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@ -153,7 +153,7 @@ Let $\rho(\prec) \coloneqq \sup \{\rho_{\prec}(x) + 1 : x \in X\}$.
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\begin{fact}[{\cite[Appendix B]{kechris}}]
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\begin{fact}[{\cite[Appendix B]{kechris}}]
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Since $\rho_\prec\colon X \to \rho(\prec)$ is surjective,
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Since $\rho_\prec\colon X \to \rho(\prec)$ is surjective,
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we have that $\rho(\prec) \le |X|^+$.%
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we have that $\rho(\prec) < |X|^+$.%
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\gist{\footnote{Here, $|X|^+$ denotes the successor cardinal.}}{}
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\gist{\footnote{Here, $|X|^+$ denotes the successor cardinal.}}{}
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\end{fact}
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\end{fact}
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\begin{theorem}[{Kunen-Martin, \cite[(31.1)]{kechris}}]
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\begin{theorem}[{Kunen-Martin, \cite[(31.1)]{kechris}}]
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@ -265,20 +265,20 @@ $f$ and $g$ are bijections between
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$X_\omega \coloneqq \bigcap X_i$ and $Y_\omega \coloneqq \bigcap Y_i$.
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$X_\omega \coloneqq \bigcap X_i$ and $Y_\omega \coloneqq \bigcap Y_i$.
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% https://q.uiver.app/#q=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\adjustbox{scale=0.7,center}{%
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\adjustbox{scale=0.7,center}{%
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\begin{tikzcd}
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\[\begin{tikzcd}
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{X \setminus X_\omega =} & {(X_0 \setminus X_1)} & \cup & {(X_0 \setminus X_1)} & \cup & {(X_0 \setminus X_1)} & \cdots & {} \\
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{X \setminus X_\omega =} & {(X_0 \setminus X_1)} & \cup & {(X_1 \setminus X_2)} & \cup & {(X_2 \setminus X_3)} & \cdots & {} \\
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{Y\setminus Y_\omega =} & {(Y_0 \setminus Y_1)} & \cup & {(Y_0 \setminus Y_1)} & \cup & {(Y_0 \setminus Y_1)} & \cdots & {}
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{Y\setminus Y_\omega =} & {(Y_0 \setminus Y_1)} & \cup & {(Y_1 \setminus Y_2)} & \cup & {(Y_2 \setminus Y_3)} & \cdots & {}
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\arrow["f"'{pos=0.7}, from=1-2, to=2-4]
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\arrow["f"'{pos=0.7}, from=1-2, to=2-4]
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\arrow["g"{pos=0.1}, from=2-2, to=1-4]
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\arrow["g"{pos=0.1}, from=2-2, to=1-4]
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\arrow["f"{pos=0.8}, from=1-6, to=2-8]
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\arrow["f"{pos=0.8}, from=1-6, to=2-8]
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\arrow["g"{pos=0.1}, from=2-6, to=1-8]
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\arrow["g"{pos=0.1}, from=2-6, to=1-8]
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\end{tikzcd}
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\end{tikzcd}\]
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}
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}
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By \autoref{thm:lusinsouslin}
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By \autoref{thm:lusinsouslin}
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the injective image via a Borel set of a Borel set is Borel.
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the injective image via a Borel function of a Borel set is Borel.
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\autoref{thm:lusinsouslin} also gives that the inverse
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\autoref{thm:lusinsouslin} also gives that the inverse
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of a bijective Borel map is Borel.
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of a bijective Borel map is Borel.
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