w23-logic-3/inputs/tutorial_05.tex

\tutorial{05}{}{}

% Sheet 5 - 18.5 / 20

\subsection{Exercise 1}

Let $B \subseteq C$ be comeager.
Then $B = B_1 \cup B_2$,
where $B_1$ is dense $G_\delta$
and $B_2$ is meager.


\begin{fact}
     $X$ is Baire iff every non-empty open set is non-meager.

     In particular, let $X$ be Baire,
     then $U \overset{\text{open}}{\subseteq} X$
     is Baire.
\end{fact}

\subsection{Exercise 4}

\begin{enumerate}[(i)]
    \item $|B| = \fc$, since $B$ contains a comeager
        $G_\delta$ set, $B'$:
        $B'$ is Polish,
        hence $B' =  P \cup C$
        for $P$ perfect and $C$ countable,
        and $|P| \in \{\fc, 0\}$.
        But $B'$ can't contain isolated point$.
\end{enumerate}