# 初拓扑

## 定义

${\displaystyle I\,{\overset {Y}{\cong }}\,{\mathcal {Y}}}$
${\displaystyle I\,{\overset {\tau }{\cong }}\,{\mathfrak {T}}}$
${\displaystyle (\forall i\in I)\left[\tau _{i}{\text{ is a topology of }}Y_{i}\right]}$
${\displaystyle I\,{\overset {f}{\cong }}\,{\mathcal {F}}}$
${\displaystyle (\forall i\in I)\left(f_{i}:X\to Y_{i}\right)}$

${\displaystyle X}$  上關於 ${\displaystyle {\mathcal {F}}}$ 初拓扑 ${\displaystyle \tau _{\mathcal {F}}}$ ，定義為「對所有 ${\displaystyle i\in I}$ ${\displaystyle f(i)}$ ${\displaystyle \tau _{\mathcal {F}}}$  - ${\displaystyle \tau _{i}}$  连续」的最粗糙拓扑。

${\displaystyle {\mathcal {B}}:=\left\{B\,|\,(\exists i\in I)(\exists O\in \tau _{i})[B={f_{i}}^{-1}(O)]\right\}}$

${\displaystyle {\mathcal {B}}}$ ${\displaystyle X}$ 拓撲基，且 ${\displaystyle \tau _{\mathcal {F}}}$  就是由 ${\displaystyle {\mathcal {B}}}$  所生成的拓扑。

${\displaystyle (\forall i\in I)\{[X=f^{-1}(Y_{i})]\wedge (Y_{i}\in \tau _{i})\}}$

${\displaystyle \left(X=\bigcup \{X\}\right)\wedge (\{X\}\subseteq {\mathcal {B}})}$

${\displaystyle {f_{i}}^{-1}(V\cap W)={f_{i}}^{-1}(V)\cap {f_{i}}^{-1}(W)}$
${\displaystyle {f_{i}}^{-1}(V)\cap {f_{i}}^{-1}(W)\in {\mathcal {B}}}$

「對所有 ${\displaystyle i\in I}$  ，和所有 ${\displaystyle O\in \tau _{i}}$ ${\displaystyle {f_{i}}^{-1}(O)\in \tau }$

${\displaystyle {\mathcal {B}}\subseteq \tau }$

## 性质

### 特征性质

${\displaystyle Z}$ ${\displaystyle X}$ 的映射${\displaystyle g}$ 是连续的，当且仅当 ${\displaystyle f_{i}\circ g}$  是连续的。

### 从闭集分离点

${\displaystyle f_{i}:X\rightarrow Y_{i}}$ 从闭集分离点，如果${\displaystyle X}$ 中任意闭集${\displaystyle A}$ ，与任意不属于${\displaystyle A}$ 的点${\displaystyle x}$ ${\displaystyle \exists i\in I}$ ，使得
${\displaystyle f_{i}(x)\notin \operatorname {cl} (f_{i}(A))}$