Cantor Space is Nowhere Dense/Proof 1

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Theorem

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


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


Proof

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.

$\blacksquare$


Sources

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