# 积空间

## 無窮積空間

${\displaystyle I\,{\overset {x}{\cong }}\,{\mathcal {X}}}$

${\displaystyle I\,{\overset {\tau }{\cong }}\,{\mathcal {T}}}$
${\displaystyle (\forall i\in I)[\tau (i){\text{ is topology of }}x(i)]}$

${\displaystyle \tau _{\pi }}$ 就是无穷乘积 ${\displaystyle \prod _{x}{\mathcal {X}}}$滿足需求的那個拓撲，那對於任意指標 ${\displaystyle j\in I}$ ，以下的第 ${\displaystyle j}$ 投影映射：

${\displaystyle \pi _{j}:\prod _{x}{\mathcal {X}}\to x(j)}$
${\displaystyle \left(\forall f\in \prod _{x}{\mathcal {X}}\right)[\pi _{j}(f)=f(j)]}$

${\displaystyle {(\pi _{j})}^{-1}(o_{j})\in \tau _{\pi }}$

${\displaystyle \left[f\in {(\pi _{j})}^{-1}(o_{j})\right]\Leftrightarrow \left\{\left(\forall f\in \prod _{x}{\mathcal {X}}\right)\wedge {\big [}f(j)\in o_{j}{\big ]}\right\}}$

${\displaystyle (\forall i\in I)\left\{(i\neq j)\Rightarrow {\big [}o(i)=x(i){\big ]}\right\}}$
${\displaystyle o(j)=o_{j}}$

${\displaystyle {(\pi _{j})}^{-1}(o_{j})=\prod _{o}o(I)\in \tau _{\pi }}$

${\displaystyle \left\{\prod _{o}o(I)\,{\Bigg |}\,(\exists j\in I)\left\{\left(o:I\to \bigcup {\mathcal {T}}\right)\wedge (\forall i\in I)\left\{(i\neq j)\Rightarrow {\big [}o(i)=x(i){\big ]}\right\}\wedge {\big [}o(j)\in \tau (j){\big ]}\right\}\right\}}$

${\displaystyle I\,{\overset {x}{\cong }}\,{\mathcal {X}}}$

${\displaystyle I\,{\overset {\tau }{\cong }}\,{\mathcal {T}}}$
${\displaystyle (\forall i\in I)[\tau (i){\text{ is topology of }}x(i)]}$

${\displaystyle {\mathcal {C}}=\left\{\prod _{o}o(I)\,{\Bigg |}\,(\exists j\in I)\left\{\left(o:I\to \bigcup {\mathcal {T}}\right)\wedge (\forall i\in I)\left\{(i\neq j)\Rightarrow {\big [}o(i)=x(i){\big ]}\right\}\wedge {\big [}o(j)\in \tau (j){\big ]}\right\}\right\}}$

## 有限積空間

${\displaystyle (X_{1},\,\tau _{1}),\,(X_{2},\,\tau _{2}),\,\dots ,\,(X_{n},\,\tau _{n})}$ 都是拓扑空间，若對任意自然数指標 ${\displaystyle j\leq n}$ 來說，以下的投影映射 ${\displaystyle \pi _{j}}$

${\displaystyle \pi _{j}:\prod _{i=1}^{n}X_{i}\to x(j)}$
${\displaystyle \pi _{j}(a_{1},\,a_{2},\,\dots ,\,a_{n})=a_{j}}$

${\displaystyle {(\pi _{j})}^{-1}(O_{j})\in \tau _{\pi }}$

${\displaystyle \left[p\in {(\pi _{j})}^{-1}(O_{j})\right]\Leftrightarrow (\forall i\in \mathbb {N} )\left\{(p_{i}\in X_{i})\wedge (p_{j}\in O_{j})\right\}}$

${\displaystyle {(\pi _{j})}^{-1}(O_{j})=X_{1}\times \dots \times X_{j-1}\times O_{j}\times X_{j+1}\times \dots \times X_{n}\in \tau _{\pi }}$${\displaystyle 1
${\displaystyle {(\pi _{j})}^{-1}(O_{1})=O_{1}\times X_{2}\times \dots \times X_{n}\in \tau _{\pi }}$

${\displaystyle V:\mathbb {N} \to \bigcup \{\tau _{1},\,\tau _{2},\,\dots ,\,\tau _{n}\}}$
${\displaystyle V(i)=V_{i}\in \tau _{i}}$

${\displaystyle \prod _{i=1}^{n}V_{i}=V_{1}\times \dots \times V_{n}\in \tau _{\pi }}$

${\displaystyle V_{i}=V_{1}\times \dots \times V_{n}=\bigcap _{j=1}^{n}X_{1}\times \dots \times X_{j-1}\times V_{j}\times X_{j+1}\times \dots \times X_{n}}$

${\displaystyle {\mathcal {C}}=\left\{\prod _{i=1}^{n}V_{i}\,{\Bigg |}\,V_{i}\in \tau _{i}\right\}}$