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\subsubsection*{* Non-examinable material}

The similarity between \nameref{baire_category} on
zero measure is immediately apparent. However they
give very different kinds of genericity.

** This is the end of the non-examinable comments.

Now let us prove the existence of subsets of $[0,
1]$ which are closed, have empty interior and have
no isolated points. (Using standard Euclidean
metric).

We prove that for Hausdorff metric
\[ d(E, F) = \sup_{e \in E} \inf_{f \in F} |e - f|
+ \sup_{f \in F} \inf_{e \in E} |e - f| ,\]
except for a collection of category 1, all sets
have this property.

Sufficient to prove:
\begin{enumerate}[(1)]
  \item The collection of non-empty compact sets
    with no isolated points is category $1$.
  \item The collection of non-empty compact sets
    with empty interior is of category $1$.
\end{enumerate}
To show collection with no isolated points is of
first category, consider
\[ \mathcal{E}_n = \{\text{$E$ compact} : \exists
x \in E, E \cap B(x, 1 / n) = \{x\}\} .\]
Then:
\begin{enumerate}[(1)]
  \item $\mathcal{E}_n$ is closed in Hausdorff
    metric: Suppose $E_j$ such that there exists
    $x_j \in E$ with $B(x, 1/n) = \{x_j\}$. Then
    there exists $j(m) \to \infty$ such that
    $x_{j(m)} \to x^*$. By extracting to a
    subsequence, may suppose $x_j \to x^*$. If
    $\eta > 0$ and $n$ large enough, $|x_j - x^*|
    < \eta$ for all $j \le n$, and if $m \ge n$
    then $|x^* - y| \ge \frac{1}{n} - 2\eta$ for
    all $y \in E_j$, $y \neq x_j$. Thus $|x^* - y|
    \ge \frac{1}{n} - 2\eta$ for all $y \in E$, $y
    \neq x_n$. But $\eta$ arbitrary, so $|x^* - y|
    \ge \frac{1}{n} - 2 \eta$ for all $\eta > 0$,
    $y \neq x$. So $E \in \mathcal{E}$.

    On the other hand if $E \neq \emptyset$ is
    closed then writing
    \[ F_m = \left\{ \frac{r}{m} : d \left(
    \frac{r}{m}, E \right) \le \frac{1}{m} \right\} \]
    we know $d(F_n, E) \le \frac{2}{n}$ and
    provided $n \ge 4\delta^{-1} + 1$, then there
    does not exist $y \in F_n$ such that $B(y,
    \delta) \cap E = \{y\}$.
\end{enumerate}

So (1) is true.

To prove (2), let
\[ \mathcal{E}_{r, n} = \left\{\text{$E$ closed such
that $E \supseteq \left[ \frac{r}{n}, \frac{r +
1}{n} \right]$}\right\} .\]
Then $\mathcal{E}_{r, n}$ is closed: $E_m
\supseteq \left[ \frac{r}{n}, \frac{r + 1}{n}
\right]$, $E_m \dto E$, $E \supseteq \left[
\frac{r}{n}, \frac{r + 1}{n} \right]$ but as
before, if $E$ closed and non-empty, and $\delta >
0$, then there exists $F$ finite such that $d(E,
F) < \delta$ and $F \subseteq \mathcal{E}_{r, n}$. 

If $E \notin \bigcup_{n = 1}^\infty \bigcup_{r = 0}^{n -
1} \mathcal{E}_{r, n}$ then $E$ contains non
non-trivial interval.

Finally we give a positive use of
\nameref{baire_category}.
