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z&&WMo@^A?V-}{3l8e0qpv8xKFmyI*pJsbxSnei=1mtnDS4g5W{iajtN8;72kfh0wz z3cNw<&@%U5=k(=MbDN~;mE-#75G14e?7?4Lrx-@|dG7c~A?Pd!T92mYK}>r_xgbt> zt!CqTq*;^A0!~D&2#d)O=w1l1XzUQ2_<9Bspn*1hV3|4&>Nf_BOl^jrjateUS50Gk z_^HjvUbN=UAuD?=btNaHGWS=klb)Ilz;{XeE3mP>?fQ}v!}POcxhIJCWY?#Sv4qET zd!=z_@R7s-1h1hYYaULRaLvPb-!?m z54|5ILcL$V{6BO2e)7k6`;-$d7rD32p4QpxpRKJdy#8+M_n!D)6Sq=-%}y6?c0{6tor_a zi>Yq#>G(_6dVJnm+?q9Y<@DOv=whR%9BTyfdv07q9i5 zXHy~jj#&=I8H{Tms93Q2HJe@FvpOJofpeFG zxdqPNGotgHz~lpMXJG3i%}(HsQDD8D!2YHw`+?OC#=NHJ2WC5%za0RLi!wHuPku1} zpYHU;$L_h)10K5<%f44Xc)uQq)qnhEskirkW`BR)o)115f9!2e%O9Wq=CQj1m#L94 Lm#V6(zZ(|-M*^?y diff --git a/documents/GeoTopo/Kapitel3.tex b/documents/GeoTopo/Kapitel3.tex index 13530e5..6b66f0b 100644 --- a/documents/GeoTopo/Kapitel3.tex +++ b/documents/GeoTopo/Kapitel3.tex @@ -490,21 +490,16 @@ Wenn $\pi_1(X,x) = \Set{e}$ für ein $x \in X$ gilt, dann wegen Überlagerungen sind surjektiv. \end{bemerkung} -\begin{beweis}durch Widerspruch\\ - Sei $p$ eine Überlagerung. +\begin{beweis} + Sei $p: Y \rightarrow X$ eine Überlagerung und $x \in X$ beliebig. + Dann existiert eine offene Umgebung $U(x) \subseteq X$ und offene + Teilmengen $V_j \subseteq X$ mit + $p^{-1}(U) = \Dcup V_j$ und + $p|_{V_j}: V_j \rightarrow U$ ist Homöomorphismus. - \underline{Annahme}: $p$ ist nicht surjektiv - - Dann $\exists x \in X$ mit $U=U(x): p^{-1}(U) = \emptyset$. - Da $p$ eine Überlagerung ist, existiert eine offene Umgebung $U$, - sodass $p^{-1}(U)$ eine disjunkte Vereinigung von offenen Teilmengen - $V_j \subseteq Y$ ist und $p|_{V_j}: V_j \rightarrow U$ ein - Homöomorphismus ist. - - Da jedes $x$ eine solche Umgebung $U$ besitzt, ist $U \neq \emptyset$. - Da $p|_{V_j}: V_j \rightarrow U$ ein Homöomorphismus ist, kann also - auch $V_j$ nicht leer sein. $\Rightarrow$ Widerspruch zur Annahme. - $\qed$ + D.~h. es existiert ein $y \in V_j$, so dass $p|_{V_j}(y) = x$. + Da $x \in X$ beliebig war und ein $y \in Y$ existiert, mit + $p(y) = x$, ist $p$ surjektiv. $\qed$ \end{beweis} \begin{definition}\xindex{Abbildung!offene} diff --git a/documents/GeoTopo/Readme.md b/documents/GeoTopo/Readme.md index a9ca0ad..09ab7d5 100644 --- a/documents/GeoTopo/Readme.md +++ b/documents/GeoTopo/Readme.md @@ -55,6 +55,15 @@ Was noch kommen soll 3. A5-Version drucken * In `GeoTopo.tex`: `\AFivefalse` → `\AFivetrue` * Momentan sind es ca. 89 Seiten in A4. In A5 sind es ca. 142 Seiten. + * Druckereien + * An der Uni + * http://www.epubli.de/ (ca. 9 Euro SW, 26 Euro farbig) + * https://www.viaprinto.de/ (ca. 15 Euro SW, 35 Euro farbig) + * http://shop.kopie.de/article/show/diplomarbeit + * http://www.drucksofa.com/ + * http://www.mein-druck.de/category.htm?c=15510 + * http://www.epubli.de/ + * http://www.1buch.de/preisuebersicht/ 4. Version für Sehgeschädigte: * min `12pt`, besser `14pt` * nicht `article`, `book`, `report` sondern `extarticle` diff --git a/documents/GeoTopo/shortcuts.sty b/documents/GeoTopo/shortcuts.sty index 94c7c50..a694318 100644 --- a/documents/GeoTopo/shortcuts.sty +++ b/documents/GeoTopo/shortcuts.sty @@ -73,6 +73,7 @@ \def\PSL{\ensuremath{\mathrm{PSL}}} \newcommand\mapsfrom{\mathrel{\reflectbox{\ensuremath{\mapsto}}}} \newcommand\dcup{\mathbin{\dot{\cup}}} +\newcommand\Dcup{\mathop{\dot{\bigcup}}} \newcommand{\id}{\textnormal{id}} \DeclareMathOperator{\Deck}{Deck} \DeclareMathOperator{\Fix}{Fix}