From a7c3cd0d3b1d6fc8072f6c40519ab0ab68c748e0 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Thu, 19 Oct 2023 16:59:53 +0200 Subject: [PATCH] tutorial 1 --- inputs/intro.tex | 2 +- inputs/tutorial_01.tex | 70 ++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 71 insertions(+), 1 deletion(-) create mode 100644 inputs/tutorial_01.tex diff --git a/inputs/intro.tex b/inputs/intro.tex index 8f35813..787e8d1 100644 --- a/inputs/intro.tex +++ b/inputs/intro.tex @@ -4,7 +4,7 @@ in the summer term 2023 at the University Münster. \begin{warning} This is not an official script. - The official lecture notes can be found on + The official lecture notes can be found \href{https://sites.google.com/site/akwiatkmath/teaching/logic-3-abstract-topological-dynamics-and-descriptive-set-theory}{here}. \end{warning} diff --git a/inputs/tutorial_01.tex b/inputs/tutorial_01.tex new file mode 100644 index 0000000..bce909e --- /dev/null +++ b/inputs/tutorial_01.tex @@ -0,0 +1,70 @@ +\tutorial{01}{202-10-17}{} + +% TODO MAIL + +\begin{fact} + A countable product of separable spaces $(X_n)_{n \in \N}$ is separable. +\end{fact} +\begin{proof} + Choose a countable dense subset $D_n \subseteq X_n$ + Fix some point $(a_1,a_2,\ldots) \in \prod_n X_n$ + and consider $\bigcup_{i \in \N} \prod_{n \le i} D_n \times \prod_{n > i} \{a_n\}$. +\end{proof} + +\begin{fact} + \begin{itemize} + \item Let $X$ be a topological space. + Then $X$ 2nd countable $\implies$ X separable. + \item If $X$ is a metric space and separable, + then $X$ is 2nd countable. + \end{itemize} +\end{fact} +\begin{proof} + For the first point, choose some point from every basic open set. + + For the second point consider balls of rational radius + around the points of a countable dense subset. +\end{proof} + +\begin{definition} + A topological space is \vocab{Lindelöf} + if every open cover has a countable subcover. +\end{definition} +\begin{fact} + Let $X$ be a metric space. + If $X$ is Lindelöf, + then it is 2nd countable. +\end{fact} +\begin{proof} + For all $q \in \Q$ + Consider the cover $B_q(x), x \in X$ + and choose a countable subcover. + The union of these subcovers is + a countable base. +\end{proof} +\begin{fact} + Let $X$ be a topological space. + If $X$ is 2nd countable, + then it is Lindelöff. +\end{fact} +\begin{proof} + Let $A_0, A_1,\ldots$ + be a countable base. + + Let $\{U_i\}_{i \in I}$ + be a cover. + Consider $J \coloneqq \{j : \exists i \in I.~A_j \in U_i\}$. + For every $j \in J$ choose a $U_i$ such that + $A_j \subseteq U_j$. + Let $I' \subseteq I$ be the subset of chosen indices. + Then $\{U_i\}_{i \in I'}$ is a countable subcover. +\end{proof} +\begin{remark} + For metric spaces the notions + of being 2nd countable, separable + and Lindelöf coincide. + + In arbitrary topological spaces, + Lindelöf is the strongest of these notions. + +\end{remark}