diff options
Diffstat (limited to 'ts/n3.lyx')
| -rw-r--r-- | ts/n3.lyx | 115 |
1 files changed, 33 insertions, 82 deletions
@@ -714,7 +714,7 @@ Llamamos \series bold unión disjunta \series default - de dos objetos + de dos conjuntos \begin_inset Formula $X$ \end_inset @@ -722,45 +722,8 @@ unión disjunta \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 $(X\times\{0\})\cup(Y\times\{1\})$ + a +\begin_inset Formula $X\amalg Y:=(X\times\{0\})\cup(Y\times\{1\})$ \end_inset . @@ -773,7 +736,7 @@ unión disjunta \end_inset son espacios topológicos, definimos la topología -\begin_inset Formula ${\cal T}_{X\amalg Y}:=\{U\subseteq X\amalg Y:L^{-1}(U)\in{\cal T}_{X}\land R^{-1}(U)\in{\cal T}_{Y}\}$ +\begin_inset Formula ${\cal T}_{X\amalg Y}:=\{U\subseteq X\amalg Y:\{x:(x,0)\in U\}\in{\cal T}_{X}\land\{y:(y,1)\in U\}\in{\cal T}_{Y}\}$ \end_inset . @@ -785,11 +748,11 @@ Vemos que \end_inset es continua si y sólo si lo son -\begin_inset Formula $f\circ L$ +\begin_inset Formula $f|_{X\times\{0\}}$ \end_inset y -\begin_inset Formula $f\circ R$ +\begin_inset Formula $f|_{Y\times\{1\}}$ \end_inset , y que @@ -797,11 +760,11 @@ Vemos que \end_inset es continua si y sólo si -\begin_inset Formula $f|_{f^{-1}(L(X))}$ +\begin_inset Formula $f|_{f^{-1}(X\times\{1\})}$ \end_inset y -\begin_inset Formula $f|_{f^{-1}(R(Y))}$ +\begin_inset Formula $f|_{f^{-1}(Y\times\{0\})}$ \end_inset lo son. @@ -843,11 +806,11 @@ Sea \end_inset dada por -\begin_inset Formula $f(L(x)):=e^{x_{1}}v_{1}+\sum_{k=2}^{n}x_{k}v_{k}$ +\begin_inset Formula $f(x,0):=e^{x_{1}}v_{1}+\sum_{k=2}^{n}x_{k}v_{k}$ \end_inset y -\begin_inset Formula $f(R(x)):=-e^{x_{1}}v_{1}+\sum_{k=2}^{n}x_{k}v_{k}$ +\begin_inset Formula $f(y,0):=-e^{x_{1}}v_{1}+\sum_{k=2}^{n}x_{k}v_{k}$ \end_inset es un homeomorfismo. @@ -861,8 +824,8 @@ Sea \begin_inset Formula \[ f^{-1}\left(\sum_{k=1}^{n}x_{k}v_{k}\right)=\begin{cases} -L(\log x_{1},x_{2},\dots,x_{n}) & \text{si }x_{1}>0,\\ -R(\log(-x_{1}),x_{2},\dots,x_{n}) & \text{si }x_{1}<0, +((\log x_{1},x_{2},\dots,x_{n}),0), & x_{1}>0;\\ +((\log(-x_{1}),x_{2},\dots,x_{n}),1), & x_{1}<0; \end{cases} \] @@ -890,11 +853,11 @@ Basta tomar el homeomorfismo \end_inset dado por -\begin_inset Formula $f(L(A))=A$ +\begin_inset Formula $f(A,0)=A$ \end_inset , -\begin_inset Formula $f(R(A))=-A$ +\begin_inset Formula $f(A,1)=-A$ \end_inset , y @@ -971,7 +934,7 @@ Sea \end_inset , -\begin_inset Formula $\{L^{-1}(A_{i})\}_{i\in I}$ +\begin_inset Formula $\{U_{i}:=\{x:(x,0)\in A_{i}\}\}_{i\in I}$ \end_inset lo es de @@ -979,16 +942,16 @@ Sea \end_inset y por tanto admite un subrecubrimiento finito -\begin_inset Formula $L^{-1}(A_{i_{1}}),\dots,L^{-1}(A_{i_{n}})$ +\begin_inset Formula $U_{i_{1}},\dots,U_{i_{n}}$ \end_inset . Del mismo modo -\begin_inset Formula $\{R^{-1}(A_{i})\}_{i\in I}$ +\begin_inset Formula $\{V_{j}:=\{y:(y,1)\in A_{i}\}\}_{j\in I}$ \end_inset admite un subrecubrimiento finito -\begin_inset Formula $R^{-1}(A_{j_{1}}),\dots,R^{-1}(A_{j_{m}})$ +\begin_inset Formula $V_{j_{1}},\dots,V_{j_{m}}$ \end_inset de @@ -1029,7 +992,7 @@ Sea \end_inset , -\begin_inset Formula $\{L(A_{i})\}_{i\in I}\cup Y$ +\begin_inset Formula $\{A_{i}\times\{0\}\}_{i\in I}\cup(Y\times\{1\})$ \end_inset es un recubrimiento por abiertos de @@ -1037,7 +1000,7 @@ Sea \end_inset que admite pues un subrecubrimiento finito -\begin_inset Formula $L(A_{1}),\dots,L(A_{n}),Y$ +\begin_inset Formula $A_{1}\times\{0\},\dots,A_{n}\times\{0\},Y\times\{1\}$ \end_inset , con lo que @@ -1092,23 +1055,19 @@ status open \end_inset Sean -\begin_inset Formula $p,q\in X\amalg Y$ +\begin_inset Formula $(p,i),(q,j)\in X\amalg Y$ \end_inset , -\begin_inset Formula $p\neq q$ +\begin_inset Formula $(p,i)\neq(q,j)$ \end_inset . Si -\begin_inset Formula $p,q\in L(X)$ -\end_inset - - o -\begin_inset Formula $p,q\in R(Y)$ +\begin_inset Formula $i=j$ \end_inset - basta tomar los abiertos en +, basta tomar los abiertos en \begin_inset Formula $X$ \end_inset @@ -1117,20 +1076,12 @@ Sean \end_inset . - Si -\begin_inset Formula $p\in L(X)$ -\end_inset - - y -\begin_inset Formula $q\in R(Y)$ + De lo contrario basta tomar +\begin_inset Formula $X\times\{0\}$ \end_inset -, basta tomar -\begin_inset Formula $L(X)$ -\end_inset - - y -\begin_inset Formula $R(Y)$ + e +\begin_inset Formula $Y\times\{1\}$ \end_inset . @@ -1163,19 +1114,19 @@ Sean \end_inset entornos respectivos de -\begin_inset Formula $p$ +\begin_inset Formula $(p,0)$ \end_inset y -\begin_inset Formula $q$ +\begin_inset Formula $(q,0)$ \end_inset disjuntos, y basta tomar -\begin_inset Formula $U\cap X$ +\begin_inset Formula $\{x:(x,0)\in U\}$ \end_inset y -\begin_inset Formula $V\cap X$ +\begin_inset Formula $\{x:(x,0)\in V\}$ \end_inset . @@ -1206,7 +1157,7 @@ Si status open \begin_layout Plain Layout -\begin_inset Formula $\{L(X),R(Y)\}$ +\begin_inset Formula $\{X\times\{0\},Y\times\{1\}\}$ \end_inset es una separación por abiertos. |