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-rw-r--r--ts/n1.lyx70
1 files changed, 35 insertions, 35 deletions
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
generated by cgit v1.2.3 (git 2.46.0) at 2025-10-30 23:57:46 +0000