcation on a Metric Space appoarch to proving the Cantor Tentary Set is nowhere dense?

Question

$Propostion$: The Cantor Tenetary Set donated in 1.) is nowhere near dense.

$1.)$ $$ C = \Big\{ X \in [0,1]: X = \sum_{n=1}^{\infty}\frac{I_n}{j_n}\Big\} \, for \, I_n=0 \, \, or \, \, 2$$

$Lemma:1.1$

Let $X$ be a metric space. A Subset C within X is considered nowhere dense in X if the closure has an empty interior

Applying and considering $Lemma:1.1$ to our initial proposition in $1.)$ I yielded the following conclusions

Since C withen X is a subset, the intial interior of the set can be taken as follows in 2.)

$2.)$ $$int(C) = \Big\{ c \in C \, \,| \, \, some \, \, B_{ra}(a) A, \, r_a > 0 \Big\}$$

Following from $2.)$ the closure of $C$ can be shown as follows.

$$ 2.) \, \, \, C = \Big\{x \in X | x = \lim_{x\to\infty} C_c \,\, with \, \, C_c \in C \, \, for \, \, \, all \, \, \, c \, \, \Big\}$$

Initially where my question arises is following from $2.)$ how would one appaorch showing the Closure of $C$ ?

Answer 1

You can write the Cantor set $C$ as a countable intersection of closed sets:

$C = \bigcap_{i = 0}^{\infty} C_i$

where $C_0 = [0;1]$, $C_1 = [0; \frac{1}{3}] \cup [\frac{2}{3}; 1]$ and every successive set is formed by removing the "open middle third" of each interval in the set.

Hence $C$ is closed and therefore $(\bar{C})^{\circ} = C^{\circ}$

Now let $x \in C$: For all $\epsilon > 0$, $B(x, \epsilon) \cap C^C \neq \emptyset$ (argue by ternary expansion) hence no open set is contained in the interior of $C$ and so $ (\bar{C})^{\circ} = \emptyset$ . Therefore $C$ is nowhere dense.

Note that for subsets $A \subseteq X$ of a topological space in general $ (\bar{A})^{\circ} \neq \overline{(A^{\circ})}$ i.e. the closure of the interior is not the interior of the closure. For example let $X = \mathbb{R} $ and $ (\bar{\mathbb{Q}})^{\circ} = \mathbb{R}^\circ = \mathbb{R}$ but $ \overline{(\mathbb{Q}^{\circ})} = \bar{\emptyset} = \emptyset$

Hence I would guess your approach does not work (you should clarify your approach. Your notation is pretty messed up. E.g. you write $x \in X$ and then $\lim_{x \to \infty} $. What does that mean?)