From 3df55b651630eb7b5bd3dd933cd069a2f28b424e Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Sat, 10 Feb 2024 05:20:35 +0100 Subject: [PATCH] some small changes --- inputs/lecture_02.tex | 1 - inputs/lecture_11.tex | 2 +- inputs/tutorial_05.tex | 2 -- inputs/tutorial_08.tex | 2 +- 4 files changed, 2 insertions(+), 5 deletions(-) diff --git a/inputs/lecture_02.tex b/inputs/lecture_02.tex index 403d9c7..114b7e9 100644 --- a/inputs/lecture_02.tex +++ b/inputs/lecture_02.tex @@ -91,7 +91,6 @@ because $\tilde{f_U}$ is continuous. It is closed}{} in $X \times \R$ \gist{because $\tilde{f_U} \to \infty$ for $d(x, U^c) \to 0$}{}. - \todo{Make this precise} Therefore we identified $U$ with a closed subspace of the Polish space $(X \times \R, d_1)$. diff --git a/inputs/lecture_11.tex b/inputs/lecture_11.tex index e4193dc..150a6f8 100644 --- a/inputs/lecture_11.tex +++ b/inputs/lecture_11.tex @@ -96,7 +96,7 @@ Polish space and $\cC = 2^\omega$ the Cantor space, then they are Borel isomorphic. There is $2^\omega \hookrightarrow X$ Borel - (continuous wrt.~to the topology of $X$) + (continuous wrt.~the topology of $X$) On the other hand \[ X \hookrightarrow\cN \overset{\text{continuous embedding\footnotemark}}{\hookrightarrow}\cC diff --git a/inputs/tutorial_05.tex b/inputs/tutorial_05.tex index a0fdcad..4be235a 100644 --- a/inputs/tutorial_05.tex +++ b/inputs/tutorial_05.tex @@ -119,5 +119,3 @@ Then $f(B) = U_{(x_0,\ldots,x_{n-1})}$ is open, hence $f$ is open. \end{enumerate} - - diff --git a/inputs/tutorial_08.tex b/inputs/tutorial_08.tex index 813ceae..3df1ad1 100644 --- a/inputs/tutorial_08.tex +++ b/inputs/tutorial_08.tex @@ -95,7 +95,7 @@ for some $B_i \in \cB(Y_i)$. Then $\bigcap A_i$ is the image of $D$ under $Z \xrightarrow{(y_n) \mapsto f_0(y_0)} X$. - \paragraph{Other solution} + \emph{Other solution:} Let $F_n \subseteq X \times \cN$ be closed, and $C \subseteq X \times \cN^{\N}$ defined by