From 0ec04223340c96cf00383a263a4293b5e1d89269 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Thu, 13 Feb 2014 11:50:23 +0100 Subject: [PATCH] =?UTF-8?q?Verbesserungsvorschl=C3=A4ge=20von=20J=C3=A9r?= =?UTF-8?q?=C3=B4me=20Urhausen,=20Email=20vom=2013.02.2014,=20umgesetzt.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- documents/GeoTopo/Arbeitszeit.md | 3 ++- documents/GeoTopo/GeoTopo.pdf | Bin 5142716 -> 5143339 bytes documents/GeoTopo/Kapitel3.tex | 23 +++++++++--------- documents/GeoTopo/Kapitel4.tex | 8 +++--- documents/GeoTopo/Makefile | 15 +++++++++--- documents/GeoTopo/Vorwort.tex | 12 ++++++--- documents/GeoTopo/definitions/Makefile | 4 +-- .../GeoTopo/definitions/definitionen.pdf | Bin 356878 -> 356872 bytes .../GeoTopo/other-formats/GeoTopo-A5.pdf | Bin 5191227 -> 5194263 bytes 9 files changed, 40 insertions(+), 25 deletions(-) diff --git a/documents/GeoTopo/Arbeitszeit.md b/documents/GeoTopo/Arbeitszeit.md index c105546..48b78db 100644 --- a/documents/GeoTopo/Arbeitszeit.md +++ b/documents/GeoTopo/Arbeitszeit.md @@ -76,4 +76,5 @@ in dem Erstellen dieses Skripts steckt: |10.02.2014 | 11:40 - 13:20 | Verbesserungsvorschläge von Jérôme Urhausen, Email 1 vom 10.02.2014, umgesetzt. |11.02.2014 | 05:30 - 06:00 | TikZ'en eines Bildes mit Hilfe von Jérôme Urhausen (Email 1 vom 10.02.2014) |11.02.2014 | 06:30 - 07:00 | Verbesserungsvorschläge von Jérôme Urhausen, Email 2 vom 10.02.2014, umgesetzt. -|11.02.2014 | 09:45 - 12:20 | Digitalisieren der Vorlesung von 11.02.2014 \ No newline at end of file +|11.02.2014 | 09:45 - 12:20 | Digitalisieren der Vorlesung von 11.02.2014 +|13.02.2014 | 10:00 - 11:00 | Verbesserungsvorschläge von Jérôme Urhausen, Email vom 13.02.2014, umgesetzt. \ No newline at end of file diff --git a/documents/GeoTopo/GeoTopo.pdf b/documents/GeoTopo/GeoTopo.pdf index 77520d42903d654b813614be6898c90bd9f0138f..a7de2bc9ad157d1c8ed37134cb10d79346a03a7e 100644 GIT binary patch delta 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z#Dfgf<&Jb#Y@iLnrIfmbDY^rqWW0sJDF9uO25w=(3i@%Ept#g&nwhV~hVHHd(PYk@fm=&<>MvcJO5m();OUSqun`xfbi?XuNCwe?5f03V;Dr(1wxddR z+;8eOfR(quH)7E^MN2aMGWJ}gw0n%Ud)z9DOr(s^8csgv#D{&}#gg39QS4FtH`HJB zYa1sPucUvFs~KUSlpgtC9$rF_hV-(}k{&sd{Ps{&FSR^L-;#V$%pElhkVepzbKTTb zo?wgWnI?v!N9n(t0sc#+KA}054;3&QqB$1r8T|t!{$l^t^hzyAQRdN~M<1SY2-jRo W&cnTNnUh3HL5hTrPwR>{$^Qel$)5@U diff --git a/documents/GeoTopo/Kapitel3.tex b/documents/GeoTopo/Kapitel3.tex index 7b47d3d..59b869e 100644 --- a/documents/GeoTopo/Kapitel3.tex +++ b/documents/GeoTopo/Kapitel3.tex @@ -587,16 +587,15 @@ Wenn $\pi_1(X,x) = \Set{e}$ für ein $x \in X$ gilt, dann wegen $\Rightarrow p^{-1}(U_1)$ und $p^{-1}(U_2)$ sind disjunkte Umgebungen von $y_1$ und $y_2$. - \item Sei $y \in Y$ + \item Sei $x \in X$ beliebig, aber fest. - \underline{1. Fall}: $y \in p^{-1}(x)$ + \underline{Zu zeigen}: $\forall y_i \in p^{-1}(x): \exists V_i \in \fT_Y \text{ mit } y_i \in V_i \text{, sodass gilt:} i \neq j \Rightarrow V_i \cap V_j = \emptyset$. - Finde $v_j$, sodass kein \dots - \todo[inline]{...} - - \underline{2. Fall}: $y \notin p^{-1}(x)$ - - \todo[inline]{...} + Die $V_i$ existieren wegen der Definition einer Überlagerung: + $p$ heißt Überlagerung $:\gdw \forall x \in X \exists U=U(x) \in \fT_X: p^{-1}(U) = \Dcup_{V_i \in \fT_Y} V_i \text{ und } p|_{V_i} \text{ ist Homöomorphismus}$.\\ + $\Rightarrow (p|_{V_i})^{-1}(x) = \Set{y_i}$\\ + $\Rightarrow$ Alle $y_i$ liegen diskret in $Y$, da Häufungspunkte unendlich + viele Elemente in jeder Umgebung benötigen. $\qed$ \end{enumerate} \end{beweis} @@ -716,7 +715,7 @@ $p|_{V_j}: V_j \rightarrow U$ Homöomorphismus. Dann gilt: \begin{enumerate}[label=(\roman*)] \item $\tilde{H}$ ist stetig (Beweis wie für \cref{kor:12.5}) - \item $\tilde{H}(t,0) = \tilde{\gamma_0}(t) = \tilde{H}(t,1) = \tilde{\gamma_1}(t)$ + \item $\tilde{H}(t,0) = \tilde{\gamma_0}(t), \;\;\; \tilde{H}(t,1) = \tilde{\gamma_1}(t)$ \item $\tilde{H}(0,s) = \tilde{\gamma_s}(0) = \tilde{a}$ \item $\tilde{H}(1,s) \in p^{-1}(b)$ \end{enumerate} @@ -973,7 +972,7 @@ der folgende Satz: Zu $i \in \Set{0, \dots, d-1}$ gibt es höchstens ein $f \in \Deck(Y/X)$ mit $f(y_0) = y_1$, denn ist - $f(y_0) = g(y_0)$, so ist \todo{Was steht hier?}{$(g^{-1} - f) y_0 = y_0$}, + $f(y_0) = g(y_0)$, so ist $(g^{-1} \circ f)(y_0) = y_0$, also nach \cref{kor:12.14c} $g^{-1} \circ f = \id_Y$. \end{enumerate} \end{beweis} @@ -1104,8 +1103,8 @@ und der Fundamentalgruppe herstellen: \begin{bemerkung}%In Vorlesung: Bemerkung 13.2 Jede stetige Gruppenoperation ist eine Gruppenoperation durch Homöomorphismen. \end{bemerkung} -\begin{beweis} - Nach Voraussetzung ist $\circ |_{\Set{g} \times X} : X \rightarrow X, x \mapsto g \circ x$ stetig. +\begin{beweis}\leavevmode + Nach Voraussetzung ist $m_g := \circ |_{\Set{g} \times X} : X \rightarrow X, x \mapsto g \circ x$ stetig. Die Umkehrabbildung zu $m_g$ ist $m_{g^{-1}}$: \begin{align*} diff --git a/documents/GeoTopo/Kapitel4.tex b/documents/GeoTopo/Kapitel4.tex index 215c7b3..8f7913e 100644 --- a/documents/GeoTopo/Kapitel4.tex +++ b/documents/GeoTopo/Kapitel4.tex @@ -40,7 +40,7 @@ aufgestellt. \item Je zwei rechte Winkel sind gleich (Isometrie, Bewegung) \item Parallelenaxiom von Euklid:\xindex{Parallelenaxiom}\\ Wird eine Gerade so von zwei Geraden geschnitten, dass die - Summe der Innenwinkel zwei Rechte ist, dann schneiden sich + Summe der Innenwinkel kleiner als zwei Rechte ist, dann schneiden sich diese Geraden auf der Seite dieser Winkel.\\ \\ Man mache sich klar, dass das nur dann nicht der Fall ist, @@ -150,7 +150,7 @@ aufgestellt. mit $\varphi_i (P) = P'$ und $\varphi_i(Q) = Q', i=1,2$\footnote{Die \enquote{Verschiebung} von $P'Q'$ nach $PQ$ und die Isometrie, die zusätzlich an der Gerade durch $P$ und $Q$ spiegelt.} \item \label{axiom:5}\textbf{Parallelenaxiom}\xindex{Parallele}: Für jedes $g \in G$ und jedes - $P \in X \setminus g$ gibt es höchstens ein $h \in G$ mit + $P \in X \setminus g$ gibt es höchstens ein $h \in G$ mit $P \in h$ und $h \cap g = \emptyset$. $h$ heißt \textbf{Parallele zu $g$ durch $P$}. \end{enumerate} \end{definition} @@ -168,7 +168,7 @@ aufgestellt. Dieser Satz besagt, dass Geraden, die eine Seite eines Dreiecks (also nicht nur eine Ecke) schneiden, auch eine weitere Seite -scheiden. +schneiden. \begin{beweis} $g \cap \overline{PQ} \neq \emptyset$\\ @@ -194,7 +194,7 @@ scheiden. \label{fig:geometry-5} \end{figure} -Auch \cref{kor:beh3} lässt sich Umgangssprachlich sehr viel +Auch \cref{kor:beh3} lässt sich umgangssprachlich sehr viel einfacher ausdrücken: Die Diagonalen eines konvexen Vierecks schneiden sich. diff --git a/documents/GeoTopo/Makefile b/documents/GeoTopo/Makefile index 40087e6..674f72f 100644 --- a/documents/GeoTopo/Makefile +++ b/documents/GeoTopo/Makefile @@ -2,10 +2,10 @@ DOKUMENT = GeoTopo make: sketch figures/torus.sketch > figures/torus.tex - pdflatex $(DOKUMENT).tex -output-format=pdf # aux-files for makeindex / makeglossaries + pdflatex $(DOKUMENT).tex -interaction=batchmode -output-format=pdf 1>/dev/null # aux-files for makeindex / makeglossaries makeindex $(DOKUMENT) - pdflatex $(DOKUMENT).tex -output-format=pdf # include index - pdflatex $(DOKUMENT).tex -output-format=pdf # include symbol table + pdflatex $(DOKUMENT).tex -interaction=batchmode -output-format=pdf 1>/dev/null # include index + pdflatex $(DOKUMENT).tex -interaction=batchmode -output-format=pdf 1>/dev/null # include symbol table make clean # remove intermediate files like *.log and *.aux ebook: @@ -13,5 +13,14 @@ ebook: latexmlpost -dest=$(DOKUMENT).html $(DOKUMENT).xml ebook-convert $(DOKUMENT).html $(DOKUMENT).epub --language de --no-default-epub-cover +all: + cd definitions;make + sed -i 's/\\newif\\ifAFive\\AFivefalse/\\newif\\ifAFive\\AFivetrue/' GeoTopo.tex + make + mv GeoTopo.pdf other-formats/GeoTopo-A5.pdf + sed -i 's/\\newif\\ifAFive\\AFivetrue/\\newif\\ifAFive\\AFivefalse/' GeoTopo.tex + make + + clean: rm -rf $(TARGET) *.class *.html *.log *.aux *.out *.thm *.idx *.toc *.ind *.ilg figures/torus.tex *.glg *.glo *.gls *.ist *.xdy *.fdb_latexmk diff --git a/documents/GeoTopo/Vorwort.tex b/documents/GeoTopo/Vorwort.tex index 3b7734c..795a76d 100644 --- a/documents/GeoTopo/Vorwort.tex +++ b/documents/GeoTopo/Vorwort.tex @@ -5,6 +5,11 @@ von Martin Thoma geschrieben. Es beinhaltet die Mitschriften aus der Vorlesung von Prof.~Dr.~Herrlich sowie die Mitschriften einiger Übungen und Tutorien. +Das Skript ist kostenlos über \href{http://martin-thoma.com/geotopo/}{martin-thoma.com/geotopo} +verfügbar. Wer es gerne in A5 (Schwarz-Weiß, Klebebindung) für ca. 10 Euro hätte, +kann mir eine Email schicken (info@martin-thoma.de). + +\section*{Danksagungen} An dieser Stelle möchte ich Herrn~Prof.~Dr.~Herrlich für einige Korrekturvorschläge und einen gut strukturierten Tafelanschrieb danken, der als Vorlage für dieses Skript diente. Tatsächlich basiert @@ -16,9 +21,10 @@ zu dürfen! Vielen Dank auch an Frau Lenz und Frau Randecker, die es mir erlaubt haben, ihre Übungsaufgaben und Lösungen zu benutzen. -Das Skript ist kostenlos über \href{http://martin-thoma.com/geotopo/}{martin-thoma.com/geotopo} -verfügbar. Wer es gerne in A5 (Schwarz-Weiß, Klebebindung) für ca. 10 Euro hätte, -kann mir eine Email schicken (info@martin-thoma.de). +Jérôme Urhausen hat durch Verbesserungsvorschläge und Beweise zu einer +Qualitätssteigerung am Skript beigetragen und meine Tutorin Sarah hat mir +viele Fragen per Email und nach dem Tutorium beantwortet. Danke! + \section*{Was ist Topologie?} diff --git a/documents/GeoTopo/definitions/Makefile b/documents/GeoTopo/definitions/Makefile index de93414..5cee9e0 100644 --- a/documents/GeoTopo/definitions/Makefile +++ b/documents/GeoTopo/definitions/Makefile @@ -1,8 +1,8 @@ SOURCE = definitionen make: ./generateDefinitions.py - pdflatex $(SOURCE).tex -output-format=pdf - pdflatex $(SOURCE).tex -output-format=pdf + pdflatex $(SOURCE).tex -interaction=batchmode -output-format=pdf 1>/dev/null + pdflatex $(SOURCE).tex -interaction=batchmode -output-format=pdf 1>/dev/null make clean clean: diff --git a/documents/GeoTopo/definitions/definitionen.pdf b/documents/GeoTopo/definitions/definitionen.pdf index 57a1a72213f28360a5fcf800acbae17c46b628b3..ca53bc27c8b91b05a53e95d22f8a4a1a6402a02e 100644 GIT binary patch delta 4195 zcmai0S2Wy>x5Z!>$zTvABM5_tHqk|AbP{C{JxFvhI>C<+y~XGyh>0>vbfZKMAxb1# z3{j&-mmq}uee1vOeYtn7`>@wOd#`iWe%LQ3X_YBvm8oVwk?funHawEQo-N4)0>q3V 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