diff --git a/inputs/lecture_12.tex b/inputs/lecture_12.tex index 6f88217..2eaf19e 100644 --- a/inputs/lecture_12.tex +++ b/inputs/lecture_12.tex @@ -20,7 +20,7 @@ T \in \IF &\iff& \exists \beta \in \cN .~\forall n \in \N.~T(\beta\defon{n}) = 1. \end{IEEEeqnarray*} - Consider $\{(T, \beta) \in \Tr \times \cN : \forall n.~ T(\beta\defon{n}) = 1\}$. + Consider $D \coloneqq \{(T, \beta) \in \Tr \times \cN : \forall n.~ T(\beta\defon{n}) = 1\}$. Note that this set is closed in $\Tr \times \cN$, since it is a countable intersection of clopen sets. % TODO Why clopen? @@ -123,9 +123,9 @@ For the proof we need some prerequisites: $\IF$ is $\Sigma^1_1$-complete. \end{corollary} \begin{proof} - Let $A \subseteq X$ is analytic - and $X$ Polish and uncountable, - then + Let $X$ be Polish. + Suppose that $A \subseteq X$ is analytic and uncountable. + Then % https://q.uiver.app/#q=WzAsNSxbMCwwLCJYIl0sWzEsMCwiXFxjTiJdLFsyLDAsIlxcVHIiXSxbMCwxLCJBIl0sWzEsMSwiYihBKSJdLFsxLDIsImYiXSxbMCwxLCJiIl0sWzMsMCwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbNCwxLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dXQ== \[\begin{tikzcd} X & \cN & \Tr \\ @@ -135,6 +135,7 @@ For the proof we need some prerequisites: \arrow[hook, from=2-1, to=1-1] \arrow[hook, from=2-2, to=1-2] \end{tikzcd}\] + where $f$ is chosen as in \yaref{thm:lec12:1}. If $X$ is Polish and countable and $A \subseteq X$ analytic, just consider