ts/n3

3 files changed, 2931 insertions, 44 deletions

diff --git a/ts/n.lyx b/ts/n.lyx
index 51ce287..9ee935a 100644
--- a/ts/n.lyx
+++ b/ts/n.lyx
@@ -155,6 +155,21 @@ Essential Topology, Martin D.
  Crossley (2005), Springer.
 \end_layout
 
+\begin_layout Itemize
+Wikipedia, the Free Encyclopedia, 
+\begin_inset Flex URL
+status open
+
+\begin_layout Plain Layout
+
+https://en.wikipedia.org/wiki/Coproduct
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
 \begin_layout Chapter
 Espacios topológicos
 \end_layout
@@ -183,5 +198,19 @@ filename "n2.lyx"
 
 \end_layout
 
+\begin_layout Chapter
+Homeomorfismos y construcciones topológicas
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n3.lyx"
+
+\end_inset
+
+
+\end_layout
+
 \end_body
 \end_document
diff --git a/ts/n2.lyx b/ts/n2.lyx
index 19703bf..b70277e 100644
--- a/ts/n2.lyx
+++ b/ts/n2.lyx
@@ -1864,50 +1864,6 @@ compacto
  Así:
 \end_layout
 
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-\begin_inset Formula $\mathbb{S}^{1}$
-\end_inset
-
-, 
-\begin_inset Formula $[a,b]$
-\end_inset
-
-, no 
-\begin_inset Formula $\mathbb{R}$
-\end_inset
-
-, espacio topológico finito, discreto, 
-\begin_inset Formula $(a,b)$
-\end_inset
-
-, 
-\begin_inset Formula $(a,b]$
-\end_inset
-
-, 
-\begin_inset Formula $[a,b)$
-\end_inset
-
-, 
-\begin_inset Formula $(a,+\infty)$
-\end_inset
-
-, 
-\begin_inset Formula $(-\infty,b)$
-\end_inset
-
-.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
 \begin_layout Enumerate
 Todo espacio topológico finito es compacto.
 \begin_inset Note Comment
diff --git a/ts/n3.lyx b/ts/n3.lyx
new file mode 100644
index 0000000..bf5be1d
--- /dev/null
+++ b/ts/n3.lyx
@@ -0,0 +1,2902 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Un 
+\series bold
+homeomorfismo
+\series default
+ es una biyección 
+\begin_inset Formula $f:X\to Y$
+\end_inset
+
+ continua con inversa continua.
+ Si existe, 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ son 
+\series bold
+homeomorfos
+\series default
+ (
+\begin_inset Formula $X\cong Y$
+\end_inset
+
+).
+ Esta relación es de equivalencia.
+ Una función 
+\begin_inset Formula $f:X\to Y$
+\end_inset
+
+ es un 
+\series bold
+embebimiento
+\series default
+ si su restricción de rango 
+\begin_inset Formula $\hat{f}:X\to f(X)$
+\end_inset
+
+ es un homeomorfismo.
+\end_layout
+
+\begin_layout Standard
+Algunos homeomorfismos:
+\end_layout
+
+\begin_layout Enumerate
+Los isomorfismos lineales entre 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathbb{R}^{m}$
+\end_inset
+
+ son homeomorfismos.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Son biyecciones que, al ser lineales en 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+, son continuas, y como su inversa es también lineal, es continua.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Los intervalos abiertos y acotados de 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ son homeomorfos entre sí, al igual que los cerrados y acotados.
+ 
+\end_layout
+
+\begin_layout Enumerate
+Las funciones 
+\begin_inset Formula $f,g:(-1,1)\to\mathbb{R}$
+\end_inset
+
+ dadas por 
+\begin_inset Formula $f(x):=\tan(\frac{\pi}{2}x)$
+\end_inset
+
+ y 
+\begin_inset Formula $g(x):=\frac{x}{1-x^{2}}$
+\end_inset
+
+ son homeomorfismos.
+\end_layout
+
+\begin_layout Enumerate
+Sea 
+\begin_inset Formula $N:=(0,\dots,0,1)\in\mathbb{R}^{n+1}$
+\end_inset
+
+, la 
+\series bold
+proyección estereográfica
+\series default
+, que asigna a cada 
+\begin_inset Formula $x\in\mathbb{S}^{n}\setminus N$
+\end_inset
+
+ el punto de corte de 
+\begin_inset Formula $\mathbb{R}^{n}\times\{-1\}$
+\end_inset
+
+ con la recta 
+\begin_inset Formula $\overrightarrow{Nx}$
+\end_inset
+
+, es un homeomorfismo.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Sean 
+\begin_inset Formula $\pi:=\mathbb{R}^{n}\times\{-1\}$
+\end_inset
+
+ y 
+\series bold
+
+\begin_inset Formula $g:\mathbb{S}^{n}\setminus N\to\pi$
+\end_inset
+
+
+\series default
+ la proyección estereográfica.
+ Si 
+\begin_inset Formula $y:=g(x)$
+\end_inset
+
+, 
+\begin_inset Formula $(y_{1},\dots,y_{n},-1)=(0,\dots,0,1)+\mu(x_{1},\dots,x_{n},x_{n+1}-1)$
+\end_inset
+
+, y como 
+\begin_inset Formula $-1=1+\mu(x_{n+1}-1)$
+\end_inset
+
+, 
+\begin_inset Formula $\mu=\frac{1}{1-x_{n+1}}$
+\end_inset
+
+ e 
+\begin_inset Formula $(y_{1},\dots,y_{n})=\frac{1}{1-x_{n+1}}(x_{1},\dots,x_{n})$
+\end_inset
+
+, y 
+\begin_inset Formula $g$
+\end_inset
+
+ está bien definida.
+ Partiendo de 
+\begin_inset Formula $(y_{1},\dots,y_{n})$
+\end_inset
+
+, 
+\begin_inset Formula $(x_{1},\dots,x_{n+1})=(ty_{1},\dots,ty_{n},1-t)$
+\end_inset
+
+ para un cierto 
+\begin_inset Formula $t$
+\end_inset
+
+, pero como 
+\begin_inset Formula $x_{1}^{2}+\dots+x_{n}^{2}+x_{n+1}^{2}=t^{2}(y_{1}^{2}+\dots+y_{n}^{2})+(1-t)^{2}=(1+y_{1}^{2}+\dots+y_{n}^{2})t^{2}-2t+1=1$
+\end_inset
+
+, se tiene 
+\begin_inset Formula $t\in\{0,\frac{2}{1+y_{1}^{2}+\dots+y_{n}^{2}}\}$
+\end_inset
+
+, y como 
+\begin_inset Formula $t\neq0$
+\end_inset
+
+ porque 
+\begin_inset Formula $x\neq N$
+\end_inset
+
+ es 
+\begin_inset Formula $t=\frac{2}{1+y_{1}^{2}+\dots+y_{n}^{2}}$
+\end_inset
+
+, y 
+\begin_inset Formula $g$
+\end_inset
+
+ es biyectiva.
+ A partir de las fórmulas obtenidas es claro que 
+\begin_inset Formula $g$
+\end_inset
+
+ y 
+\begin_inset Formula $g^{-1}$
+\end_inset
+
+ son continuas, luego 
+\begin_inset Formula $g$
+\end_inset
+
+ es un homeomorfismo.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{S}^{n}\setminus\{p\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ son homeomorfos.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\mathbb{S}^{n}\setminus\{N:=(0,\dots,0,1)\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathbb{S}^{n}\setminus\{p\}$
+\end_inset
+
+, así como 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $\pi:=\mathbb{R}^{n}\times\{-1\}$
+\end_inset
+
+, son linealmente isomorfos, por lo que son homeomorfos y 
+\begin_inset Formula $\mathbb{S}^{n}\setminus\{p\}\cong S\setminus\{N\}\cong\pi\cong\mathbb{R}^{n}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+El disco 
+\begin_inset Formula $\mathbb{D}^{n}:=\overline{B}_{d_{2}}(0;1)\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ es homeomorfo a 
+\begin_inset Formula $[-1,1]^{n}$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Sea 
+\begin_inset Formula $f:\mathbb{D}^{n}\to[-1,1]^{n}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $f(x):=x\frac{\Vert x\Vert_{\infty}}{\Vert x\Vert_{2}}$
+\end_inset
+
+ para 
+\begin_inset Formula $x\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $f(0):=0$
+\end_inset
+
+, queremos ver que 
+\begin_inset Formula $f$
+\end_inset
+
+ es biyectiva con inversa 
+\begin_inset Formula $g(y):=y\frac{\Vert y\Vert_{2}}{\Vert y\Vert_{\infty}}$
+\end_inset
+
+ para 
+\begin_inset Formula $y\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $g(0)=0$
+\end_inset
+
+.
+ En efecto, para 
+\begin_inset Formula $x\neq0$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+g(f(x))=g\left(x\frac{\Vert x\Vert_{\infty}}{\Vert x\Vert_{2}}\right)=x\frac{\Vert x\Vert_{\infty}}{\Vert x\Vert_{2}}\frac{\Vert x\Vert_{2}\frac{\Vert x\Vert_{\infty}}{\Vert x\Vert_{2}}}{\Vert x\Vert_{\infty}\frac{\Vert x\Vert_{\infty}}{\Vert x\Vert_{2}}}=x,
+\]
+
+\end_inset
+
+y 
+\begin_inset Formula $g(f(0))=0$
+\end_inset
+
+, luego 
+\begin_inset Formula $g\circ f=1_{\mathbb{D}^{n}}$
+\end_inset
+
+, y análogamente 
+\begin_inset Formula $f\circ g=1_{[-1,1]^{n}}$
+\end_inset
+
+.
+ La continuidad en puntos distintos de 0 es clara.
+ Para el 0, 
+\begin_inset Formula $\lim_{x\to0}\Vert f(x)\Vert_{2}=\lim_{x\to0}\Vert x\Vert_{2}\frac{\Vert x\Vert_{\infty}}{\Vert x\Vert_{2}}=\lim_{x\to0}\Vert x\Vert_{\infty}=0=\Vert f(0)\Vert$
+\end_inset
+
+, y análogamente 
+\begin_inset Formula $\lim_{g\to0}\Vert g(x)\Vert_{2}=\Vert g(0)\Vert$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Dados dos espacios topológicos 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+, si para cierto 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ y para todo 
+\begin_inset Formula $y\in Y$
+\end_inset
+
+, 
+\begin_inset Formula $X\setminus\{x\}$
+\end_inset
+
+ e 
+\begin_inset Formula $Y\setminus\{y\}$
+\end_inset
+
+ no son homeomorfos, 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ tampoco lo son.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Si lo fueran, sea 
+\begin_inset Formula $f:X\to Y$
+\end_inset
+
+ el homeomorfismo, 
+\begin_inset Formula $f$
+\end_inset
+
+ también sería un homeomorfismo entre 
+\begin_inset Formula $X\setminus\{x\}$
+\end_inset
+
+ e 
+\begin_inset Formula $Y\setminus\{f(x)\}\#$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+invariante topológico
+\series default
+ es una propiedad que pueden tener espacios topológicos que se conserva
+ por homeomorfismos, y que podemos usar para saber si dos espacios son o
+ no homeomorfos.
+ Como 
+\series bold
+teorema
+\series default
+, la conexión, la conexión por caminos, la compacidad, ser Hausdorff y los
+ axiomas de numerabilidad 
+\begin_inset Formula $\text{1A}\mathbb{N}$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{2A}\mathbb{N}$
+\end_inset
+
+ son invariantes topológicos.
+ Así:
+\end_layout
+
+\begin_layout Enumerate
+Un intervalo cerrado y uno abierto no son homeomorfos.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Al quitar un punto al abierto este queda disconexo, pero no al quitar un
+ extremo del cerrado.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{S}^{1}$
+\end_inset
+
+ y un intervalo (incluyendo 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+) no son homeomorfos.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Si el intervalo es unipuntual esto es obvio.
+ En otro caso, al quitar un punto intermedio al intervalo este queda disconexo,
+ lo que no ocurre con ningún punto de 
+\begin_inset Formula $\mathbb{S}^{1}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Una función 
+\begin_inset Formula $f:X\to Y$
+\end_inset
+
+ es 
+\series bold
+abierta
+\series default
+ si transforma abiertos de 
+\begin_inset Formula $X$
+\end_inset
+
+ en abiertos de 
+\begin_inset Formula $Y$
+\end_inset
+
+, y es 
+\series bold
+cerrada
+\series default
+ si transforma cerrados de 
+\begin_inset Formula $X$
+\end_inset
+
+ en cerrados de 
+\begin_inset Formula $Y$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $f$
+\end_inset
+
+ es biyectiva, 
+\begin_inset Formula $f$
+\end_inset
+
+ es abierta si y sólo si es cerrada, y es un homeomorfismo si y sólo si
+ es continua y abierta.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $f:X\to Y$
+\end_inset
+
+ es continua con 
+\begin_inset Formula $X$
+\end_inset
+
+ compacto e 
+\begin_inset Formula $Y$
+\end_inset
+
+ Hausdorff, 
+\begin_inset Formula $f$
+\end_inset
+
+ es cerrada.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, dado 
+\begin_inset Formula $U\subseteq X$
+\end_inset
+
+ cerrado, 
+\begin_inset Formula $U$
+\end_inset
+
+ es compacto, luego 
+\begin_inset Formula $f(U)$
+\end_inset
+
+ también y, por ser 
+\begin_inset Formula $Y$
+\end_inset
+
+ Hausdorff, 
+\begin_inset Formula $f(U)$
+\end_inset
+
+ es cerrado.
+\end_layout
+
+\end_inset
+
+ En particular, si además 
+\begin_inset Formula $f$
+\end_inset
+
+ biyectiva, es un homeomorfismo.
+\end_layout
+
+\begin_layout Standard
+Un espacio 
+\begin_inset Formula $Y$
+\end_inset
+
+ compacto y Hausdorff es una 
+\series bold
+compactificación
+\series default
+ de un subespacio 
+\begin_inset Formula $X\subseteq Y$
+\end_inset
+
+ si 
+\begin_inset Formula $\overline{X}=Y$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $Y\setminus X$
+\end_inset
+
+ es unipuntual, llamamos a este punto 
+\begin_inset Formula $\infty$
+\end_inset
+
+ y decimos que 
+\begin_inset Formula $Y$
+\end_inset
+
+ es la 
+\series bold
+compactificación por un punto
+\series default
+ de 
+\begin_inset Formula $X$
+\end_inset
+
+.
+ Por ejemplo, 
+\begin_inset Formula $\mathbb{R}^{n}\cup\{\infty\}\cong\mathbb{S}^{n}$
+\end_inset
+
+.
+ Llamamos 
+\series bold
+esfera de Riemann
+\series default
+ a 
+\begin_inset Formula $\mathbb{R}^{2}\cup\{\infty\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Uniones disjuntas
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+unión disjunta
+\series default
+ de dos objetos 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+, 
+\begin_inset Formula $X\amalg Y$
+\end_inset
+
+, a un objeto para el que existen 
+\begin_inset Formula $L:X\to X\amalg Y$
+\end_inset
+
+ y 
+\begin_inset Formula $R:Y\to X\amalg Y$
+\end_inset
+
+ tales que para cada objeto 
+\begin_inset Formula $Z$
+\end_inset
+
+, 
+\begin_inset Formula $f_{L}:X\to Z$
+\end_inset
+
+ y 
+\begin_inset Formula $f_{R}:Y\to Z$
+\end_inset
+
+, existe una única 
+\begin_inset Formula $f:X\amalg Y\to Z$
+\end_inset
+
+ tal que 
+\begin_inset Formula $f_{L}=f\circ L$
+\end_inset
+
+ y 
+\begin_inset Formula $f_{R}:=f\circ R$
+\end_inset
+
+.
+ Se puede construir como 
+\begin_inset Formula $(X\times\{0\})\cup(Y\times\{1\})$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $(X,{\cal T}_{X})$
+\end_inset
+
+ e 
+\begin_inset Formula $(Y,{\cal T}_{Y})$
+\end_inset
+
+ son espacios topológicos, definimos la topología 
+\begin_inset Formula ${\cal T}_{X\amalg Y}:=\{U\subseteq X\amalg Y:L^{-1}(U)\in{\cal T}_{X}\land R^{-1}(U)\in{\cal T}_{Y}\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Vemos que 
+\begin_inset Formula $f:X\amalg Y\to Z$
+\end_inset
+
+ es continua si y sólo si lo son 
+\begin_inset Formula $f\circ L$
+\end_inset
+
+ y 
+\begin_inset Formula $f\circ R$
+\end_inset
+
+, y que 
+\begin_inset Formula $f:Z\to X\amalg Y$
+\end_inset
+
+ es continua si y sólo si 
+\begin_inset Formula $f|_{f^{-1}(L(X))}$
+\end_inset
+
+ y 
+\begin_inset Formula $f|_{f^{-1}(R(Y))}$
+\end_inset
+
+ lo son.
+ Entonces:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un hiperplano de 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{R}^{n}\amalg\mathbb{R}^{n}\cong\mathbb{R}^{n}\setminus H$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Sea 
+\begin_inset Formula $(v_{1},\dots,v_{n})$
+\end_inset
+
+ una base de 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ donde 
+\begin_inset Formula $H=\langle v_{2},\dots,v_{n}\rangle$
+\end_inset
+
+, queremos ver que 
+\begin_inset Formula $f:\mathbb{R}^{n}\amalg\mathbb{R}^{n}\to\mathbb{R}^{n}\setminus H$
+\end_inset
+
+ dada por 
+\begin_inset Formula $f(L(x)):=e^{x_{1}}v_{1}+\sum_{k=2}^{n}x_{k}v_{k}$
+\end_inset
+
+ y 
+\begin_inset Formula $f(R(x)):=-e^{x_{1}}v_{1}+\sum_{k=2}^{n}x_{k}v_{k}$
+\end_inset
+
+ es un homeomorfismo.
+ Al ser la composición de una transformación ortonormal con una exponencial,
+ 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua.
+ Su inversa viene dada por 
+\begin_inset Formula 
+\[
+f^{-1}\left(\sum_{k=1}^{n}x_{k}v_{k}\right)=\begin{cases}
+L(\log x_{1},x_{2},\dots,x_{n}) & \text{si }x_{1}>0,\\
+R(\log(-x_{1}),x_{2},\dots,x_{n}) & \text{si }x_{1}<0,
+\end{cases}
+\]
+
+\end_inset
+
+que es claramente continua.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal SO}(3)\amalg{\cal SO}(3)\cong{\cal O}(3)$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Basta tomar el homeomorfismo 
+\begin_inset Formula $f$
+\end_inset
+
+ dado por 
+\begin_inset Formula $f(L(A))=A$
+\end_inset
+
+, 
+\begin_inset Formula $f(R(A))=-A$
+\end_inset
+
+, y 
+\begin_inset Formula $f^{-1}(A)$
+\end_inset
+
+ es 
+\begin_inset Formula $A$
+\end_inset
+
+ si 
+\begin_inset Formula $|A|=1$
+\end_inset
+
+ o 
+\begin_inset Formula $-A$
+\end_inset
+
+ si 
+\begin_inset Formula $|A|=-1$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teoremas
+\series default
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ son compactos si y sólo si lo es 
+\begin_inset Formula $X\amalg Y$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $\{A_{i}\}_{i\in I}$
+\end_inset
+
+ un recubrimiento por abiertos de 
+\begin_inset Formula $X\amalg Y$
+\end_inset
+
+, 
+\begin_inset Formula $\{L^{-1}(A_{i})\}_{i\in I}$
+\end_inset
+
+ lo es de 
+\begin_inset Formula $X$
+\end_inset
+
+ y por tanto admite un subrecubrimiento finito 
+\begin_inset Formula $L^{-1}(A_{i_{1}}),\dots,L^{-1}(A_{i_{n}})$
+\end_inset
+
+.
+ Del mismo modo 
+\begin_inset Formula $\{R^{-1}(A_{i})\}_{i\in I}$
+\end_inset
+
+ admite un subrecubrimiento finito 
+\begin_inset Formula $R^{-1}(A_{j_{1}}),\dots,R^{-1}(A_{j_{m}})$
+\end_inset
+
+ de 
+\begin_inset Formula $Y$
+\end_inset
+
+, luego 
+\begin_inset Formula $A_{i_{1}},\dots,A_{i_{n}},A_{j_{1}},\dots,A_{j_{m}}$
+\end_inset
+
+ es un subrecubrimiento finito de 
+\begin_inset Formula $X\amalg Y$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $\{A_{i}\}_{i\in I}$
+\end_inset
+
+ un recubrimiento por abiertos de 
+\begin_inset Formula $X$
+\end_inset
+
+, 
+\begin_inset Formula $\{L(A_{i})\}_{i\in I}\cup Y$
+\end_inset
+
+ es un recubrimiento por abiertos de 
+\begin_inset Formula $X\amalg Y$
+\end_inset
+
+ que admite pues un subrecubrimiento finito 
+\begin_inset Formula $L(A_{1}),\dots,L(A_{n}),Y$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $A_{1},\dots,A_{n}$
+\end_inset
+
+ es un subrecubrimiento finito de 
+\begin_inset Formula $X$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $Y$
+\end_inset
+
+ es análogo.
+\end_layout
+
+\end_deeper
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ son Hausdorff si y sólo si lo es 
+\begin_inset Formula $X\amalg Y$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $p,q\in X\amalg Y$
+\end_inset
+
+, 
+\begin_inset Formula $p\neq q$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $p,q\in L(X)$
+\end_inset
+
+ o 
+\begin_inset Formula $p,q\in R(Y)$
+\end_inset
+
+ basta tomar los abiertos en 
+\begin_inset Formula $X$
+\end_inset
+
+ o en 
+\begin_inset Formula $Y$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $p\in L(X)$
+\end_inset
+
+ y 
+\begin_inset Formula $q\in R(Y)$
+\end_inset
+
+, basta tomar 
+\begin_inset Formula $L(X)$
+\end_inset
+
+ y 
+\begin_inset Formula $R(Y)$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $p,q\in X$
+\end_inset
+
+, 
+\begin_inset Formula $p\neq q$
+\end_inset
+
+, existen 
+\begin_inset Formula $U,V\subseteq X\amalg Y$
+\end_inset
+
+ entornos respectivos de 
+\begin_inset Formula $p$
+\end_inset
+
+ y 
+\begin_inset Formula $q$
+\end_inset
+
+ disjuntos, y basta tomar 
+\begin_inset Formula $U\cap X$
+\end_inset
+
+ y 
+\begin_inset Formula $V\cap X$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $Y$
+\end_inset
+
+ es análogo.
+\end_layout
+
+\end_deeper
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $X,Y\neq\emptyset$
+\end_inset
+
+, 
+\begin_inset Formula $X\amalg Y$
+\end_inset
+
+ es disconexo.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\{L(X),R(Y)\}$
+\end_inset
+
+ es una separación por abiertos.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $Z$
+\end_inset
+
+ es disconexo, 
+\begin_inset Formula $Z\cong X\amalg Y$
+\end_inset
+
+ para ciertos 
+\begin_inset Formula $X,Y\neq\emptyset$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Sea 
+\begin_inset Formula $\{U,V\}$
+\end_inset
+
+ una separación por abiertos de 
+\begin_inset Formula $Z$
+\end_inset
+
+, basta tomar 
+\begin_inset Formula $U\amalg V$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Espacios producto
+\end_layout
+
+\begin_layout Standard
+Dados dos espacios topológicos 
+\begin_inset Formula $(X,{\cal T}_{X})$
+\end_inset
+
+ e 
+\begin_inset Formula $(Y,{\cal T}_{Y})$
+\end_inset
+
+, llamamos 
+\series bold
+topología producto
+\series default
+ en 
+\begin_inset Formula $X\times Y$
+\end_inset
+
+ a la generada por la base 
+\begin_inset Formula ${\cal T}_{X}\times{\cal T}_{Y}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{R}^{m}\times\mathbb{R}^{n}\cong\mathbb{R}^{m+n}$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Claramente 
+\begin_inset Formula $f:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{m+n}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $f((x_{1},\dots,x_{m}),(y_{1},\dots,y_{n})):=(x_{1},\dots,x_{m},y_{1},\dots,y_{n})$
+\end_inset
+
+ es biyectiva.
+ Dados abiertos 
+\begin_inset Formula $U=:\bigcup_{x\in U}B_{\infty}(x,\varepsilon_{x})\subseteq\mathbb{R}^{m}$
+\end_inset
+
+ y 
+\begin_inset Formula $V=:\bigcup_{y\in V}B_{\infty}(y,\delta_{y})\subseteq\mathbb{R}^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $f(U\times V)=\bigcup_{x\in U}\bigcup_{y\in V}f(B_{\infty}(x,\varepsilon_{x}),B_{\infty}(y,\delta_{y}))$
+\end_inset
+
+, y basta ver que los productos de bolas son abiertos, pero para 
+\begin_inset Formula $(z,w)\in f(B_{\infty}(x,\varepsilon_{x}),B_{\infty}(y,\delta_{y}))$
+\end_inset
+
+, sean 
+\begin_inset Formula $a:=d_{\infty}(z,x)<\varepsilon_{x}$
+\end_inset
+
+ y 
+\begin_inset Formula $b:=d_{\infty}(w,y)<\delta_{y}$
+\end_inset
+
+, la bola 
+\begin_inset Formula $B((z,w),\min\{\varepsilon_{x}-a,\delta_{y}-b\})\subseteq f(B_{\infty}(x,\varepsilon_{x}),B_{\infty}(y,\delta_{y}))$
+\end_inset
+
+, luego 
+\begin_inset Formula $f$
+\end_inset
+
+ es abierta.
+ Por otro lado, sea 
+\begin_inset Formula $W=:\bigcup_{(z,w)\in W}B_{\infty}((z,w),\varepsilon_{z,w})\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ abierto, 
+\begin_inset Formula $f^{-1}(W)=\bigcup_{(z,w)\in W}f^{-1}(B_{\infty}((z,w),\varepsilon_{z,w}))=\bigcup_{(z,w)\in W}(B_{\infty}(z,\varepsilon_{z,w})\times B_{\infty}(w,\varepsilon_{z,w}))$
+\end_inset
+
+, luego 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{R}\amalg\mathbb{R}\cong\mathbb{R}\times\mathbb{S}^{0}$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Claramente 
+\begin_inset Formula $f:\mathbb{R}\amalg\mathbb{R}\to\mathbb{R}\times\mathbb{S}^{0}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $f(L(x)):=(x,-1)$
+\end_inset
+
+ y 
+\begin_inset Formula $f(R(y)):=(y,1)$
+\end_inset
+
+ es biyectiva.
+ Además, 
+\begin_inset Formula $f\circ L$
+\end_inset
+
+, 
+\begin_inset Formula $f\circ R$
+\end_inset
+
+, 
+\begin_inset Formula $f^{-1}|_{f(L(X))}$
+\end_inset
+
+ y 
+\begin_inset Formula $f^{-1}|_{f(R(X))}$
+\end_inset
+
+ son continuas, luego 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua y abierta.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teoremas
+\series default
+:
+\end_layout
+
+\begin_layout Enumerate
+Las 
+\series bold
+proyecciones
+\series default
+ 
+\begin_inset Formula $\pi_{1}:X\times Y\to X$
+\end_inset
+
+ y 
+\begin_inset Formula $\pi_{2}:X\times Y\to Y$
+\end_inset
+
+ dadas por 
+\begin_inset Formula $\pi_{1}(a,b):=a$
+\end_inset
+
+ y 
+\begin_inset Formula $\pi_{2}(a,b):=b$
+\end_inset
+
+ son continuas.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Dado un abierto 
+\begin_inset Formula $U\subseteq X$
+\end_inset
+
+, 
+\begin_inset Formula $\pi_{1}^{-1}(U)=U\times Y$
+\end_inset
+
+, y para 
+\begin_inset Formula $\pi_{2}$
+\end_inset
+
+ es análogo.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $a:X\to Y$
+\end_inset
+
+ y 
+\begin_inset Formula $b:X\to Z$
+\end_inset
+
+, 
+\begin_inset Formula $f:X\to Y\times Z$
+\end_inset
+
+ dada por 
+\begin_inset Formula $f(x):=(a(x),b(x))$
+\end_inset
+
+ es continua si y sólo si lo son 
+\begin_inset Formula $a$
+\end_inset
+
+ y 
+\begin_inset Formula $b$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Dado un abierto 
+\begin_inset Formula $U\subseteq Y$
+\end_inset
+
+, 
+\begin_inset Formula $a^{-1}(U)=\{x\in X:a(x)\in U\}=f^{-1}(U\times Y)$
+\end_inset
+
+, que es abierto por la hipótesis.
+ Para 
+\begin_inset Formula $b$
+\end_inset
+
+ es análogo.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Dado un elemento básico 
+\begin_inset Formula $U\times V\subseteq Y\times Z$
+\end_inset
+
+, 
+\begin_inset Formula $f^{-1}(U\times)=\{x\in X:a(x)\in U,b(x)\in V\}=a^{-1}(U)\cap b^{-1}(V)$
+\end_inset
+
+, que es abierto.
+\end_layout
+
+\end_deeper
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $k\in\{0,1,2\}$
+\end_inset
+
+ y 
+\begin_inset Formula $X,Y\neq\emptyset$
+\end_inset
+
+, 
+\begin_inset Formula $X\times Y$
+\end_inset
+
+ es 
+\begin_inset Formula $T_{k}$
+\end_inset
+
+ si y sólo si lo son 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $p,p'\in X$
+\end_inset
+
+, 
+\begin_inset Formula $p\neq p'$
+\end_inset
+
+, y 
+\begin_inset Formula $q\in Y$
+\end_inset
+
+ cualquiera, existen entornos de 
+\begin_inset Formula $(p,q)$
+\end_inset
+
+ y 
+\begin_inset Formula $(p',q)$
+\end_inset
+
+ que cumplen la propiedad y podemos suponer básicos.
+ Sean 
+\begin_inset Formula $U\times V$
+\end_inset
+
+ y 
+\begin_inset Formula $U'\times V'$
+\end_inset
+
+ estos entornos.
+ Para 
+\begin_inset Formula $k=2$
+\end_inset
+
+, si hubiera un 
+\begin_inset Formula $x\in U\cap U'$
+\end_inset
+
+ sería 
+\begin_inset Formula $(x,q)\in(U\times V)\cap(U'\times V')\#$
+\end_inset
+
+, y para 
+\begin_inset Formula $k\in\{0,1\}$
+\end_inset
+
+, si por ejemplo 
+\begin_inset Formula $(p',q)\notin U\times V$
+\end_inset
+
+, entonces 
+\begin_inset Formula $p'\notin U$
+\end_inset
+
+.
+ En cualquier caso podemos tomar 
+\begin_inset Formula $U$
+\end_inset
+
+ y 
+\begin_inset Formula $U'$
+\end_inset
+
+ y tenemos que 
+\begin_inset Formula $X$
+\end_inset
+
+ es 
+\begin_inset Formula $T_{k}$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $Y$
+\end_inset
+
+ es análogo.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $(p,q),(p',q')\in X\times Y$
+\end_inset
+
+, 
+\begin_inset Formula $(p,q)\neq(p',q')$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $p\neq p'$
+\end_inset
+
+, existen entornos 
+\begin_inset Formula $U\in{\cal E}(p)$
+\end_inset
+
+ y 
+\begin_inset Formula $V\in{\cal E}(q)$
+\end_inset
+
+ que cumplen la propiedad correspondiente según 
+\begin_inset Formula $k$
+\end_inset
+
+, y basta tomar 
+\begin_inset Formula $U\times Y$
+\end_inset
+
+ y 
+\begin_inset Formula $V\times Y$
+\end_inset
+
+, y de lo contrario 
+\begin_inset Formula $q\neq q'$
+\end_inset
+
+ y se hace el mismo argumento.
+\end_layout
+
+\end_deeper
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X\times Y$
+\end_inset
+
+ es conexo si y sólo si lo son 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Por la continuidad de las proyecciones.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Dado 
+\begin_inset Formula $y_{0}\in Y$
+\end_inset
+
+, 
+\begin_inset Formula $X\times\{y_{0}\}$
+\end_inset
+
+ es conexo por homeomorfismo con 
+\begin_inset Formula $X$
+\end_inset
+
+, y análogamente, para 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, 
+\begin_inset Formula $\{x\}\times Y$
+\end_inset
+
+ es conexo, y por el criterio del peine, 
+\begin_inset Formula $X\times Y=(X\times\{y_{0}\})\cup\bigcup_{x\in X}(\{x\}\times Y)$
+\end_inset
+
+ es conexo.
+\end_layout
+
+\end_deeper
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X\times Y$
+\end_inset
+
+ es 
+\begin_inset Formula $\text{1A}\mathbb{N}$
+\end_inset
+
+ si y sólo si lo son 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $(x,y)\in X\times Y$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ una base numerable de entornos numerable de 
+\begin_inset Formula $(x,y)$
+\end_inset
+
+, que podemos suponer básicos, para todo 
+\begin_inset Formula $U\in{\cal E}(x)$
+\end_inset
+
+ existe 
+\begin_inset Formula $V\times W\in{\cal B}$
+\end_inset
+
+ contenido en 
+\begin_inset Formula $U\times Y$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $V\subseteq U$
+\end_inset
+
+ y 
+\begin_inset Formula $\{V\}_{V\times W\in{\cal B}}$
+\end_inset
+
+ es una base numerable de entornos de 
+\begin_inset Formula $x\in X$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $Y$
+\end_inset
+
+ es análogo.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $(x,y)\in X\times Y$
+\end_inset
+
+, 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ una base numerable de entornos de 
+\begin_inset Formula $x$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal B}'$
+\end_inset
+
+ una base numerable de entornos de 
+\begin_inset Formula $y$
+\end_inset
+
+, dado 
+\begin_inset Formula $U\in{\cal E}(x,y)$
+\end_inset
+
+, existen 
+\begin_inset Formula $V\in{\cal E}(x)$
+\end_inset
+
+ y 
+\begin_inset Formula $W\in{\cal E}(y)$
+\end_inset
+
+ con 
+\begin_inset Formula $V\times W\subseteq U$
+\end_inset
+
+, 
+\begin_inset Formula $V_{0}\in{\cal B}$
+\end_inset
+
+ contenida en 
+\begin_inset Formula $V$
+\end_inset
+
+ y 
+\begin_inset Formula $W_{0}\in{\cal B}'$
+\end_inset
+
+ contenida en 
+\begin_inset Formula $W$
+\end_inset
+
+, luego 
+\begin_inset Formula $V_{0}\times W_{0}\subseteq U$
+\end_inset
+
+ y 
+\begin_inset Formula $\{B\times B'\}_{B\in{\cal B},B'\in{\cal B}'}$
+\end_inset
+
+ es base numerable de entornos de 
+\begin_inset Formula $X\times Y$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X\times Y$
+\end_inset
+
+ es 
+\begin_inset Formula $\text{2A}\mathbb{N}$
+\end_inset
+
+ si y sólo si lo son 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ una base numerable de 
+\begin_inset Formula $X\times Y$
+\end_inset
+
+, todo 
+\begin_inset Formula $U\in{\cal B}$
+\end_inset
+
+ se puede escribir como 
+\begin_inset Formula $U=:\bigcup_{i\in I}(V_{i}\times W_{i})$
+\end_inset
+
+, luego 
+\begin_inset Formula $\pi_{1}(U)=\bigcup_{i\in I}\pi_{1}(V_{i}\times W_{i})=\bigcup_{i\in I}V_{i}$
+\end_inset
+
+ es abierto.
+ Entonces, dado un abierto 
+\begin_inset Formula $A\subseteq X$
+\end_inset
+
+, existe 
+\begin_inset Formula ${\cal U}\subseteq{\cal B}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\bigcup_{U\in{\cal {\cal U}}}U=A\times Y$
+\end_inset
+
+, luego 
+\begin_inset Formula $A=\pi_{1}(A\times Y)=\bigcup_{U\in{\cal U}}\pi_{1}(U)$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\pi_{1}(U)\}_{U\in{\cal B}}$
+\end_inset
+
+ es base de 
+\begin_inset Formula $X$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $Y$
+\end_inset
+
+ es análogo.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal B}'$
+\end_inset
+
+ bases respectivas de 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+, todo abierto en 
+\begin_inset Formula $X\times Y$
+\end_inset
+
+ se puede escribir de la forma 
+\begin_inset Formula $\bigcup_{i\in I}(V_{i}\times W_{i})$
+\end_inset
+
+, pero si 
+\begin_inset Formula ${\cal V}_{i}\subseteq{\cal B}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal W}_{i}\subseteq{\cal B}'$
+\end_inset
+
+ son tales que 
+\begin_inset Formula $\bigcup{\cal V}_{i}=V_{i}$
+\end_inset
+
+ y 
+\begin_inset Formula $\bigcup{\cal W}'_{i}=W_{i}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $V_{i}\times W_{i}=\bigcup_{A\in{\cal V}_{i},B\in{\cal W}_{i}}(A\times B)$
+\end_inset
+
+, luego 
+\begin_inset Formula $\bigcup_{i\in I}(V_{i}\times W_{i})=\bigcup_{i\in I,A\in{\cal V}_{i},B\in{\cal W}_{i}}(A\times B)$
+\end_inset
+
+ y 
+\begin_inset Formula $\{A\times B\}_{A\in{\cal B},B\in{\cal B}'}$
+\end_inset
+
+ es base numerable de 
+\begin_inset Formula $X\times Y$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Lema del tubo:
+\series default
+ Si 
+\begin_inset Formula $Y$
+\end_inset
+
+ es compacto, sea 
+\begin_inset Formula $W\subseteq X\times Y$
+\end_inset
+
+ abierto con 
+\begin_inset Formula $\{x_{0}\}\times Y\subseteq W$
+\end_inset
+
+ para cierto 
+\begin_inset Formula $x_{0}$
+\end_inset
+
+, existe 
+\begin_inset Formula $U\in{\cal E}(x_{0})$
+\end_inset
+
+ tal que 
+\begin_inset Formula $U\times Y\subseteq W$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\series default
+ Para 
+\begin_inset Formula $y\in Y$
+\end_inset
+
+, existe un entorno básico 
+\begin_inset Formula $U_{y}\times V_{y}\in{\cal E}(x_{0},y)$
+\end_inset
+
+ contenido en 
+\begin_inset Formula $W$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\{V_{y}\}_{y\in Y}$
+\end_inset
+
+ es un recubrimiento abierto de 
+\begin_inset Formula $Y$
+\end_inset
+
+ que admite un subrecubrimiento finito 
+\begin_inset Formula $\{V_{y_{1}},\dots,V_{y_{n}}\}$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $U:=\bigcap_{k=1}^{n}U_{y_{k}}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $U\times Y=U\times\bigcup_{k}V_{y_{k}}\subseteq\bigcup_{k}(U_{y_{k}}\times V_{y_{k}})\subseteq W$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Tychonov:
+\series default
+ 
+\begin_inset Formula $X\times Y$
+\end_inset
+
+ es compacto si y sólo si lo son 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+.
+ Por ejemplo, el cilindro 
+\begin_inset Formula $C\cong\mathbb{S}^{1}\times[0,1]$
+\end_inset
+
+ y el toro 
+\begin_inset Formula $\mathbb{T}\cong\mathbb{S}^{1}\times\mathbb{S}^{1}$
+\end_inset
+
+ son compactos.
+\begin_inset Note Comment
+status open
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $\{A_{i}\}_{i\in I}$
+\end_inset
+
+ un recubrimiento de 
+\begin_inset Formula $X$
+\end_inset
+
+, 
+\begin_inset Formula $\{A_{i}\times Y\}_{i\in I}$
+\end_inset
+
+ es un recubrimiento de 
+\begin_inset Formula $X\times Y$
+\end_inset
+
+ que admite un subrecubrimiento finito 
+\begin_inset Formula $\{A_{i_{1}}\times Y,\dots,A_{i_{n}}\times Y\}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\{A_{i_{1}},\dots,A_{i_{n}}\}$
+\end_inset
+
+ es un subrecubrimiento finito de 
+\begin_inset Formula $\{A_{i}\}_{i\in I}$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $Y$
+\end_inset
+
+ es análogo.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $\{A_{i}\}_{i\in I}$
+\end_inset
+
+ un recubrimiento de 
+\begin_inset Formula $X\times Y$
+\end_inset
+
+, que por ahora supondremos formado por elementos básicos 
+\begin_inset Formula $A_{i}=:U_{i}\times V_{i}$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, sea 
+\begin_inset Formula $I_{x}:=\{i\in I:x\in U_{i}\}$
+\end_inset
+
+, 
+\begin_inset Formula $\bigcup_{i\in I_{x}}V_{i}=Y$
+\end_inset
+
+, luego 
+\begin_inset Formula $\{V_{i}\}_{i\in I_{x}}$
+\end_inset
+
+ es un recubrimiento abierto de 
+\begin_inset Formula $Y$
+\end_inset
+
+ que admite un subrecubrimiento finito 
+\begin_inset Formula $\{A_{i_{x,1}},\dots,V_{i_{x,p_{x}}}\}$
+\end_inset
+
+.
+ Por el lema del tubo, como 
+\begin_inset Formula $\{x\}\times Y\subseteq\bigcup_{k=1}^{p_{x}}A_{i_{x,k}}$
+\end_inset
+
+, existe 
+\begin_inset Formula $W_{x}\in{\cal E}(x)$
+\end_inset
+
+ tal que 
+\begin_inset Formula $W_{x}\times Y\subseteq\bigcup_{k=1}^{p_{x}}A_{i_{x,k}}$
+\end_inset
+
+.
+ Así, 
+\begin_inset Formula $\{W_{x}\}_{x\in X}$
+\end_inset
+
+ es un recubrimiento de 
+\begin_inset Formula $X$
+\end_inset
+
+ que admite un subrecubrimiento finito 
+\begin_inset Formula $\{W_{x_{1}},\dots,W_{x_{n}}\}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\bigcup_{j=1}^{n}\bigcup_{k=1}^{p_{x_{j}}}A_{i_{x_{j},k}}\supseteq\bigcup_{j=1}^{n}(W_{x_{j}}\times Y)=X\times Y$
+\end_inset
+
+, y 
+\begin_inset Formula $\{A_{i_{x_{j},k}}\}_{j\in\{1,\dots,n\},k\in\{1,\dots,p_{x_{k}}\}}$
+\end_inset
+
+ es un subrecubrimiento finito de 
+\begin_inset Formula $\{A_{i}\}_{i\in I}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Espacios cociente
+\end_layout
+
+\begin_layout Standard
+Dado un espacio topológico 
+\begin_inset Formula $(X,{\cal T})$
+\end_inset
+
+ y una relación de equivalencia 
+\begin_inset Formula $\sim$
+\end_inset
+
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+, llamamos 
+\series bold
+topología cociente
+\series default
+ en 
+\begin_inset Formula $X/\sim$
+\end_inset
+
+ a 
+\begin_inset Formula $\{V\subseteq(X/\sim):p^{-1}(V)\in{\cal T}\}$
+\end_inset
+
+, donde 
+\begin_inset Formula $p:X\to X/\sim$
+\end_inset
+
+ es la 
+\series bold
+proyección canónica
+\series default
+ o 
+\series bold
+aplicación cociente
+\series default
+ 
+\begin_inset Formula $p(x):=\overline{x}:=[x]$
+\end_inset
+
+ que a cada 
+\begin_inset Formula $x$
+\end_inset
+
+ le asigna su clase de equivalencia u 
+\series bold
+órbita
+\series default
+.
+\end_layout
+
+\begin_layout Standard
+Toda aplicación cociente es continua, por lo que si 
+\begin_inset Formula $X$
+\end_inset
+
+ es compacto, conexo o conexo por caminos, 
+\begin_inset Formula $X/\sim$
+\end_inset
+
+ también.
+ Ser Hausdorff no se conserva, pues 
+\begin_inset Formula $\mathbb{R}\amalg\mathbb{R}$
+\end_inset
+
+ es Hausdorff pero 
+\begin_inset Formula $(\mathbb{R}\amalg\mathbb{R})/\sim$
+\end_inset
+
+ con 
+\begin_inset Formula $L(x)\sim L(y):\iff x=y$
+\end_inset
+
+, 
+\begin_inset Formula $R(x)\sim R(y):\iff x=y$
+\end_inset
+
+, 
+\begin_inset Formula $L(x)\sim R(y):\iff x=y\neq0$
+\end_inset
+
+, no lo es.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $A\subseteq X$
+\end_inset
+
+, llamamos 
+\begin_inset Formula $X/A:=X/\sim_{A}$
+\end_inset
+
+ donde 
+\begin_inset Formula $a\sim_{A}b:\iff a=b\lor a,b\in A$
+\end_inset
+
+.
+ En el espacio cociente, llamamos 
+\begin_inset Formula $*$
+\end_inset
+
+ a la clase 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $a$
+\end_inset
+
+ a la clase 
+\begin_inset Formula $\{a\}$
+\end_inset
+
+ para cada 
+\begin_inset Formula $a\in X\setminus A$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+Ejemplos:
+\end_layout
+
+\begin_layout Enumerate
+Sea 
+\begin_inset Formula $X:=\mathbb{D}^{2}$
+\end_inset
+
+, 
+\begin_inset Formula $X/\partial X\cong\mathbb{S}^{2}$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Vemos que 
+\begin_inset Formula $\partial X=\mathbb{S}^{1}$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $f:\mathbb{D}^{2}/\mathbb{S}^{1}\to\mathbb{S}^{2}$
+\end_inset
+
+ dada por 
+\begin_inset Formula 
+\[
+f(x,y):=\left(x\frac{\sin(\pi\Vert(x,y)\Vert)}{\Vert(x,y)\Vert},y\frac{\sin(\pi\Vert(x,y)\Vert)}{\Vert(x,y)\Vert},\cos(\pi\Vert(x,y)\Vert)\right)
+\]
+
+\end_inset
+
+para 
+\begin_inset Formula $(x,y)\in B(0,1)\setminus\{0\}$
+\end_inset
+
+, 
+\begin_inset Formula $f(0):=(0,0,1)$
+\end_inset
+
+ y 
+\begin_inset Formula $f(*):=(0,0,-1)$
+\end_inset
+
+.
+ Es claro que 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $B(0,1)\setminus\{0\}$
+\end_inset
+
+.
+ Para ver que lo es en 0 y en 
+\begin_inset Formula $*$
+\end_inset
+
+, vemos que
+\begin_inset Formula 
+\begin{align*}
+\lim_{(x,y)\to0}f(x,y) & =\lim_{(x,y)\to0}\left(x\frac{\sin(\pi\Vert(x,y)\Vert)}{\Vert(x,y)\Vert},y\frac{\sin(\pi\Vert(x,y)\Vert)}{\Vert(x,y)\Vert},\cos(\pi\Vert(x,y)\Vert)\right)\\
+ & =\lim_{(x,y)\to0}(\pi x,\pi y,\cos(\pi\Vert(x,y)\Vert))=(0,0,1);\\
+\lim_{\Vert(x,y)\Vert\to1}f(x,y) & =\lim_{\Vert(x,y)\Vert\to1}\left(x\frac{\sin(\pi\Vert(x,y)\Vert)}{\Vert(x,y)\Vert},y\frac{\sin(\pi\Vert(x,y)\Vert)}{\Vert(x,y)\Vert},\cos(\pi\Vert(x,y)\Vert)\right)=(0,0,-1).
+\end{align*}
+
+\end_inset
+
+Sea 
+\begin_inset Formula $(a,b,c)\in\mathbb{S}^{1}\setminus\{(0,0,1),(0,0,-1)\}$
+\end_inset
+
+, queremos encontrar 
+\begin_inset Formula $(x,y)$
+\end_inset
+
+ con 
+\begin_inset Formula $f(x,y)=(a,b,c)$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $c=\cos(\pi\Vert(x,y)\Vert)$
+\end_inset
+
+, tenemos 
+\begin_inset Formula $\Vert(x,y)\Vert=\frac{1}{\pi}\arccos c\in(0,1)$
+\end_inset
+
+.
+ De aquí podemos hallar 
+\begin_inset Formula $\frac{\sin(\pi\Vert(x,y)\Vert)}{\Vert(x,y)\Vert}$
+\end_inset
+
+ y despejar 
+\begin_inset Formula $x$
+\end_inset
+
+ e 
+\begin_inset Formula $y$
+\end_inset
+
+, con 
+\begin_inset Formula $x=a\frac{\Vert(x,y)\Vert}{\sin(\pi\Vert(x,y)\Vert)}=\frac{a}{\pi}\frac{\arccos c}{\sin\arccos c}$
+\end_inset
+
+ y, análogamente, 
+\begin_inset Formula $y=\frac{b\arccos c}{\pi\sin\arccos c}$
+\end_inset
+
+.
+ Para ver que efectivamente 
+\begin_inset Formula $(x,y)\in B(0,1)\setminus\{0\}$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\left\Vert \left(\frac{a\arccos c}{\pi\sin\arccos c},\frac{b\arccos c}{\pi\sin\arccos c}\right)\right\Vert ^{2}=\frac{(a^{2}+b^{2})\arccos^{2}c}{\pi^{2}\sin^{2}\arccos c}=\frac{(1-c^{2})\arccos^{2}c}{\pi^{2}(1-c^{2})}\in(0,1).
+\]
+
+\end_inset
+
+Por tanto 
+\begin_inset Formula $f$
+\end_inset
+
+ es biyectiva, y claramente las fórmulas dadas para 
+\begin_inset Formula $x$
+\end_inset
+
+ e 
+\begin_inset Formula $y$
+\end_inset
+
+ son continuas.
+ Para ver que 
+\begin_inset Formula $f^{-1}$
+\end_inset
+
+ es continua también en 0 y 
+\begin_inset Formula $*$
+\end_inset
+
+, tomando límites,
+\begin_inset Formula 
+\begin{align*}
+\lim_{(a,b,c)\to(0,0,1)}\left\Vert f^{-1}(a,b,c)\right\Vert  & =\lim_{c\to1}\sqrt{\frac{\arccos^{2}c}{\pi^{2}}}=\lim_{c\to1}\left|\frac{\arccos c}{\pi}\right|=\frac{0}{\pi}=0;\\
+\lim_{(a,b,c)\to(0,0,-1)}\left\Vert f^{-1}(a,b,c)\right\Vert  & =\lim_{c\to-1}\left|\frac{\arccos c}{\pi}\right|=\left|\frac{\pi}{\pi}\right|=1.
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $n\geq1$
+\end_inset
+
+, sea 
+\begin_inset Formula $X:=[0,1]^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $X/\partial X\cong\mathbb{S}^{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $X:=\mathbb{R}^{3}\setminus\{0\}$
+\end_inset
+
+ y 
+\begin_inset Formula $x\sim y:\iff\exists\lambda\in\mathbb{R}:y=\lambda x$
+\end_inset
+
+, 
+\begin_inset Formula $X/\sim$
+\end_inset
+
+ es homeomorfo al plano proyectivo 
+\begin_inset Formula $\mathbb{RP}^{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Además, sea 
+\begin_inset Formula $X:=[0,1]\times[0,1]$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $(x,y)\sim(x',y'):\iff x-x'\in\mathbb{Z}\land y=y'$
+\end_inset
+
+, 
+\begin_inset Formula $X/\sim$
+\end_inset
+
+ es homeomorfo al cilindro.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $(x,y)\sim(x',y'):\iff(x,y)=(x',y')\lor(x-x'\in\{\pm1\}\land y=1-y')$
+\end_inset
+
+, 
+\begin_inset Formula $X/\sim$
+\end_inset
+
+ es homeomorfo a la cinta de Möbius.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $(x,y)\sim(x',y'):\iff(x-x'\in\mathbb{Z}\land y=y')\lor(x=1-x'\land y-y'\in\{\pm1\})$
+\end_inset
+
+, 
+\begin_inset Formula $X/\sim$
+\end_inset
+
+ es homeomorfo a la botella de Klein.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, sean 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ espacios topológicos, 
+\begin_inset Formula $\sim$
+\end_inset
+
+ una relación de equivalencia en 
+\begin_inset Formula $X$
+\end_inset
+
+ y 
+\begin_inset Formula $g:X\to Y$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall x,y\in X,(x\sim y\implies g(x)=g(y))$
+\end_inset
+
+, 
+\begin_inset Formula $g$
+\end_inset
+
+ induce una única función 
+\begin_inset Formula $f:{X/\sim}\to Y$
+\end_inset
+
+ tal que 
+\begin_inset Formula $f\circ p=g$
+\end_inset
+
+, y 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua si y sólo si lo es 
+\begin_inset Formula $g$
+\end_inset
+
+.
+ Por tanto existe una biyección entre las funciones continuas 
+\begin_inset Formula ${X/\sim}\to Y$
+\end_inset
+
+ y las funciones continuas 
+\begin_inset Formula $X\to Y$
+\end_inset
+
+ constantes en las órbitas.
+\end_layout
+
+\begin_layout Standard
+Una relación de equivalencia 
+\begin_inset Formula $\sim$
+\end_inset
+
+ en 
+\begin_inset Formula $(X,{\cal T})$
+\end_inset
+
+ es 
+\series bold
+abierta
+\series default
+ si 
+\begin_inset Formula $\forall U\in{\cal T},p^{-1}(p(U))\in{\cal T}$
+\end_inset
+
+.
+ Entonces:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\sim$
+\end_inset
+
+ es abierta si y sólo si 
+\begin_inset Formula $p:X\to X/\sim$
+\end_inset
+
+ es abierta.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\sim$
+\end_inset
+
+ es abierta y 
+\begin_inset Formula $X$
+\end_inset
+
+ es 
+\begin_inset Formula $\text{2A}\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $X/\sim$
+\end_inset
+
+ es 
+\begin_inset Formula $\text{2A}\mathbb{N}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\sim$
+\end_inset
+
+ es abierta, 
+\begin_inset Formula $X/\sim$
+\end_inset
+
+ es Hausdorff si y sólo si 
+\begin_inset Formula $\{(x,y)\in X\times X:x\sim y\}$
+\end_inset
+
+ es cerrado en 
+\begin_inset Formula $X\times X$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document