fixed some typos; currying

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}.
+        \item $\cN \coloneqq \N^{\N}$ is called the \vocab{Baire space}.
-            ($\omega$ with descrete topology)
+            ($\N$ with descrete topology)
-        \item $[0,1]^{\omega}$ is called the \vocab{Hilbert cube}.
+        \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}

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 $((y_n), x) \in \cU$
+    we set $((y_{m,n}), x) \in \cU$
-    iff $\exists n.~(y_n, x) \in U_{\xi_n}$,
+    iff $\exists n.~((y_{m,n})_{m < \omega}, x) \in U_{\xi_n}$,
-    i.e.~iff $\exists n.~x \in (U_{\xi_n})_{y_n}$.
+    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 $\cU \in \Sigma^0_{\xi}((2^{\omega \times \omega} \times X)$.
-    Furthermore $\cU \in \Sigma^0_{\xi}((2^\omega)^\omega \times X)$.
 \end{proof}
 \begin{remark}
     Since $2^{\omega}$ embeds

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
     $A = \bigcup_{m < \omega} A_{\underbrace{\langle m \rangle}_{\in \omega^1}}$

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

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}