diff --git a/inputs/lecture_01.tex b/inputs/lecture_01.tex index c465d79..ce9d7a8 100644 --- a/inputs/lecture_01.tex +++ b/inputs/lecture_01.tex @@ -61,7 +61,9 @@ However the converse of this does not hold. Then $\{x_0\}$ is dense in $X$, but $X$ is not second countable. \end{example} \begin{example}[Sorgenfrey line] - \todo{Counterexamples in Topology} + Consider $\R$ with the topology given by the basis $\{[a,b) : a,b \in \R\}$. + This is T3, but not second countable + and not metrizable. \end{example} \begin{fact} @@ -98,6 +100,7 @@ However the converse of this does not hold. then $X$ is metrisable. \end{theorem} +\begin{absolutelynopagebreak} \begin{fact} If $X$ is a compact Hausdorff space, the following are equivalent: @@ -107,6 +110,7 @@ However the converse of this does not hold. \item $X$ is second countable. \end{itemize} \end{fact} +\end{absolutelynopagebreak} \subsection{Some facts about polish spaces} @@ -175,6 +179,7 @@ suffices to show that open balls in one metric are unions of open balls in the o \end{proof} \begin{definition}[Our favourite Polish spaces] + \leavevmode \begin{itemize} \item $2^{\omega}$ is called the \vocab{Cantor set}. (Consider $2$ with the discrete topology) @@ -185,7 +190,7 @@ suffices to show that open balls in one metric are unions of open balls in the o \end{itemize} \end{definition} \begin{proposition} - Let $X$ be a separable, metrisable topological space. + Let $X$ be a separable, metrisable topological space\footnote{e.g.~Polish, but we don't need completeness.}. Then $X$ topologically embeds into the \vocab{Hilbert cube}, i.e. there is an injective $f: X \hookrightarrow [0,1]^{\omega}$ @@ -227,7 +232,9 @@ suffices to show that open balls in one metric are unions of open balls in the o $f^{-1}$ is continuous. \end{claim} \begin{subproof} - \todo{Exercise!} + Consider $B_{\epsilon}(x_n) \subseteq X$ for some $n \in \N$, $\epsilon > 0$. + Then $f(U) = f(X) \cap [0,1]^{n} \times [0,\epsilon) \times [0,1]^{\omega}$ + is open\footnote{as a subset of $f(X)$!}. \end{subproof} \end{proof} \begin{proposition} diff --git a/inputs/lecture_02.tex b/inputs/lecture_02.tex index c2b9144..3c6a526 100644 --- a/inputs/lecture_02.tex +++ b/inputs/lecture_02.tex @@ -13,7 +13,6 @@ \begin{proof} Let $C \subseteq X$ be closed. Let $U_{\frac{1}{n}} \coloneqq \{x | d(x, C) < \frac{1}{n}\}$. - \todo{Exercise} Clearly $C \subseteq \bigcap U_{\frac{1}{n}}$. Let $x \in \bigcap U_{\frac{1}{n}}$. Then $\forall n .~ \exists x_n\in C.~d(x,x_n) < \frac{1}{n}$. @@ -54,7 +53,7 @@ then there exists a complete metric on $Y$. \end{claim} \begin{refproof}{psubspacegdelta:c1} - Let $Y = U$be open in $X$. + Let $Y = U$ be open in $X$. Consider the map \begin{IEEEeqnarray*}{rCl} f_U\colon U &\longrightarrow & diff --git a/inputs/lecture_03.tex b/inputs/lecture_03.tex index 1f02625..e376cbe 100644 --- a/inputs/lecture_03.tex +++ b/inputs/lecture_03.tex @@ -1,6 +1,6 @@ \lecture{03}{2023-10-17}{Embedding of the cantor space into polish spaces} -% ? \subsection{Trees} TODO +\subsection{Trees} \begin{notation} @@ -255,9 +255,7 @@ - [To be continued] \phantom\qedhere - \end{refproof} diff --git a/inputs/lecture_04.tex b/inputs/lecture_04.tex index 2defe1b..51a8ec6 100644 --- a/inputs/lecture_04.tex +++ b/inputs/lecture_04.tex @@ -1,6 +1,7 @@ \lecture{04}{2023-10-20}{} + \begin{remark} - Some of $F_s$ might be empty. + Some of the $F_s$ might be empty. \end{remark} \begin{refproof}{thm:bairetopolish} @@ -29,7 +30,6 @@ \begin{refproof}{thm:bairetopolish:c1} Let $x_n$ be a series in $D$ converging to $x$ in $\cN$. - Then $x \in \cN$. \begin{claim} $(f(x_n))$ is Cauchy. \end{claim} @@ -40,21 +40,19 @@ $x_m\defon{N} = x\defon{N}$. Then for all $m, n \ge M$, we have that $f(x_m), f(x_n) \in F_{x\defon{N}}$. - So $d(f(x_m), f(x_n)) < \epsilon$ - we have that $(f(x_n))$ is Cauchy. - - Since $(X,d)$ is complete, - there exists $y = \lim_n f(x_n)$. + So $d(f(x_m), f(x_n)) < \epsilon$, i.e.~$(f(x_n))$ is Cauchy. - Since for all $m \ge M$, $f(x_m) \in F_{x\defon{N}}$, - we get that $y \in \overline{F_{x\defon{N}}}$. - - Note that for $N' > N$ by the same argument - we get $y \in \overline{F_{x\defon{N'}}}$. - Hence - \[y \in \bigcap_{n} \overline{F_{x\defon{n}}} = \bigcap_{n} F_{x\defon{n}},\] - i.e.~$y \in D$ and $y = f(x)$. \end{subproof} + Since $(X,d)$ is complete, + there exists $y = \lim_n f(x_n)$. + Since for all $m \ge M$, $f(x_m) \in F_{x\defon{N}}$, + we get that $y \in \overline{F_{x\defon{N}}}$. + + Note that for $N' > N$ by the same argument + we get $y \in \overline{F_{x\defon{N'}}}$. + Hence + \[y \in \bigcap_{n} \overline{F_{x\defon{n}}} = \bigcap_{n} F_{x\defon{n}},\] + i.e.~$y \in D$ and $y = f(x)$. \end{refproof} We extend $f$ to $g\colon\cN \to X$ @@ -70,7 +68,7 @@ (i.e.~$r = \id$ on $D$ and $r$ is a continuous surjection). Then $g \coloneqq f \circ r$. - To construct $r$, we will define by induction + To construct $r$, we will define $\phi\colon \N^{<\N} \to S$ by induction on the length such that \begin{itemize} diff --git a/inputs/tutorial_01.tex b/inputs/tutorial_01.tex index a6a4894..2ef9ca6 100644 --- a/inputs/tutorial_01.tex +++ b/inputs/tutorial_01.tex @@ -13,9 +13,9 @@ \begin{fact} \begin{itemize} \item Let $X$ be a topological space. - Then $X$ 2nd countable $\implies$ X separable. + Then $X$ \nth{2} countable $\implies$ X separable. \item If $X$ is a metric space and separable, - then $X$ is 2nd countable. + then $X$ is \nth{2} countable. \end{itemize} \end{fact} \begin{proof} @@ -32,18 +32,18 @@ \begin{fact} Let $X$ be a metric space. If $X$ is Lindelöf, - then it is 2nd countable. + then it is \nth{2} countable. \end{fact} \begin{proof} For all $q \in \Q$ - Consider the cover $B_q(x), x \in X$ + 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, + If $X$ is \nth{2} countable, then it is Lindelöff. \end{fact} \begin{proof} @@ -60,7 +60,7 @@ \end{proof} \begin{remark} For metric spaces the notions - of being 2nd countable, separable + of being \nth{2} countable, separable and Lindelöf coincide. In arbitrary topological spaces, diff --git a/jrpie-math.sty b/jrpie-math.sty index ea1aca6..17e3080 100644 --- a/jrpie-math.sty +++ b/jrpie-math.sty @@ -50,3 +50,13 @@ \RequirePackage{mkessler-mathfixes} % Load this last since it renews behaviour \DeclareMathOperator{\inter}{int} % interior \newcommand{\defon}[1]{|_{#1}} % TODO + +\RequirePackage[super]{nth} + +% TODO MOVE +% https://tex.stackexchange.com/a/94702 +\newenvironment{absolutelynopagebreak} + {\par\nobreak\vfil\penalty0\vfilneg + \vtop\bgroup} + {\par\xdef\tpd{\the\prevdepth}\egroup + \prevdepth=\tpd} diff --git a/logic3.tex b/logic3.tex index f0d961a..52f3cab 100644 --- a/logic3.tex +++ b/logic3.tex @@ -61,6 +61,7 @@ \input{inputs/tutorial_07} \input{inputs/tutorial_08} \input{inputs/tutorial_09} +\input{inputs/tutorial_10} \section{Facts} \input{inputs/facts}