From f4d64527c46eb94ce486cde7d4ced48991214c45 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Mon, 5 Feb 2024 15:07:13 +0100 Subject: [PATCH] fixed some typos; currying --- inputs/lecture_01.tex | 8 ++++---- inputs/lecture_08.tex | 13 ++++++------- inputs/lecture_10.tex | 7 ++++--- inputs/lecture_13.tex | 2 +- inputs/lecture_15.tex | 2 +- 5 files changed, 16 insertions(+), 16 deletions(-) diff --git a/inputs/lecture_01.tex b/inputs/lecture_01.tex index df7eb8c..acab3f3 100644 --- a/inputs/lecture_01.tex +++ b/inputs/lecture_01.tex @@ -185,11 +185,11 @@ suffices to show that open balls in one metric are unions of open balls in the o \begin{definition}[Our favourite Polish spaces] \leavevmode \begin{itemize} - \item $2^{\omega}$ is called the \vocab{Cantor set}. + \item $2^{\N}$ is called the \vocab{Cantor set}. (Consider $2$ with the discrete topology) - \item $\omega^{\omega}$ is called the \vocab{Baire space}. - ($\omega$ with descrete topology) - \item $[0,1]^{\omega}$ is called the \vocab{Hilbert cube}. + \item $\cN \coloneqq \N^{\N}$ is called the \vocab{Baire space}. + ($\N$ with descrete topology) + \item $\mathbb{H} \coloneqq [0,1]^{\N}$ is called the \vocab{Hilbert cube}. ($[0,1] \subseteq \R$ with the usual topology) \end{itemize} \end{definition} diff --git a/inputs/lecture_08.tex b/inputs/lecture_08.tex index 97d494d..c58fca2 100644 --- a/inputs/lecture_08.tex +++ b/inputs/lecture_08.tex @@ -78,18 +78,17 @@ where $X$ is a metrizable, usually second countable space. we also have $A = \bigcup_n A_n'$ with $A'_n \in \Pi^0_{\xi_n}(X)$. - We construct a $(2^\omega)^\omega \cong 2^\omega$-universal set + We construct a $(2^{\omega \times \omega}) \cong 2^\omega$-universal set for $\Sigma^0_\xi(X)$. - For $(y_n) \in (2^\omega)^\omega$ + For $(y_{m,n}) \in (2^{\omega \times \omega})$ and $x \in X$ - we set $((y_n), x) \in \cU$ - iff $\exists n.~(y_n, x) \in U_{\xi_n}$, - i.e.~iff $\exists n.~x \in (U_{\xi_n})_{y_n}$. + we set $((y_{m,n}), x) \in \cU$ + iff $\exists n.~((y_{m,n})_{m < \omega}, x) \in U_{\xi_n}$, + i.e.~iff $\exists n.~x \in (U_{\xi_n})_{(y_{m,n})_{m < \omega}}$. Let $A \in \Sigma^0_\xi(X)$. Then $A = \bigcup_{n} B_n$ for some $B_n \in \Pi^0_{\xi_n}(X)$. - % TODO - Furthermore $\cU \in \Sigma^0_{\xi}((2^\omega)^\omega \times X)$. + Furthermore $\cU \in \Sigma^0_{\xi}((2^{\omega \times \omega} \times X)$. \end{proof} \begin{remark} Since $2^{\omega}$ embeds diff --git a/inputs/lecture_10.tex b/inputs/lecture_10.tex index a8fb5ea..514ee1f 100644 --- a/inputs/lecture_10.tex +++ b/inputs/lecture_10.tex @@ -34,7 +34,8 @@ we need the following definition: \begin{lemma} \label{lem:lusinsephelp} If $P = \bigcup_{m < \omega} P_m$, $Q = \bigcup_{n < \omega} Q_n$ are such that - for any $m, n$ the sets $P_m$ and $Q_n$ are Borel separable. + for any $m, n$ the sets $P_m$ and $Q_n$ are Borel separable, + then $P$ and $Q$ are Borel separable. \end{lemma} \begin{proof} For all $m, n$ pick $R_{m,n}$ Borel, @@ -42,7 +43,7 @@ we need the following definition: and $Q_n \cap R_{m,n} = \emptyset$. Then $R = \bigcup_m \bigcap_n R_{m,n}$ has the desired property - that $R \subseteq R$ and $R \cap Q = \emptyset$. + that $P \subseteq R$ and $R \cap Q = \emptyset$. \end{proof} \begin{notation} @@ -60,7 +61,7 @@ we need the following definition: Write $A_s \coloneqq f(\cN_s)$ and $B_s \coloneqq g(\cN_s)$. Note that $A_s = \bigcup_m A_{s\concat m}$ - and $B_ns = \bigcup_{n < \omega} B_{s\concat n}$. + and $B_s = \bigcup_{n < \omega} B_{s\concat n}$. In particular $A = \bigcup_{m < \omega} A_{\underbrace{\langle m \rangle}_{\in \omega^1}}$ diff --git a/inputs/lecture_13.tex b/inputs/lecture_13.tex index a495a93..019002f 100644 --- a/inputs/lecture_13.tex +++ b/inputs/lecture_13.tex @@ -183,7 +183,7 @@ i.e.}{} Let $X$ be Polish and $C \subseteq X$ coanalytic. Then $\phi\colon C \to \Ord$ is a \vocab[Rank!$\Pi^1_1$-rank]{$\Pi^1_1$-rank} - provided that $\le^\ast$ and $<^\ast$ are coanalytic subsets of $X \times X$, + provided that $\le^\ast_\phi$ and $<^\ast_\phi$ are coanalytic subsets of $X \times X$, where $x \le^\ast_{\phi} y$ iff diff --git a/inputs/lecture_15.tex b/inputs/lecture_15.tex index decf836..fa9c486 100644 --- a/inputs/lecture_15.tex +++ b/inputs/lecture_15.tex @@ -55,7 +55,7 @@ then $\xi = \alpha$ and \[ - E_\xi = E_\alpha = \bigcup_{\eta < \alpha} + E_\xi = E_\alpha = \bigcup_{\eta < \alpha} E_\eta \] is a countable union of Borel sets by the previous case. \end{itemize}