Josia Pietsch
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35 lines
701 B
TeX
35 lines
701 B
TeX
\tutorial{05}{}{}
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% Sheet 5 - 18.5 / 20
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\subsection{Exercise 1}
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Let $B \subseteq C$ be comeager.
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Then $B = B_1 \cup B_2$,
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where $B_1$ is dense $G_\delta$
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and $B_2$ is meager.
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\begin{fact}
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$X$ is Baire iff every non-empty open set is non-meager.
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In particular, let $X$ be Baire,
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then $U \overset{\text{open}}{\subseteq} X$
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is Baire.
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\end{fact}
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\subsection{Exercise 4}
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\begin{enumerate}[(i)]
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\item $|B| = \fc$, since $B$ contains a comeager
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$G_\delta$ set, $B'$:
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$B'$ is Polish,
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hence $B' = P \cup C$
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for $P$ perfect and $C$ countable,
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and $|P| \in \{\fc, 0\}$.
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But $B'$ can't contain isolated point$.
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\end{enumerate}
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