From af97180d9adf23e7de4f5ae166f8ccb359a04f86 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Fri, 21 Feb 2014 07:11:54 +0100 Subject: [PATCH] =?UTF-8?q?Textsetzung;=20Definition=20von=20'Zusammenh?= =?UTF-8?q?=C3=A4ngende=20Menge'?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- documents/GeoTopo/Abkuerzungen.tex | 1 + documents/GeoTopo/GeoTopo.pdf | Bin 5223784 -> 5224935 bytes documents/GeoTopo/Kapitel1.tex | 49 ++++++++---------- documents/GeoTopo/Kapitel2.tex | 2 +- documents/GeoTopo/Kapitel4.tex | 2 +- .../definitions/{mycards.cfg => a7cards.cfg} | 0 .../GeoTopo/definitions/definitionen.pdf | Bin 361437 -> 361595 bytes .../GeoTopo/definitions/flashcards-try.tex | 2 +- .../GeoTopo/other-formats/GeoTopo-A5.pdf | Bin 5278301 -> 5278432 bytes 9 files changed, 26 insertions(+), 30 deletions(-) rename documents/GeoTopo/definitions/{mycards.cfg => a7cards.cfg} (100%) diff --git a/documents/GeoTopo/Abkuerzungen.tex b/documents/GeoTopo/Abkuerzungen.tex index 4b059e4..a63203d 100644 --- a/documents/GeoTopo/Abkuerzungen.tex +++ b/documents/GeoTopo/Abkuerzungen.tex @@ -9,6 +9,7 @@ \acro{d. h.}{das heißt} \acro{Def.}{Definition} \acro{etc.}{et cetera} + \acro{ex.}{existieren} \acro{Hom.}{Homomorphismus} \acro{o. B. d. A.}{ohne Beschränkung der Allgemeinheit} \acro{Prop.}{Proposition} diff --git a/documents/GeoTopo/GeoTopo.pdf b/documents/GeoTopo/GeoTopo.pdf index 8b78e296147fa811250685a5087338bee9eb7fec..33f7c867b7e521e0a536d58360525562f91b9592 100644 GIT binary patch delta 552771 zcmV)5K*_)8>b?T!^S%O*CIK^-fm|Ockx(ZB>tK<2gamn=oEnn>0W37*>TPyoE+pw= zV?sxc$0QvgkJ+*y@f0MXf3vWvU~&ejeq6;0!r?c|p1bK#B%tG;&+7n%ju5o>@giFCY*(yxg`8=kSd= zehhQ4y)p-N%yD8l04O4YW}%3ZI4`g5!~_^*gFS>ZqkEDkQLJ+?Y2S=k$x!%={U6!^ zsFMhk9e>%-f||BWa$>=(oxy@!K*iAdge8!!JSFMNojMJbz71UgrK+ISk- zxY`<0OySox;M<3jhj=jX&1htD>&~g|+Uw7GRG2L7!!IyB=uVNNeXPWT zc^~hG*#CRMz6Ce&C~goeipd^$m2YAy{FB6eUfwN$oVw4oVQ$--J>H$d7zYC zvwB^Mfvh`b1OVyTj9a(-ZXs2`Ksui z5w8Dj2IB6)=Hx^AMU*VEeA%!b}yd=X$K zlNsmwBbg*^CU)EOI)0!n*kUb_E=46yetpkEpd&{%BgeO$`_LI%1i;<1i`@edOUjHg zlA4qtX(*Q@9TkdXK|%^rl7x|@6bWldsl%94bvhMut_4Mm%i1_bVD10D1rWzZ`~ zxr~6Ie}y8WNCU~>c5cX6GD1*}+ggosLB^q{DTgF(=LY`{f0pusEF`TY3oBNUg&0iQ z0{@mA9DxyM

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z_M|AK=yG-NCtXj`k$Q&GdvBUD$ACT0-(eObu!Yr4>3zh%#q@|DKp7{`Sv94EBg|4N z@E12Cgmjx=qVlxeXgw}UVkv~2AXUMS%1xaMYe;#ICdZb^>o)nFlBWt6wM=0=eHutA z(R}u{Fc&g$4-9*_z{6+K9&{N-qsP-?askVOh$H7_3jezT@K1#PhcduF=;JT-LI`oC z(+JzJ2$%qwfCT+T9`Fsjzvha6lj-@De>JDaSCYtRkh#ZbJIC}?Qapfuw>w1tWO)3@ s 0 \Rightarrow \exists \varepsilon > 0: \fB_\varepsilon(x) \cap \fB_\varepsilon(y) = \emptyset\] + Metrische Räume sind hausdorffsch, wegen + \[d(x, y) > 0 \Rightarrow \exists \varepsilon > 0: \fB_\varepsilon(x) \cap \fB_\varepsilon(y) = \emptyset\] \end{bemerkung} \begin{beispiel}[Topologische Räume und Hausdorff-Räume] \begin{bspenum} \item $(\mdr, \fT_Z)$ ist ein topologischer Raum, der nicht hausdorffsch ist. - \item $(\mdr, \fT)$ ist ein topologischer Raum, der hausdorffsch ist. + \item $(\mdr, \fT_{\ts{Euklid}})$ ist ein topologischer Hausdorff-Raum. \end{bspenum} \end{beispiel} @@ -402,7 +402,7 @@ Die Teilraumtopologie wird auch \textit{Spurtopologie} oder \begin{bspenum} \item Für jeden topologischen Raum $X$ gilt: $\id_X : X \rightarrow X$ ist Homöomorphismus. - \item Ist $Y$ trivialer topologischer Raum, d.~h. $\fT = \fT_\text{triv}$, + \item Ist $(Y, \fT_Y)$ trivialer topologischer Raum, d.~h. $\fT_Y = \fT_\text{triv}$, so ist jede Abbildung $f:X \rightarrow Y$ stetig. \item Ist $X$ diskreter topologischer Raum, so ist $f:X \rightarrow Y$ stetig für jeden topologischen Raum $Y$ und jede Abbildung $f$. @@ -466,8 +466,8 @@ Die Teilraumtopologie wird auch \textit{Spurtopologie} oder \end{bemerkung} \begin{beweis} - Sei $U \subseteq X$ offen $\Rightarrow \pi_x^{-1} (U) = U \times Y$ - ist offen in $X \times Y$. $\qed$ + Sei $U \subseteq X$ offen\\ + $\Rightarrow \pi_X^{-1} (U) = U \times Y$ ist offen in $X \times Y$. $\qed$ \end{beweis} \begin{bemerkung}\xindex{Quotiententopologie}% @@ -527,10 +527,14 @@ sodass $\pi$ stetig wird. % Mitschrieb vom 31.10.2013 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Zusammenhang}\index{Zusammenhang|(} -\begin{definition}\xindex{zusammenhaengend@zusammenhängend}% - Ein Raum $X$ heißt \textbf{zusammenhängend}, wenn es keine offenen, - nichtleeren Teilmengen $U_1, U_2$ von $X$ gibt mit - $U_1 \cap U_2 = \emptyset$ und $U_1 \cup U_2 = X$. +\begin{definition}\xindex{Raum!zusammenhaengender@zusammenhängender}\xindex{Menge!zusammenhaengende@zusammenhängende}% + \begin{defenum} + \item Ein Raum $X$ heißt \textbf{zusammenhängend}, wenn es keine offenen, + nichtleeren Teilmengen $U_1, U_2$ von $X$ gibt mit + $U_1 \cap U_2 = \emptyset$ und $U_1 \cup U_2 = X$. + \item Eine Teilmenge $Y \subseteq X$ heißt zusammenhängend, wenn $Y$ + als topologischer Raum mit der Teilraumtopologie zusammenhängend ist. + \end{defenum} \end{definition} \begin{bemerkung} @@ -539,15 +543,6 @@ sodass $\pi$ stetig wird. und $A_1 \cup A_2 = X$. \end{bemerkung} -\begin{bemerkung} - Eine Teilmenge $Y \subseteq X$ heißt zusammenhängend, wenn $Y$ - als topologischer Raum mit der Teilraumtopologie zusammenhängend ist. -\end{bemerkung} - -%\begin{beispiel} -% -%\end{beispiel} - \begin{beispiel}[Zusammenhang von Räumen] \begin{bspenum} \item $(\mdr^n, \fT_{\ts{Euklid}})$ ist zusammenhängend, denn: @@ -623,7 +618,7 @@ sodass $\pi$ stetig wird. $Z(x)$ heißt \textbf{Zusammenhangskomponente}. \end{definition} -\begin{bemerkung} +\begin{bemerkung}[Eigenschaften von Zusammenhangskomponenten] Sei $X$ ein topologischer Raum. Dann gilt: \begin{bemenum} \item $Z(x)$ ist die größte zusammenhängende Teilmenge von $X$, @@ -779,7 +774,7 @@ $\qed$ Sei ${\color{blue} U_{x_0}} := \bigcap_{i=1}^{m(x)} U_{x_0, y_i}$. Da $X$ kompakt ist, gibt es $x_1, \dots, x_n \in X$ mit $\bigcup_{j=1}^n U_{x_j} = X$\\ - $\Rightarrow \bigcup_{j=1}^k \bigcup_{i=1}^{m(x_j)} \underbrace{\left ( U_{x_j, y_i} \times V_{x_j, y_i} \right)}_{\text{Ein grün-oranges Kästchen}} \supseteq X \times Y$\\ + $\Rightarrow \bigcup_{j=1}^k \bigcup_{i=1}^{m(x_j)} \underbrace{\left ( U_{x_j, y_i} \times V_{x_j, y_i} \right)}_{\mathclap{\text{Ein grün-oranges Kästchen}}} \supseteq X \times Y$\\ $\Rightarrow \bigcup_j \bigcup_i W_i (x_j, y_i) = X \times Y \qed$ \end{beweis} @@ -814,7 +809,7 @@ $\qed$ \end{beweis} \begin{bemerkung}\label{kor:5.6}%In Vorlesung: Bemerkung 5.6 - Seien $X, Y$ topologische Räume, $f: X \rightarrow Y$ stetig. + Seien $X, Y$ topologische Räume, $f: X \rightarrow Y$ stetig.\\ Ist $K \subseteq X$ kompakt, so ist $f(K) \subseteq Y$ kompakt. \end{bemerkung} diff --git a/documents/GeoTopo/Kapitel2.tex b/documents/GeoTopo/Kapitel2.tex index e66acf4..21bacf6 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/documents/GeoTopo/Kapitel2.tex @@ -177,7 +177,7 @@ Mannigfaltigkeiten können beliebig viele Elemente haben. \begin{figure}[htp] \centering \includegraphics[width=0.2\linewidth, keepaspectratio]{figures/Double-torus-illustration.png} - \caption{Zweifachtorus} + \caption{Durch Verklebung zweier Tori entsteht ein Zweifachtorus.} \label{fig:double-torus} \end{figure} \end{beispiel} diff --git a/documents/GeoTopo/Kapitel4.tex b/documents/GeoTopo/Kapitel4.tex index 69dfb9d..3a8e314 100644 --- a/documents/GeoTopo/Kapitel4.tex +++ b/documents/GeoTopo/Kapitel4.tex @@ -379,7 +379,7 @@ schneiden sich. \begin{beweis} Sei $\varphi$ die Isometrie mit $\varphi(A') = A$, $\varphi(A'C'^+) = AC^+$ - und $\varphi(AB^+) = AB^+$. Diese Isometrie existiert wegen \cref{axiom:4}. + und $\varphi(A'B'^+) = AB^+$. 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