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【拓扑空间】示例及详解1

news 2025/8/27 0:12:27

例1

度量空间 $(X,d)$ 的任意两球形邻域的交集是若干球形邻域的并集

Proof：

任取空间 $(X,d)$ 的两个球形邻域 $B(x_1,\varepsilon _1)$ 、 $B(x_2,\varepsilon _2)$ ，令 $U=B(x_1,\varepsilon _1)\cap B(x_2,\varepsilon _2)$

任取 $x\in U$ ,令 $\varepsilon_x=min\left \{ \varepsilon_1-d(x_1,x), \varepsilon_2-d(x_2,x) \right \}$

$\Rightarrow B(x,\varepsilon_x)\subseteq U$

$\Rightarrow U=\bigcup_{x\in U}B(x,\varepsilon_x)$

球形领域 $B(x_0,\varepsilon )=\left \{ x \in X : d(x,x_0)< \varepsilon,x_0\in X,\varepsilon >0 \right \}$

例2

规定X的子集族 $\tau_d=\left \{ U:U\ is \ union \ of\ spherical \ neighborhoods \right \}$ ,证明 $\tau_d$ 是X上的一个拓扑

Proof：

1. $\varnothing \in \tau_d$

$X=\bigcup_{x\in X}B(x,\varepsilon_x) \in \tau_d$

2. $\forall u_1,u_2 \in \tau_d, u_1\cup u_2 \ is \ union\ of\ spherical\ neighboorhoods$ , $u_1\cup u_2\in\tau_d$

（若干个球形邻域的并集都是 $\tau_d$ 的元素，元素间的任意并依旧是若干个球形邻域的并集，故对任意并封闭）

3. $\begin{gathered}\exists u_1,u_2\in\tau_d,u_1=\bigcup_{\alpha }B(x_{\alpha},\varepsilon_{\alpha}),u_2=\bigcup_{\beta}B(x_{\beta},\varepsilon_{\beta}).\end{gathered}$

$\begin{gathered} u_1\cap u_2=\left(\bigcup_\alpha B(x_\alpha,\varepsilon_\alpha)\right)\bigcap\left(\bigcup_\beta B(x_\beta,\varepsilon_\beta)\right) \\ =\bigcup_{\alpha,\beta}\left(B(x_{a},\varepsilon_{a})\bigcap B(x_{\beta},\varepsilon_{\beta})\right) \end{gathered}$

$let \ U =B(x_{a},\varepsilon_{a})\bigcap B(x_{\beta},\varepsilon_{\beta})$

$\forall x \in U,let \ \varepsilon_x=min\left \{ d(x,x_\alpha),d(x,x_\beta) \right \}$

$then,U=\bigcup_{x\in U}B(x,\varepsilon_x)$

$\begin{gathered} then,u_1\cap u_2=\bigcup_{\alpha,\beta}\left(\bigcup_{x\in U}B(x,\varepsilon_x)\right) \end{gathered},so \ u_1\cap u_2 \in \tau_d$

$therefore \ \tau_d \ is \ a \ topo \ in\ X$

拓扑：=

1. $X,\varnothing \in \tau$

2.任意并封闭

3.有限交封闭

$\left(\bigcup_\alpha B(x_\alpha,\varepsilon_\alpha)\right)\bigcap\left(\bigcup_\beta B(x_\beta,\varepsilon_\beta)\right) =\bigcup_{\alpha,\beta}\left(B(x_{a},\varepsilon_{a})\bigcap B(x_{\beta},\varepsilon_{\beta})\right)$