Cantor Space is Nowhere Dense

Theorem

Let $T = \struct {\CC, \tau_d}$ be the Cantor space.


Then $T$ is nowhere dense in $\closedint 0 1$.


Proof 1

From Cantor Set is Closed in Real Number Space, $\CC$ is closed.

So from Closed Set equals its Closure:

$\CC^- = \CC$

where $\CC^-$ denotes the closure of $\CC$.


Let $0 \le a < b \le 1$.

Then $I = \openint a b$ is an open interval of $\closedint 0 1$.

Let $\epsilon = b - a$.

Clearly $\epsilon > 0$.

Let $n \in \N$ such that $3^{-n} < \epsilon$.


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So there exists an open interval of $\closedint 0 1$ which has been deleted from $\closedint 0 1$ during the process of creating $\CC$.

Thus no open interval of $\closedint 0 1$ is disjoint from all the open intervals deleted from $\closedint 0 1$.

So an open interval of $\closedint 0 1$ can not be a subset of $\CC = \CC^-$.

Hence the result, by definition of nowhere dense.

$\blacksquare$


Proof 2

Let $S_n$ and $C_n$ be as in the definition of the Cantor set as a limit of a decreasing sequence.

Then the length of every interval in $S_n$ is seen to be $\dfrac 1 {3^n} = 3^{-n}$.


Let $0 \le a < b \le 1$.

Then $\openint a b \subseteq \closedint 0 1$ is an open interval.

Let $n \in \N$ such that $3^{-n} < b - a$, so that the length of every interval in $S_n$ is $3^{-n} < b - a$.


Therefore, as the intervals in $S_n$ do not overlap, no interval of length $b - a$ is contained in $C_n = \ds \bigcup S_n$.

Consequently, as $\CC \subseteq C_n$, no interval of length $b - a$ is contained in $\CC$.

Since the interval $\openint a b$ was of arbitrary length, there do not exist any open intervals in $\CC$.

Hence the result, by definition of nowhere dense.

$\blacksquare$


Sources

  • 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 7$: Problem $10 \ \text{(v)}$
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