diff options
Diffstat (limited to 'ts/n3.lyx')
| -rw-r--r-- | ts/n3.lyx | 70 |
1 files changed, 35 insertions, 35 deletions
@@ -165,11 +165,11 @@ Las funciones \end_inset dadas por -\begin_inset Formula $f(x):=\tan(\frac{\pi}{2}x)$ +\begin_inset Formula $f(x)\coloneqq \tan(\frac{\pi}{2}x)$ \end_inset y -\begin_inset Formula $g(x):=\frac{x}{1-x^{2}}$ +\begin_inset Formula $g(x)\coloneqq \frac{x}{1-x^{2}}$ \end_inset son homeomorfismos. @@ -177,7 +177,7 @@ Las funciones \begin_layout Enumerate Sea -\begin_inset Formula $N:=(0,\dots,0,1)\in\mathbb{R}^{n+1}$ +\begin_inset Formula $N\coloneqq (0,\dots,0,1)\in\mathbb{R}^{n+1}$ \end_inset , la @@ -202,7 +202,7 @@ status open \begin_layout Plain Layout Sean -\begin_inset Formula $\pi:=\mathbb{R}^{n}\times\{-1\}$ +\begin_inset Formula $\pi\coloneqq \mathbb{R}^{n}\times\{-1\}$ \end_inset y @@ -215,7 +215,7 @@ Sean \series default la proyección estereográfica. Si -\begin_inset Formula $y:=g(x)$ +\begin_inset Formula $y\coloneqq g(x)$ \end_inset , @@ -309,7 +309,7 @@ Sean status open \begin_layout Plain Layout -\begin_inset Formula $\mathbb{S}^{n}\setminus\{N\mid =(0,\dots,0,1)\}$ +\begin_inset Formula $\mathbb{S}^{n}\setminus\{N\coloneqq (0,\dots,0,1)\}$ \end_inset y @@ -321,7 +321,7 @@ status open \end_inset y -\begin_inset Formula $\pi:=\mathbb{R}^{n}\times\{-1\}$ +\begin_inset Formula $\pi\coloneqq \mathbb{R}^{n}\times\{-1\}$ \end_inset , son linealmente isomorfos, por lo que son homeomorfos y @@ -338,7 +338,7 @@ status open \begin_layout Enumerate El disco -\begin_inset Formula $\mathbb{D}^{n}:=\overline{B}_{d_{2}}(0;1)\subseteq\mathbb{R}^{n}$ +\begin_inset Formula $\mathbb{D}^{n}\coloneqq \overline{B}_{d_{2}}(0;1)\subseteq\mathbb{R}^{n}$ \end_inset es homeomorfo a @@ -355,7 +355,7 @@ Sea \end_inset dada por -\begin_inset Formula $f(x):=x\frac{\Vert x\Vert_{\infty}}{\Vert x\Vert_{2}}$ +\begin_inset Formula $f(x)\coloneqq x\frac{\Vert x\Vert_{\infty}}{\Vert x\Vert_{2}}$ \end_inset para @@ -363,7 +363,7 @@ Sea \end_inset y -\begin_inset Formula $f(0):=0$ +\begin_inset Formula $f(0)\coloneqq 0$ \end_inset , queremos ver que @@ -371,7 +371,7 @@ Sea \end_inset es biyectiva con inversa -\begin_inset Formula $g(y):=y\frac{\Vert y\Vert_{2}}{\Vert y\Vert_{\infty}}$ +\begin_inset Formula $g(y)\coloneqq y\frac{\Vert y\Vert_{2}}{\Vert y\Vert_{\infty}}$ \end_inset para @@ -723,7 +723,7 @@ unión disjunta \end_inset a -\begin_inset Formula $X\amalg Y:=(X\times\{0\})\cup(Y\times\{1\})$ +\begin_inset Formula $X\amalg Y\coloneqq (X\times\{0\})\cup(Y\times\{1\})$ \end_inset . @@ -736,7 +736,7 @@ unión disjunta \end_inset son espacios topológicos, definimos la topología -\begin_inset Formula ${\cal T}_{X\amalg Y}:=\{U\subseteq X\amalg Y\mid \{x\mid (x,0)\in U\}\in{\cal T}_{X}\land\{y\mid (y,1)\in U\}\in{\cal T}_{Y}\}$ +\begin_inset Formula ${\cal T}_{X\amalg Y}\coloneqq \{U\subseteq X\amalg Y\mid \{x\mid (x,0)\in U\}\in{\cal T}_{X}\land\{y\mid(y,1)\in U\}\in{\cal T}_{Y}\}$ \end_inset . @@ -806,11 +806,11 @@ Sea \end_inset dada por -\begin_inset Formula $f(x,0):=e^{x_{1}}v_{1}+\sum_{k=2}^{n}x_{k}v_{k}$ +\begin_inset Formula $f(x,0)\coloneqq e^{x_{1}}v_{1}+\sum_{k=2}^{n}x_{k}v_{k}$ \end_inset y -\begin_inset Formula $f(y,0):=-e^{x_{1}}v_{1}+\sum_{k=2}^{n}x_{k}v_{k}$ +\begin_inset Formula $f(y,0)\coloneqq -e^{x_{1}}v_{1}+\sum_{k=2}^{n}x_{k}v_{k}$ \end_inset es un homeomorfismo. @@ -934,7 +934,7 @@ Sea \end_inset , -\begin_inset Formula $\{U_{i}\mid =\{x\mid (x,0)\in A_{i}\}\}_{i\in I}$ +\begin_inset Formula $\{U_{i}\coloneqq \{x\mid (x,0)\in A_{i}\}\}_{i\in I}$ \end_inset lo es de @@ -947,7 +947,7 @@ Sea . Del mismo modo -\begin_inset Formula $\{V_{j}\mid =\{y\mid (y,1)\in A_{i}\}\}_{j\in I}$ +\begin_inset Formula $\{V_{j}\coloneqq \{y\mid (y,1)\in A_{i}\}\}_{j\in I}$ \end_inset admite un subrecubrimiento finito @@ -1257,7 +1257,7 @@ Demostración: \end_inset dada por -\begin_inset Formula $f((x_{1},\dots,x_{m}),(y_{1},\dots,y_{n})):=(x_{1},\dots,x_{m},y_{1},\dots,y_{n})$ +\begin_inset Formula $f((x_{1},\dots,x_{m}),(y_{1},\dots,y_{n}))\coloneqq (x_{1},\dots,x_{m},y_{1},\dots,y_{n})$ \end_inset es biyectiva. @@ -1278,11 +1278,11 @@ Demostración: \end_inset , sean -\begin_inset Formula $a:=d_{\infty}(z,x)<\varepsilon_{x}$ +\begin_inset Formula $a\coloneqq d_{\infty}(z,x)<\varepsilon_{x}$ \end_inset y -\begin_inset Formula $b:=d_{\infty}(w,y)<\delta_{y}$ +\begin_inset Formula $b\coloneqq d_{\infty}(w,y)<\delta_{y}$ \end_inset , la bola @@ -1312,11 +1312,11 @@ Demostración: \end_inset dada por -\begin_inset Formula $f(L(x)):=(x,-1)$ +\begin_inset Formula $f(L(x))\coloneqq (x,-1)$ \end_inset y -\begin_inset Formula $f(R(y)):=(y,1)$ +\begin_inset Formula $f(R(y))\coloneqq (y,1)$ \end_inset es biyectiva. @@ -1370,11 +1370,11 @@ proyecciones \end_inset dadas por -\begin_inset Formula $\pi_{1}(a,b):=a$ +\begin_inset Formula $\pi_{1}(a,b)\coloneqq a$ \end_inset y -\begin_inset Formula $\pi_{2}(a,b):=b$ +\begin_inset Formula $\pi_{2}(a,b)\coloneqq b$ \end_inset son continuas. @@ -1416,7 +1416,7 @@ Sean \end_inset dada por -\begin_inset Formula $f(x):=(a(x),b(x))$ +\begin_inset Formula $f(x)\coloneqq (a(x),b(x))$ \end_inset es continua si y sólo si lo son @@ -2144,7 +2144,7 @@ Demostración: . Sea -\begin_inset Formula $U:=\bigcap_{k=1}^{n}U_{y_{k}}$ +\begin_inset Formula $U\coloneqq \bigcap_{k=1}^{n}U_{y_{k}}$ \end_inset , entonces @@ -2269,7 +2269,7 @@ Sean \end_inset , sea -\begin_inset Formula $I_{x}:=\{i\in I\mid x\in U_{i}\}$ +\begin_inset Formula $I_{x}\coloneqq \{i\in I\mid x\in U_{i}\}$ \end_inset , @@ -2376,7 +2376,7 @@ proyección canónica aplicación cociente \series default -\begin_inset Formula $p(x):=\overline{x}:=[x]$ +\begin_inset Formula $p(x)\coloneqq \overline{x}\coloneqq [x]$ \end_inset que a cada @@ -2429,7 +2429,7 @@ Si \end_inset , llamamos -\begin_inset Formula $X/A:=X/\sim_{A}$ +\begin_inset Formula $X/A\coloneqq X/\sim_{A}$ \end_inset donde @@ -2467,7 +2467,7 @@ Ejemplos: \begin_layout Enumerate Sea -\begin_inset Formula $X:=\mathbb{D}^{2}$ +\begin_inset Formula $X\coloneqq \mathbb{D}^{2}$ \end_inset , @@ -2501,11 +2501,11 @@ para \end_inset , -\begin_inset Formula $f(0):=(0,0,1)$ +\begin_inset Formula $f(0)\coloneqq (0,0,1)$ \end_inset y -\begin_inset Formula $f(*):=(0,0,-1)$ +\begin_inset Formula $f(*)\coloneqq (0,0,-1)$ \end_inset . @@ -2631,7 +2631,7 @@ Para \end_inset , sea -\begin_inset Formula $X:=[0,1]^{n}$ +\begin_inset Formula $X\coloneqq [0,1]^{n}$ \end_inset , @@ -2643,7 +2643,7 @@ Para \begin_layout Enumerate Sean -\begin_inset Formula $X:=\mathbb{R}^{3}\setminus\{0\}$ +\begin_inset Formula $X\coloneqq \mathbb{R}^{3}\setminus\{0\}$ \end_inset y @@ -2663,7 +2663,7 @@ Sean \begin_layout Standard Además, sea -\begin_inset Formula $X:=[0,1]\times[0,1]$ +\begin_inset Formula $X\coloneqq [0,1]\times[0,1]$ \end_inset : |