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UA MATH523A 实分析1 度量空间概念与定理总结

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UA MATH523A 實分析1 集合論基礎概念與定理總結

- 序關系
- 度量空間

limit superior and lim inferior
$lim?sup?Fn=?k=1∞?n=k∞Fnlim?inf?Fn=?k=1∞?n=k∞Fn\limsup F_n = \bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} F_n \\ \liminf F_n = \bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty} F_n$

de Morgan’s law
$(?α∈AFα)C=?α∈AFαC(?α∈AFα)C=?α∈AFαC\left( \bigcup_{\alpha \in A} F_{\alpha} \right)^C = \bigcap_{\alpha \in A} F_{\alpha}^C \\ \left( \bigcap_{\alpha \in A} F_{\alpha} \right)^C = \bigcup_{\alpha \in A} F_{\alpha}^C$

For preimage
$f?1(?α∈AEα)=?α∈Af?1(Eα)f?1(?α∈AEα)=?α∈Af?1(Eα)f?1(EαC)=(f?1(Eα))Cf^{-1}\left(\bigcup_{\alpha \in A} E_{\alpha}\right) = \bigcup_{\alpha \in A}f^{-1}(E_{\alpha}) \\ f^{-1}\left(\bigcap_{\alpha \in A} E_{\alpha}\right) = \bigcap_{\alpha \in A}f^{-1}(E_{\alpha}) \\ f^{-1}(E_{\alpha}^C) = (f^{-1}(E_{\alpha}))^C$

序關系

Partial order $?x,y∈X\forall x,y \in X$ , $R$ relation such that

x R x

，

?x∈X\forall x \in X

\Rightarrow x=y

\Rightarrow xRz

Denoted as $(X,≤)(X,\le)$ , if one of $x≤y,y≤xx\le y,y\le x$ holds, it is total order (linear order).

Axiom of Choice（by Zermelo 1904）：一列非空集合的笛卡爾積也是非空集合
Zorn’s Lemma：如果偏序集的所有全序子集都有一個上界，那么這個偏序集有最大元
Hausdorff Maximal Principle：每個偏序集都有一個最大的全序子集
Well Ordering Principle (by Cantor 1883)：任意非空集合上都可以定義一個良序使之成為良序集

度量空間

Metric Space $(X,ρ)(X,\rho)$ ， $ρ:X×X→[0,∞)\rho:X\times X \to [0,\infty)$ is metric, if

ρ(x,y)=0\rho(x,y)=0

iff

x = y

?x,y∈X\forall x,y \in X

ρ(x,y)=ρ(y,x)\rho(x,y)=\rho(y,x)

?x,y,z∈X\forall x,y,z \in X

ρ(x,z)≤ρ(x,y)+ρ(y,z)\rho(x,z) \le \rho(x,y)+\rho(y,z)

Product measure $ρ((x1,y1),(x2,y2))=max?{ρ1(x1,x2),ρ2(y1,y2)}\rho((x_1,y_1),(x_2,y_2)) = \max\{\rho_1(x_1,x_2),\rho_2(y_1,y_2)\}$

Open balls $\{z \in X:\rho(x,z)<r\}$

Interior point： $?x∈X\forall x \in X$ , if $?r>0\exists r>0$ , $\subset A$
Exterior point： $?x∈X\forall x \in X$ , if $?r>0\exists r>0$ , $\subset A^C$
Boundary point： $?x∈X\forall x \in X$ , if $?r>0\exists r>0$ , $\cap A \ne \phi$ , $B(r,x)∩AC≠?B(r,x)\cap A^C\ne \phi$

Interior $i n t (A)$ , collection of all interior points
Boundary $?A\partial A$ , collection of all boundary points
Closure $Aˉ\bar{A}$ , the smallest closed set containing $A$ , $Aˉ=intA??A\bar A = int A \sqcup \partial A$

dense in X if $Eˉ=X\bar E = X$
nowhere dense $\bar E = \phi$
Separable has countable dense subset

Proposition 0.22 Equivalent:

\in \bar E

\cap E \ne \phi, \forall r>0

?{xn}?E\exists \{x_n\} \subset E

xn→xx_n \to x

Proposition 0.23 $f:X1→X2f:X_1 \to X_2$ conti iff $f^{-1}(U)$ open in $X_1$ for all open $U$ in $X_2$ .

Cauchy $ρ(xn,xm)→0\rho(x_n,x_m) \to 0$ as $\to \infty$ .
Complete all Cauchy sequences are convergence

Proposition 0.24 A closed subset of a complete metric space is complete. A complete subset of an arbitrary metric space is closed.

Totally Bounded $?zj∈E\exists z_j \in E$ , $∪j=1∞B(?,zj)?E\cup_{j=1}^{\infty}B(\epsilon,z_j) \supset E$ .

Compact = Complete+Totally Bounded, totally bounded = bounded in real space.

Theorem 0.25 Equivalent

E

compact

Bolzano-Weierstrass: every sequence in E has a subsequence converges to a point in E

Heine-Borel: every open covering of E, exists a sub open covering of E

總結

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编程问答

UA MATH523A 实分析1 度量空间 概念与定理总结

UA MATH523A 實分析1 集合論基礎 概念與定理總結

序關系

度量空間

總結

UA MATH523A 实分析1 度量空间概念与定理总结

UA MATH523A 實分析1 集合論基礎概念與定理總結