From 4aef9647c7109c24ab795ada00e15add70ee83df Mon Sep 17 00:00:00 2001
From: Juan Marín Noguera <juan.marinn@um.es>
Date: Wed, 17 Jun 2020 14:27:31 +0200
Subject: Surface topology

---
 ts/n5.lyx | 145 ++++++++++++++------------------------------------------------
 1 file changed, 32 insertions(+), 113 deletions(-)

(limited to 'ts/n5.lyx')

diff --git a/ts/n5.lyx b/ts/n5.lyx
index 00531c4..f6928f4 100644
--- a/ts/n5.lyx
+++ b/ts/n5.lyx
@@ -2602,117 +2602,6 @@ Demostración.
 \end_inset
 
 
-\end_layout
-
-\begin_layout Standard
-Dadas dos superficies 
-\begin_inset Formula $X$
-\end_inset
-
- e 
-\begin_inset Formula $Y$
-\end_inset
-
- con subespacios respectivos 
-\begin_inset Formula $X_{0}$
-\end_inset
-
- e 
-\begin_inset Formula $Y_{0}$
-\end_inset
-
- y homeomorfos a un disco en 
-\begin_inset Formula $\mathbb{R}^{2}$
-\end_inset
-
-, dado un homeomorfismo 
-\begin_inset Formula $h:\partial X_{0}\cong\mathbb{S}^{1}\to\partial Y_{0}\cong\mathbb{S}^{1}$
-\end_inset
-
-, llamamos 
-\series bold
-suma conexa
-\series default
- de 
-\begin_inset Formula $X$
-\end_inset
-
- e 
-\begin_inset Formula $Y$
-\end_inset
-
-, 
-\begin_inset Formula $X\sharp Y$
-\end_inset
-
-, a 
-\begin_inset Formula $((X\setminus\text{\ensuremath{\mathring{X}_{0}}})\amalg(Y\setminus\mathring{Y}_{0}))/\sim$
-\end_inset
-
-, donde 
-\begin_inset Formula $x\sim y$
-\end_inset
-
- si y sólo si 
-\begin_inset Formula $x=y$
-\end_inset
-
-, o bien 
-\begin_inset Formula $x\in X_{0}$
-\end_inset
-
- e 
-\begin_inset Formula $y\in Y_{0}$
-\end_inset
-
- con 
-\begin_inset Formula $y=h(x)$
-\end_inset
-
-, o bien al revés.
- Como 
-\series bold
-teorema
-\series default
-, el grupo fundamental del 
-\series bold
-doble toro
-\series default
-, 
-\begin_inset Formula $\mathbb{T}\sharp\mathbb{T}$
-\end_inset
-
-, no es abeliano.
- 
-\begin_inset Note Comment
-status open
-
-\begin_layout Plain Layout
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-Demostración.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-Llevar este párrafo al tema 6.
-\end_layout
-
-\end_inset
-
-
 \end_layout
 
 \begin_layout Standard
@@ -2754,8 +2643,38 @@ curva
 superficie
 \series default
  es una 2-variedad.
- Así, la esfera, el toro, el plano proyectivo real y el doble toro son superfici
-es topológicamente distintas.
+\end_layout
+
+\begin_layout Standard
+Ejemplos de superficies son 
+\begin_inset Formula $\mathbb{R}^{2}$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{S}^{2}$
+\end_inset
+
+, el toro 
+\begin_inset Formula $\mathbb{T}^{2}$
+\end_inset
+
+, el 
+\series bold
+cilindro abierto
+\series default
+ 
+\begin_inset Formula $\mathbb{S}^{1}\times(0,1)$
+\end_inset
+
+, la banda de Möbius, la botella de Klein y el plano proyectivo real 
+\begin_inset Formula $\mathbb{RP}^{2}$
+\end_inset
+
+ o 
+\begin_inset Formula $\mathbb{P}^{2}$
+\end_inset
+
+.
 \end_layout
 
 \end_body
-- 
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