automatic commit jrpie-PC

inputs/lecture_08.tex

@@ -58,7 +58,7 @@
 \end{proof}
 
 \subsection{Parametrizations}
-\todo{choose better title}
+%\todo{choose better title}
 
 
 Let $\Gamma$ denote a collection of sets in some space.
@@ -109,9 +109,9 @@ where $X$ is a metrizable, usually second countable space.
     put $(y,x) \in \cU$ iff
     $x \in \bigcup \{V_n : y_n = 1\}$.
     $\cU$ is open.
-    Let $V = \bigcup \{V_n : V_n \subseteq V\}$.
-    Pick $y \in 2^\omega$
-    and let $y_n = 1$ iff $V_n \subseteq  V$.
+    For any $V \overset{\text{open}}{\subseteq} X$,
+    define $y \in 2^\omega$
+    by $y_n = 1$ iff $V_n \subseteq  V$.
     Then $\cU_y = V$.
 
 
@@ -125,7 +125,7 @@ where $X$ is a metrizable, usually second countable space.
     Recall that $\eta_1 \le  \eta_2 \implies \Pi^0_{\eta_1}(X) \subseteq \Pi^0_{\eta_2}(X)$.
 
     Note that if $A = \bigcup_n A_n$,  with $A_n \in \Pi^0_{\eta_n}(X)$
-    some $\eta_n < \xi$,
+    for some $\eta_n < \xi$,
     we also have
     $A = \bigcup_n A_n'$ with  $A'_n \in \Pi^0_{\xi_n}(X)$.
 
@@ -146,7 +146,7 @@ where $X$ is a metrizable, usually second countable space.
     Since $2^{\omega}$ embeds
     into any uncountable polish space $Y$
     such that the image is closed,
-    we can  $2^{\omega}$ by $Y$
+    we can replace $2^{\omega}$ by $Y$
     in the statement of the theorem.%
     \footnote{By definition of the subspace topology
     and transfinite induction, $\Sigma^0_\xi(Y)\defon{2^\omega} = \Sigma^0_\xi(2^\omega)$.}