diff options
Diffstat (limited to 'ts')
| -rw-r--r-- | ts/n1.lyx | 26 | ||||
| -rw-r--r-- | ts/n2.lyx | 16 | ||||
| -rw-r--r-- | ts/n3.lyx | 22 | ||||
| -rw-r--r-- | ts/n4.lyx | 2 | ||||
| -rw-r--r-- | ts/n6.lyx | 34 |
5 files changed, 66 insertions, 34 deletions
@@ -268,7 +268,7 @@ entorno \end_inset es un elemento de -\begin_inset Formula ${\cal E}(x):=\{U\in{\cal T}:x\in{\cal U}\}$ +\begin_inset Formula ${\cal E}(x):=\{U\in{\cal T}\mid x\in{\cal U}\}$ \end_inset . @@ -459,7 +459,7 @@ abierta a \begin_inset Formula \[ -B_{d}(x,\delta):=\{y\in X:d(x,y)<\varepsilon\}. +B_{d}(x,\delta):=\{y\in X\mid d(x,y)<\varepsilon\}. \] \end_inset @@ -485,7 +485,7 @@ inducida \end_inset a la topología -\begin_inset Formula ${\cal T}_{d}:=\{A\in X:\forall x\in A,\exists\delta>0:B_{d}(x,\delta)\subseteq A\}$ +\begin_inset Formula ${\cal T}_{d}:=\{A\in X\mid \forall x\in A,\exists\delta>0\mid B_{d}(x,\delta)\subseteq A\}$ \end_inset . @@ -578,7 +578,7 @@ La -esfera \series default , -\begin_inset Formula $\mathbb{S}^{n}(r):=\{(x_{1},\dots,x_{n+1})\in\mathbb{R}^{n+1}:x_{1}^{2}+\dots+x_{n+1}^{2}=r^{2}\}$ +\begin_inset Formula $\mathbb{S}^{n}(r):=\{(x_{1},\dots,x_{n+1})\in\mathbb{R}^{n+1}\mid x_{1}^{2}+\dots+x_{n+1}^{2}=r^{2}\}$ \end_inset . @@ -630,7 +630,7 @@ El cilindro \series default , -\begin_inset Formula $C:=\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}=1,0\leq z\leq1\}$ +\begin_inset Formula $C:=\{(x,y,z)\in\mathbb{R}^{3}\mid x^{2}+y^{2}=1,0\leq z\leq1\}$ \end_inset , cono de rotación sobre el eje @@ -666,7 +666,7 @@ El toro \series default , -\begin_inset Formula $\mathbb{T}:=\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}+z^{2}-4\sqrt{x^{2}+y^{2}}+3=0\}$ +\begin_inset Formula $\mathbb{T}:=\{(x,y,z)\in\mathbb{R}^{3}\mid x^{2}+y^{2}+z^{2}-4\sqrt{x^{2}+y^{2}}+3=0\}$ \end_inset , cono de rotación sobre el eje @@ -674,7 +674,7 @@ toro \end_inset de -\begin_inset Formula $\{(x,0,z):(x-2)^{2}+z^{2}=1\}$ +\begin_inset Formula $\{(x,0,z)\mid (x-2)^{2}+z^{2}=1\}$ \end_inset . @@ -695,7 +695,7 @@ status open \end_inset Tenemos -\begin_inset Formula $\{(x,0,z):(x-2)^{2}+z^{2}=1\}=\{\alpha(s):=(\cos s+2,0,\sin s)\}_{s\in[0,2\pi]}$ +\begin_inset Formula $\{(x,0,z)\mid (x-2)^{2}+z^{2}=1\}=\{\alpha(s)\mid =(\cos s+2,0,\sin s)\}_{s\in[0,2\pi]}$ \end_inset , luego el cono de rotación es @@ -1056,7 +1056,7 @@ Como los abiertos en \end_inset , -\begin_inset Formula $s^{-1}((a,b))=\{(x,y):a<s(x,y)=x+y<b\}=\{(x,y):a-x<y<b-x\}$ +\begin_inset Formula $s^{-1}((a,b))=\{(x,y)\mid a<s(x,y)=x+y<b\}=\{(x,y)\mid a-x<y<b-x\}$ \end_inset . @@ -1135,7 +1135,7 @@ Dado \end_inset , queremos ver que -\begin_inset Formula $p^{-1}((a,b))=\{(x,y):a<p(x,y)=xy<b\}$ +\begin_inset Formula $p^{-1}((a,b))=\{(x,y)\mid a<p(x,y)=xy<b\}$ \end_inset es abierto. @@ -1217,7 +1217,7 @@ Basta ver que, dada una bola , su inversa es un abierto. Tenemos -\begin_inset Formula $d^{-1}(B_{d_{\infty}}(y,r))=\{x:d_{\infty}((x,\dots,x),y)<r\}=\{t:|x-y_{1}|,\dots,|x-y_{n}|<r\}$ +\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 @@ -2043,7 +2043,7 @@ topología generada \end_inset a -\begin_inset Formula ${\cal T}_{{\cal B}}:=\{U\subseteq X:\forall x\in U,\exists B\in{\cal B}:x\in B\subseteq U\}$ +\begin_inset Formula ${\cal T}_{{\cal B}}:=\{U\subseteq X\mid \forall x\in U,\exists B\in{\cal B}\mid x\in B\subseteq U\}$ \end_inset , y se tiene que @@ -2456,7 +2456,7 @@ Dada una base \end_inset numerable, -\begin_inset Formula ${\cal B}_{x}:=\{B\in{\cal B}:x\in B\}$ +\begin_inset Formula ${\cal B}_{x}:=\{B\in{\cal B}\mid x\in B\}$ \end_inset es base de entornos de @@ -1125,7 +1125,7 @@ Ejemplos de conexión \begin_layout Enumerate La hipérbola -\begin_inset Formula $\{(x,y)\in\mathbb{R}^{2}:x^{2}-y^{2}=1\}$ +\begin_inset Formula $\{(x,y)\in\mathbb{R}^{2}\mid x^{2}-y^{2}=1\}$ \end_inset no es conexa. @@ -1134,11 +1134,11 @@ status open \begin_layout Plain Layout Sean -\begin_inset Formula $U:=\{(x,y):x>0\}$ +\begin_inset Formula $U:=\{(x,y)\mid x>0\}$ \end_inset , -\begin_inset Formula $V:=\{(x,y):x<0\}$ +\begin_inset Formula $V:=\{(x,y)\mid x<0\}$ \end_inset e @@ -1150,7 +1150,7 @@ Sean \end_inset , luego -\begin_inset Formula $Y\subseteq U\cap V=\{(x,y):x\neq0\}$ +\begin_inset Formula $Y\subseteq U\cap V=\{(x,y)\mid x\neq0\}$ \end_inset ; @@ -1351,7 +1351,7 @@ La función status open \begin_layout Plain Layout -\begin_inset Formula ${\cal GL}(3,\mathbb{R})=\{A\in{\cal M}_{3}(\mathbb{R}):\det A\neq0\}$ +\begin_inset Formula ${\cal GL}(3,\mathbb{R})=\{A\in{\cal M}_{3}(\mathbb{R})\mid \det A\neq0\}$ \end_inset , luego existe la función continua @@ -1372,7 +1372,7 @@ status open . -\begin_inset Formula ${\cal O}(3,\mathbb{K})=\{A\in{\cal M}_{3}(\mathbb{R}):\det A\in\{-1,1\}\}$ +\begin_inset Formula ${\cal O}(3,\mathbb{K})=\{A\in{\cal M}_{3}(\mathbb{R})\mid \det A\in\{-1,1\}\}$ \end_inset , luego @@ -2393,7 +2393,7 @@ Sea . Ahora bien, -\begin_inset Formula $\{U_{\delta}:=(-\infty,z-\delta)\cup(z+\delta,+\infty)\}_{\delta>0}$ +\begin_inset Formula $\{U_{\delta}\mid =(-\infty,z-\delta)\cup(z+\delta,+\infty)\}_{\delta>0}$ \end_inset es un recubrimiento de @@ -2750,7 +2750,7 @@ Sea \end_inset continua, -\begin_inset Formula $\text{fix}f:=\{x\in X:f(x)=x\}$ +\begin_inset Formula $\text{fix}f:=\{x\in X\mid f(x)=x\}$ \end_inset es cerrado en @@ -309,7 +309,7 @@ Sean status open \begin_layout Plain Layout -\begin_inset Formula $\mathbb{S}^{n}\setminus\{N:=(0,\dots,0,1)\}$ +\begin_inset Formula $\mathbb{S}^{n}\setminus\{N\mid =(0,\dots,0,1)\}$ \end_inset y @@ -736,7 +736,7 @@ unión disjunta \end_inset son espacios topológicos, definimos la topología -\begin_inset Formula ${\cal T}_{X\amalg Y}:=\{U\subseteq X\amalg Y:\{x:(x,0)\in U\}\in{\cal T}_{X}\land\{y:(y,1)\in U\}\in{\cal T}_{Y}\}$ +\begin_inset Formula ${\cal T}_{X\amalg Y}:=\{U\subseteq X\amalg Y\mid \{x\mid (x,0)\in U\}\in{\cal T}_{X}\land\{y\mid (y,1)\in U\}\in{\cal T}_{Y}\}$ \end_inset . @@ -934,7 +934,7 @@ Sea \end_inset , -\begin_inset Formula $\{U_{i}:=\{x:(x,0)\in A_{i}\}\}_{i\in I}$ +\begin_inset Formula $\{U_{i}\mid =\{x\mid (x,0)\in A_{i}\}\}_{i\in I}$ \end_inset lo es de @@ -947,7 +947,7 @@ Sea . Del mismo modo -\begin_inset Formula $\{V_{j}:=\{y:(y,1)\in A_{i}\}\}_{j\in I}$ +\begin_inset Formula $\{V_{j}\mid =\{y\mid (y,1)\in A_{i}\}\}_{j\in I}$ \end_inset admite un subrecubrimiento finito @@ -1122,11 +1122,11 @@ Sean \end_inset disjuntos, y basta tomar -\begin_inset Formula $\{x:(x,0)\in U\}$ +\begin_inset Formula $\{x\mid (x,0)\in U\}$ \end_inset y -\begin_inset Formula $\{x:(x,0)\in V\}$ +\begin_inset Formula $\{x\mid (x,0)\in V\}$ \end_inset . @@ -1449,7 +1449,7 @@ Dado un abierto \end_inset , -\begin_inset Formula $a^{-1}(U)=\{x\in X:a(x)\in U\}=f^{-1}(U\times Y)$ +\begin_inset Formula $a^{-1}(U)=\{x\in X\mid a(x)\in U\}=f^{-1}(U\times Y)$ \end_inset , que es abierto por la hipótesis. @@ -1479,7 +1479,7 @@ Dado un elemento básico \end_inset , -\begin_inset Formula $f^{-1}(U\times)=\{x\in X:a(x)\in U,b(x)\in V\}=a^{-1}(U)\cap b^{-1}(V)$ +\begin_inset Formula $f^{-1}(U\times)=\{x\in X\mid a(x)\in U,b(x)\in V\}=a^{-1}(U)\cap b^{-1}(V)$ \end_inset , que es abierto. @@ -2269,7 +2269,7 @@ Sean \end_inset , sea -\begin_inset Formula $I_{x}:=\{i\in I:x\in U_{i}\}$ +\begin_inset Formula $I_{x}:=\{i\in I\mid x\in U_{i}\}$ \end_inset , @@ -2360,7 +2360,7 @@ topología cociente \end_inset a -\begin_inset Formula $\{V\subseteq(X/\sim):p^{-1}(V)\in{\cal T}\}$ +\begin_inset Formula $\{V\subseteq(X/\sim)\mid p^{-1}(V)\in{\cal T}\}$ \end_inset , donde @@ -2832,7 +2832,7 @@ Si \end_inset es Hausdorff si y sólo si -\begin_inset Formula $\{(x,y)\in X\times X:x\sim y\}$ +\begin_inset Formula $\{(x,y)\in X\times X\mid x\sim y\}$ \end_inset es cerrado en @@ -747,7 +747,7 @@ El recíproco no se cumple: \begin_layout Enumerate La corona circular -\begin_inset Formula $\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}\in[0,1]\}$ +\begin_inset Formula $\{(x,y)\in\mathbb{R}^{2}\mid x^{2}+y^{2}\in[0,1]\}$ \end_inset es homotópicamente equivalente, pero no homeomorfa, a @@ -258,7 +258,7 @@ envoltura convexa , \begin_inset Formula \[ -\text{conv}W=\left\{ t_{1}v_{1}+\dots+t_{k}v_{k}:\sum_{i=1}^{k}t_{i}=1,t_{i}\in[0,1]\right\} . +\text{conv}W=\left\{ t_{1}v_{1}+\dots+t_{k}v_{k}\;\middle|\;\sum_{i=1}^{k}t_{i}=1,t_{i}\in[0,1]\right\} . \] \end_inset @@ -520,6 +520,22 @@ dimensión \end_layout \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{samepage} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard Ejemplos: \end_layout @@ -578,6 +594,22 @@ Añadir dibujos. \end_layout +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{samepage} +\end_layout + +\end_inset + + +\end_layout + \begin_layout Section Número de Euler \end_layout |