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Josia Pietsch 2024-02-06 15:48:39 +01:00
parent de2d240c5a
commit 8c3be56270
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2 changed files with 7 additions and 7 deletions
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2
inputs/lecture_14.tex
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@ -153,7 +153,7 @@ Let $\rho(\prec) \coloneqq \sup \{\rho_{\prec}(x) + 1 : x \in X\}$.
\begin{fact}[{\cite[Appendix B]{kechris}}]
Since $\rho_\prec\colon X \to \rho(\prec)$ is surjective,
we have that $\rho(\prec) \le |X|^+$.%
we have that $\rho(\prec) < |X|^+$.%
\gist{\footnote{Here, $|X|^+$ denotes the successor cardinal.}}{}
\end{fact}
\begin{theorem}[{Kunen-Martin, \cite[(31.1)]{kechris}}]

12
inputs/tutorial_08.tex
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@ -265,20 +265,20 @@ $f$ and $g$ are bijections between
$X_\omega \coloneqq \bigcap X_i$ and $Y_\omega \coloneqq \bigcap Y_i$.
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\adjustbox{scale=0.7,center}{%
\begin{tikzcd}
{X \setminus X_\omega =} & {(X_0 \setminus X_1)} & \cup & {(X_0 \setminus X_1)} & \cup & {(X_0 \setminus X_1)} & \cdots & {} \\
{Y\setminus Y_\omega =} & {(Y_0 \setminus Y_1)} & \cup & {(Y_0 \setminus Y_1)} & \cup & {(Y_0 \setminus Y_1)} & \cdots & {}
\[\begin{tikzcd}
{X \setminus X_\omega =} & {(X_0 \setminus X_1)} & \cup & {(X_1 \setminus X_2)} & \cup & {(X_2 \setminus X_3)} & \cdots & {} \\
{Y\setminus Y_\omega =} & {(Y_0 \setminus Y_1)} & \cup & {(Y_1 \setminus Y_2)} & \cup & {(Y_2 \setminus Y_3)} & \cdots & {}
\arrow["f"'{pos=0.7}, from=1-2, to=2-4]
\arrow["g"{pos=0.1}, from=2-2, to=1-4]
\arrow["f"{pos=0.8}, from=1-6, to=2-8]
\arrow["g"{pos=0.1}, from=2-6, to=1-8]
\end{tikzcd}
\end{tikzcd}\]
}
By \autoref{thm:lusinsouslin}
the injective image via a Borel set of a Borel set is Borel.
the injective image via a Borel function of a Borel set is Borel.
\autoref{thm:lusinsouslin} also gives that the inverse
of a bijective Borel map is Borel.