Exercise 2.17

Let $A$ be the set of sequences on the digits $0$ and $1$. We get an injection $A\to E$ by associating with a sequence $a_n$ the number whose $n$th decimal is $4$ if $a_n=1$ and $7$ otherwise. By thm. 2.14, this makes $E$ uncountable.

$E\subset [0.4,0.8]$, and is therefore not dense in $[0,1]$.

Since $E$ is bounded, compactness is equivalent to closedness. Assume $x\notin E$, and let $x_n$ be the $n$th digit in its decimal expansion. For some $N$, $x_N$ is not equal to $4$ or $7$. Choose $\mathrm{\Delta }$ such that the $N$th decimal digit of all $x$ in a $\mathrm{\Delta }$-neighborhood of $x$ is $x_n$. (Note that this can be done.) This is a neighborhood of $x$ that does not intersect $E$. Therefore $E^c$ is open and $E$ is closed, hence compact.

Let $x\in E$, choose $𝜖>0$, and choose $n$ such that $10^{-n}<𝜖$. The number obtained by changing the $n$th digit of $x$ from $4$ to $7$ or vice versa, is contained in an $𝜖$-neighborhood of $x$. This means that $x$ is a limit point, so $E$ is perfect.