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\begin_body

\begin_layout Title
第2周实分析作业
\end_layout

\begin_layout Author
姓名：
王登辉
\begin_inset Formula $\qquad$
\end_inset

学号：
2024K8009906023
\end_layout

\begin_layout Problem
习题 2.1.
 设
\begin_inset Formula $E\subset\mathbb{R}^{n},\delta>0$
\end_inset

,
 证明集合
\begin_inset Formula $\ensuremath{\mathcal{O}=\{x:d(x,E)>\delta\}}$
\end_inset

是开集.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Problem
习题 2.2.
 设
\begin_inset Formula $E\subset\mathbb{R}^{n}$
\end_inset

,
 证明
\begin_inset Formula $\ensuremath{f(x):=d(x,E)}$
\end_inset

在
\begin_inset Formula $\ensuremath{\mathbb{R}^{n}}$
\end_inset

上一致连续.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Problem
习题 2.3.
 设函数
\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$
\end_inset

.
 证明
\begin_inset Formula $\ensuremath{f(x)}$
\end_inset

连续当且仅当对
\begin_inset Formula $\ensuremath{\mathbb{R}}$
\end_inset

中任意开集
\begin_inset Formula $\mathcal{O}$
\end_inset

,
 
\begin_inset Formula $\ensuremath{f^{-1}(\mathcal{O})}为\ensuremath{\mathbb{R}^{n}}$
\end_inset

中开集.
 若定义域
\begin_inset Formula $\ensuremath{\mathbb{R}^{n}}$
\end_inset

改为开区间,
 结论是否正确?
 若定义域改为闭区间又如何?
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Problem
习题 2.4.
 设
\begin_inset Formula $\ensuremath{F_{1},F_{2}}$
\end_inset

是
\begin_inset Formula $\ensuremath{\mathbb{R}}$
\end_inset

中两个互不相交的闭集.
 证明:
 存在两个互不相交的开集
\begin_inset Formula $\mathcal{O}_{1},\mathcal{O}_{2}$
\end_inset

,
 使得
\begin_inset Formula $\mathcal{O}_{1}\supset F_{1},\mathcal{O}_{2}\supset F_{2}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Problem
习题 2.5.
 设
\begin_inset Formula $\ensuremath{\{F_{\lambda}:\lambda\in\Lambda\}}$
\end_inset

是
\begin_inset Formula $\ensuremath{\mathbb{R}^{n}}$
\end_inset

中一族有界闭集.
 若对任意有限多个
\begin_inset Formula $\ensuremath{F_{\lambda_{i}}},\ensuremath{i=1,2,\cdots,N}$
\end_inset

,
 
\begin_inset Formula $\bigcap_{i=1}^{N}F_{\lambda_{i}}\neq\varnothing$
\end_inset

,
 则
\begin_inset Formula $\bigcap_{\lambda\in\Lambda}F_{\lambda}\neq\varnothing$
\end_inset

.
\end_layout

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\begin_inset Separator plain
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\begin_layout Problem
习题 2.6.
 证明:
 代数
\begin_inset Formula $\ensuremath{\mathscr{A}}$
\end_inset

是
\begin_inset Formula $\sigma$
\end_inset

-代数当且仅当
\begin_inset Formula $\ensuremath{\mathscr{A}}$
\end_inset

关于可列递增并运算封闭 (即,
 若
\begin_inset Formula $\ensuremath{\{E_{j}\}_{1}^{\infty}\subset\mathscr{A}}$
\end_inset

且
\begin_inset Formula $E_{1}\subset E_{2}\subset\cdots$
\end_inset

,
 则
\begin_inset Formula $\bigcup_{1}^{\infty}E_{j}\in\mathscr{A}$
\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Problem
习题 2.7.
 设
\begin_inset Formula $\ensuremath{\mathscr{E}\subset\mathcal{P}(X)}.$
\end_inset

 若
\begin_inset Formula $\mathscr{M}=\sigma(\mathscr{E})$
\end_inset

,
 证明:
\begin_inset Formula 
\[
\mathscr{M}=\bigcup\{\sigma(\mathscr{F}):\mathscr{F}\text{为}\mathscr{E}\text{中可列子集}\}.
\]

\end_inset


\end_layout

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\end_body
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