From c34b47089a133e58032fe4ea52f61efacaf5f548 Mon Sep 17 00:00:00 2001
From: Juan Marin Noguera <juan@mnpi.eu>
Date: Sun, 4 Dec 2022 22:49:17 +0100
Subject: Oops

---
 ts/n1.lyx | 70 +++++++++++++++++++++++++++++++--------------------------------
 1 file changed, 35 insertions(+), 35 deletions(-)

(limited to 'ts/n1.lyx')

diff --git a/ts/n1.lyx b/ts/n1.lyx
index 4936758..5bb57f7 100644
--- a/ts/n1.lyx
+++ b/ts/n1.lyx
@@ -158,7 +158,7 @@ topología trivial
 indiscreta
 \series default
  a 
-\begin_inset Formula ${\cal T}_{\text{ind}}:=\{\emptyset,X\}$
+\begin_inset Formula ${\cal T}_{\text{ind}}\coloneqq \{\emptyset,X\}$
 \end_inset
 
  y 
@@ -166,7 +166,7 @@ indiscreta
 topología discreta
 \series default
  a 
-\begin_inset Formula ${\cal T}_{\text{dis}}:={\cal P}(X)$
+\begin_inset Formula ${\cal T}_{\text{dis}}\coloneqq {\cal P}(X)$
 \end_inset
 
 .
@@ -268,7 +268,7 @@ entorno
 \end_inset
 
  es un elemento de 
-\begin_inset Formula ${\cal E}(x):=\{U\in{\cal T}\mid x\in{\cal U}\}$
+\begin_inset Formula ${\cal E}(x)\coloneqq \{U\in{\cal T}\mid x\in{\cal U}\}$
 \end_inset
 
 .
@@ -343,7 +343,7 @@ En
 \end_inset
 
  tenemos la distancia usual 
-\begin_inset Formula $d_{u}(x,y):=|x-y|$
+\begin_inset Formula $d_{u}(x,y)\coloneqq |x-y|$
 \end_inset
 
 .
@@ -368,7 +368,7 @@ d_{p}(x,y):=\left(\sum_{k=1}^{n}d(x_{k},y_{k})^{p}\right)^{\frac{1}{p}}
 \end_inset
 
 , y 
-\begin_inset Formula $d_{\infty}(x,y):=\max_{k=1}^{n}d(x_{k},y_{k})$
+\begin_inset Formula $d_{\infty}(x,y)\coloneqq \max_{k=1}^{n}d(x_{k},y_{k})$
 \end_inset
 
 .
@@ -485,7 +485,7 @@ inducida
 \end_inset
 
  a la topología 
-\begin_inset Formula ${\cal T}_{d}:=\{A\in X\mid \forall x\in A,\exists\delta>0\mid B_{d}(x,\delta)\subseteq A\}$
+\begin_inset Formula ${\cal T}_{d}\coloneqq \{A\in X\mid \forall x\in A,\exists\delta>0\mid B_{d}(x,\delta)\subseteq A\}$
 \end_inset
 
 .
@@ -537,7 +537,7 @@ Dados un espacio topológico
 \end_inset
 
 , 
-\begin_inset Formula ${\cal T}_{Y}:=\{U\cap Y\}_{U\in{\cal T}}$
+\begin_inset Formula ${\cal T}_{Y}\coloneqq \{U\cap Y\}_{U\in{\cal T}}$
 \end_inset
 
  es una topología sobre 
@@ -578,7 +578,7 @@ La
 -esfera
 \series default
 , 
-\begin_inset Formula $\mathbb{S}^{n}(r):=\{(x_{1},\dots,x_{n+1})\in\mathbb{R}^{n+1}\mid x_{1}^{2}+\dots+x_{n+1}^{2}=r^{2}\}$
+\begin_inset Formula $\mathbb{S}^{n}(r)\coloneqq \{(x_{1},\dots,x_{n+1})\in\mathbb{R}^{n+1}\mid x_{1}^{2}+\dots+x_{n+1}^{2}=r^{2}\}$
 \end_inset
 
 .
@@ -610,7 +610,7 @@ El
 intervalo cerrado
 \series default
  
-\begin_inset Formula $I:=[0,1]\subseteq\mathbb{R}$
+\begin_inset Formula $I\coloneqq [0,1]\subseteq\mathbb{R}$
 \end_inset
 
  o el 
@@ -630,7 +630,7 @@ El
 cilindro
 \series default
 , 
-\begin_inset Formula $C:=\{(x,y,z)\in\mathbb{R}^{3}\mid x^{2}+y^{2}=1,0\leq z\leq1\}$
+\begin_inset Formula $C\coloneqq \{(x,y,z)\in\mathbb{R}^{3}\mid x^{2}+y^{2}=1,0\leq z\leq1\}$
 \end_inset
 
 , cono de rotación sobre el eje 
@@ -666,7 +666,7 @@ El
 toro
 \series default
 , 
-\begin_inset Formula $\mathbb{T}:=\{(x,y,z)\in\mathbb{R}^{3}\mid x^{2}+y^{2}+z^{2}-4\sqrt{x^{2}+y^{2}}+3=0\}$
+\begin_inset Formula $\mathbb{T}\coloneqq \{(x,y,z)\in\mathbb{R}^{3}\mid x^{2}+y^{2}+z^{2}-4\sqrt{x^{2}+y^{2}}+3=0\}$
 \end_inset
 
 , cono de rotación sobre el eje 
@@ -695,7 +695,7 @@ status open
 \end_inset
 
 Tenemos 
-\begin_inset Formula $\{(x,0,z)\mid (x-2)^{2}+z^{2}=1\}=\{\alpha(s)\mid =(\cos s+2,0,\sin s)\}_{s\in[0,2\pi]}$
+\begin_inset Formula $\{(x,0,z)\mid (x-2)^{2}+z^{2}=1\}=\{\alpha(s)\coloneqq (\cos s+2,0,\sin s)\}_{s\in[0,2\pi]}$
 \end_inset
 
 , luego el cono de rotación es 
@@ -820,7 +820,7 @@ La
 cinta de Möbius
 \series default
 , 
-\begin_inset Formula $M:=\{(\cos\theta(3-t\sin\frac{\theta}{2}),\sin\theta(3-t\sin\frac{\theta}{2}),t\cos\frac{\theta}{2})\}_{\theta\in[0,2\pi],t\in[-1,1]}$
+\begin_inset Formula $M\coloneqq \{(\cos\theta(3-t\sin\frac{\theta}{2}),\sin\theta(3-t\sin\frac{\theta}{2}),t\cos\frac{\theta}{2})\}_{\theta\in[0,2\pi],t\in[-1,1]}$
 \end_inset
 
 .
@@ -947,7 +947,7 @@ restricción del rango
 \end_inset
 
 , dada por 
-\begin_inset Formula $f'(x):=f(x)$
+\begin_inset Formula $f'(x)\coloneqq f(x)$
 \end_inset
 
 , es continua.
@@ -1039,7 +1039,7 @@ suma
 \end_inset
 
 , 
-\begin_inset Formula $s(x,y):=x+y$
+\begin_inset Formula $s(x,y)\coloneqq x+y$
 \end_inset
 
 , con la topología usual.
@@ -1065,7 +1065,7 @@ Como los abiertos en
 \end_inset
 
 , 
-\begin_inset Formula $t:=s(x_{0},y_{0})$
+\begin_inset Formula $t\coloneqq s(x_{0},y_{0})$
 \end_inset
 
  y 
@@ -1122,7 +1122,7 @@ producto
 \end_inset
 
 , 
-\begin_inset Formula $p(x,y):=xy$
+\begin_inset Formula $p(x,y)\coloneqq xy$
 \end_inset
 
 , con la topología usual.
@@ -1144,7 +1144,7 @@ Dado
 \end_inset
 
 , 
-\begin_inset Formula $t:=p(x_{0},y_{0})$
+\begin_inset Formula $t\coloneqq p(x_{0},y_{0})$
 \end_inset
 
 , 
@@ -1156,7 +1156,7 @@ Dado
 \end_inset
 
  y 
-\begin_inset Formula $\delta:=\min\{1,\frac{r}{|x_{0}|+|y_{0}|+1}\}$
+\begin_inset Formula $\delta\coloneqq \min\{1,\frac{r}{|x_{0}|+|y_{0}|+1}\}$
 \end_inset
 
 , para 
@@ -1199,7 +1199,7 @@ diagonal
 \end_inset
 
 , 
-\begin_inset Formula $d(x):=(x,\dots,x)$
+\begin_inset Formula $d(x)\coloneqq (x,\dots,x)$
 \end_inset
 
 , con la topología usual.
@@ -1217,7 +1217,7 @@ Basta ver que, dada una bola
 
 , su inversa es un abierto.
  Tenemos 
-\begin_inset Formula $d^{-1}(B_{d_{\infty}}(y,r))=\{x\mid d_{\infty}((x,\dots,x),y)<r\}=\{t\mid |x-y_{1}|,\dots,|x-y_{n}|<r\}$
+\begin_inset Formula $d^{-1}(B_{d_{\infty}}(y,r))=\{x\mid d_{\infty}((x,\dots,x),y)<r\}=\{t\mid|x-y_{1}|,\dots,|x-y_{n}|<r\}$
 \end_inset
 
 , pero 
@@ -1242,7 +1242,7 @@ Una función
 \end_inset
 
  es continua si y sólo si los componentes 
-\begin_inset Formula $f_{i}(x):=f(x)_{i}$
+\begin_inset Formula $f_{i}(x)\coloneqq f(x)_{i}$
 \end_inset
 
  lo son.
@@ -1352,15 +1352,15 @@ status open
 La función es continua porque lo es en cada componente, al serlo la suma
  y el producto.
  Dado 
-\begin_inset Formula $u:=(\cos\theta,x,y,z)$
+\begin_inset Formula $u\coloneqq (\cos\theta,x,y,z)$
 \end_inset
 
 , sean 
-\begin_inset Formula $v:=(x,y,z)$
+\begin_inset Formula $v\coloneqq (x,y,z)$
 \end_inset
 
  y 
-\begin_inset Formula $n:=\Vert v\Vert$
+\begin_inset Formula $n\coloneqq \Vert v\Vert$
 \end_inset
 
 .
@@ -1565,7 +1565,7 @@ C:=\left(\begin{array}{ccc}
 \begin_layout Enumerate
 Revertimos las dos rotaciones anteriores.
  Sea 
-\begin_inset Formula $t:=\sqrt{x^{2}+y^{2}}$
+\begin_inset Formula $t\coloneqq \sqrt{x^{2}+y^{2}}$
 \end_inset
 
 , 
@@ -2043,7 +2043,7 @@ topología generada
 \end_inset
 
  a 
-\begin_inset Formula ${\cal T}_{{\cal B}}:=\{U\subseteq X\mid \forall x\in U,\exists B\in{\cal B}\mid x\in B\subseteq U\}$
+\begin_inset Formula ${\cal T}_{{\cal B}}\coloneqq \{U\subseteq X\mid \forall x\in U,\exists B\in{\cal B}\mid x\in B\subseteq U\}$
 \end_inset
 
 , y se tiene que 
@@ -2075,7 +2075,7 @@ topología del límite inferior
 \end_inset
 
  generada por la base 
-\begin_inset Formula ${\cal B}_{\ell i}:=\{[a,b)\}_{a,b\in\mathbb{R};a<b}$
+\begin_inset Formula ${\cal B}_{\ell i}\coloneqq \{[a,b)\}_{a,b\in\mathbb{R};a<b}$
 \end_inset
 
 .
@@ -2107,7 +2107,7 @@ En efecto,
 \end_inset
 
 , pero tomando la base 
-\begin_inset Formula ${\cal B}_{u}:=\{(a,b)\}_{a,b\in\mathbb{R},a<b}$
+\begin_inset Formula ${\cal B}_{u}\coloneqq \{(a,b)\}_{a,b\in\mathbb{R},a<b}$
 \end_inset
 
  de la topología usual de 
@@ -2240,11 +2240,11 @@ Tomamos la base
 \end_inset
 
 , 
-\begin_inset Formula $(p_{n}:=\frac{\lceil an\rceil}{n})_{n\geq n_{0}}$
+\begin_inset Formula $(p_{n}\coloneqq \frac{\lceil an\rceil}{n})_{n\geq n_{0}}$
 \end_inset
 
  y 
-\begin_inset Formula $(q_{n}:=\frac{\lfloor bn\rfloor}{n})_{n\geq n_{0}}$
+\begin_inset Formula $(q_{n}\coloneqq \frac{\lfloor bn\rfloor}{n})_{n\geq n_{0}}$
 \end_inset
 
 , 
@@ -2294,7 +2294,7 @@ Sea
 \end_inset
 
 , sea 
-\begin_inset Formula $U_{x}:=[x,x+1)\in{\cal T}_{\ell i}$
+\begin_inset Formula $U_{x}\coloneqq [x,x+1)\in{\cal T}_{\ell i}$
 \end_inset
 
 , existe 
@@ -2409,7 +2409,7 @@ Ejemplos:
 status open
 
 \begin_layout Plain Layout
-\begin_inset Formula ${\cal B}_{x}:=\{[x,x+\frac{1}{n})\}_{n\in\mathbb{N}^{*}}$
+\begin_inset Formula ${\cal B}_{x}\coloneqq \{[x,x+\frac{1}{n})\}_{n\in\mathbb{N}^{*}}$
 \end_inset
 
 .
@@ -2430,7 +2430,7 @@ Todo espacio métrico
 status open
 
 \begin_layout Plain Layout
-\begin_inset Formula ${\cal B}_{x}:=\{B_{d}(x-\frac{1}{n},x+\frac{1}{n})\}_{n\in\mathbb{N}^{*}}$
+\begin_inset Formula ${\cal B}_{x}\coloneqq \{B_{d}(x-\frac{1}{n},x+\frac{1}{n})\}_{n\in\mathbb{N}^{*}}$
 \end_inset
 
 .
@@ -2456,7 +2456,7 @@ Dada una base
 \end_inset
 
  numerable, 
-\begin_inset Formula ${\cal B}_{x}:=\{B\in{\cal B}\mid x\in B\}$
+\begin_inset Formula ${\cal B}_{x}\coloneqq \{B\in{\cal B}\mid x\in B\}$
 \end_inset
 
  es base de entornos de 
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