# 拓扑比较

## 定义

### 例子

${\displaystyle X}$  的拓扑里，最粗的是由空集和全集两个元素构成的：

${\displaystyle {\mathfrak {T}}=\{X,\,\varnothing \}}$

${\displaystyle {\mathfrak {T}}_{D}={\mathcal {P}}(X)}$

## 最粗拓扑

${\displaystyle \tau _{\mathcal {F}}=\bigcap {\bigg \{}{\mathfrak {T}}\,{\bigg |}\,({\mathfrak {T}}{\text{ is a topology of }}X)\wedge ({\mathcal {F}}\subseteq {\mathfrak {T}}){\bigg \}}}$

${\displaystyle O\in \tau _{\mathcal {F}}\Leftrightarrow (\forall {\mathfrak {T}})\left\{[\,({\mathfrak {T}}{\text{ is a topology of }}X)\wedge ({\mathcal {F}}\subseteq {\mathfrak {T}})\,]\Rightarrow (O\in {\mathfrak {T}})\right\}}$  (a)

(1) ${\displaystyle X,\,\varnothing \in \tau _{\mathcal {F}}}$

${\displaystyle {\mathfrak {T}}}$  的确是 ${\displaystyle X}$ 拓扑，那由拓扑的定义可以得到 ${\displaystyle X,\,\varnothing \in {\mathfrak {T}}}$  ，这样从式(a)右方就可以得到 ${\displaystyle X,\,\varnothing \in \tau _{\mathcal {F}}}$

(2) ${\displaystyle U,\,V\in \tau _{\mathcal {F}}}$ ${\displaystyle U\cap V\in \tau _{\mathcal {F}}}$

${\displaystyle U,\,V\in \tau _{\mathcal {F}}}$  ，从式(a)左方有：

${\displaystyle (\forall {\mathfrak {T}})\left\{[\,({\mathfrak {T}}{\text{ is a topology of }}X)\wedge ({\mathcal {F}}\subseteq {\mathfrak {T}})\,]\Rightarrow (U\in {\mathfrak {T}})\right\}}$
${\displaystyle (\forall {\mathfrak {T}})\left\{[\,({\mathfrak {T}}{\text{ is a topology of }}X)\wedge ({\mathcal {F}}\subseteq {\mathfrak {T}})\,]\Rightarrow (V\in {\mathfrak {T}})\right\}}$

${\displaystyle (\forall {\mathfrak {T}})\left\{[\,({\mathfrak {T}}{\text{ is a topology of }}X)\wedge ({\mathcal {F}}\subseteq {\mathfrak {T}})\,]\Rightarrow (U,\,V\in {\mathfrak {T}})\right\}}$

${\displaystyle (\forall {\mathfrak {T}})\left\{[\,({\mathfrak {T}}{\text{ is a topology of }}X)\wedge ({\mathcal {F}}\subseteq {\mathfrak {T}})\,]\Rightarrow (U\cap V\in {\mathfrak {T}})\right\}}$

(3) ${\displaystyle {\mathcal {G}}\subseteq \tau _{\mathcal {F}}}$ ${\displaystyle \bigcup {\mathcal {G}}\in \tau _{\mathcal {F}}}$

${\displaystyle {\mathcal {G}}\subseteq \tau _{\mathcal {F}}}$  ，那对任意 ${\displaystyle g\in {\mathcal {G}}}$  ，从式(a)左方有：

${\displaystyle (\forall {\mathfrak {T}})\left\{[\,({\mathfrak {T}}{\text{ is a topology of }}X)\wedge ({\mathcal {F}}\subseteq {\mathfrak {T}})\,]\Rightarrow (g\in {\mathfrak {T}})\right\}}$

${\displaystyle (\forall {\mathfrak {T}})\left\{[\,({\mathfrak {T}}{\text{ is a topology of }}X)\wedge ({\mathcal {F}}\subseteq {\mathfrak {T}})\,]\Rightarrow ({\mathcal {G}}\subseteq {\mathfrak {T}})\right\}}$

${\displaystyle (\forall {\mathfrak {T}})\left\{[\,({\mathfrak {T}}{\text{ is a topology of }}X)\wedge ({\mathcal {F}}\subseteq {\mathfrak {T}})\,]\Rightarrow (\bigcup {\mathcal {G}}\in {\mathfrak {T}})\right\}}$

${\displaystyle \tau _{\mathcal {F}}=\bigcap {\bigg \{}{\mathfrak {T}}\,{\bigg |}\,({\mathfrak {T}}{\text{ is a topology of }}X)\wedge ({\mathcal {F}}\subseteq {\mathfrak {T}}){\bigg \}}}$

## 另见

• 初拓扑－可使集合上的一组映射皆为连续的拓扑之中，最粗糙的拓扑。
• 终拓扑－可使集合上的一组映射皆为连续的拓扑之中，最精细的拓扑。

## 参考资料

1. ^ Munkres, James R. Topology 2nd. Upper Saddle River, NJ: Prentice Hall. 2000: 77–78. ISBN 0-13-181629-2.