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-rw-r--r--ts/n3.lyx115
1 files changed, 33 insertions, 82 deletions
diff --git a/ts/n3.lyx b/ts/n3.lyx
index bf5be1d..45d5736 100644
--- a/ts/n3.lyx
+++ b/ts/n3.lyx
@@ -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.
generated by cgit v1.2.3 (git 2.46.0) at 2025-10-27 02:42:16 +0000