Point topology 定義與性質

  • 以下內容都是在 metric space (X,d)(X, d) 中討論。
  • SXS \subset X and (S,d)(S, d) is a metric space.

(集合)直徑 (Diameter)

  • 定義: The diameter of SS is diamS=sup{d(x,y)x,yS}diam_{S}=\sup\{ d(x,y) \vert \forall x,y\in S \}.
  • 由定義可知集合的diameter為集合中任兩個元素的最(大)長距離。

(集合)有界

  • 定義:SXS \subseteq X is bounded if MR\exists M \in \mathbb{R} and yXy \in X \ni d(x,y)<M, xXd(x,y) < M, \ \forall x \in X.

開球(鄰域)(Open ball (Neighborhood))

  • 定義:Let aXa \in X and r>0r>0, the open ball Na(r)={bX d(a,b)<r}N_a(r) = \{b\in X \vert \ d(a,b) < r\}.
  • Na(r)N_a(r) 包含了以aa為圓心,半徑小於rr的所有屬於集合XX中的點,但不含圓周上的點。

內點 (Interior point)

  • 定義:Let aSa \in S, aa is called an interior point of SS \Leftrightarrow r>0\exists r > 0 \ni Nr(a)SN_r(a) \subseteq S.
  • 內點會如此定義的原因是因為若點恰好在集合SS的邊界上中,會找不到符合上述定義的半徑 rr;反之若點在集合SS內,則一定可以找到滿足上述定義的半徑 rr

  • SS 中所有interior points所形成的集合記為intS\text{int} S.

開集合 (Open set)

  • 定義:SXS \subset X is an open set if all its points are interior points (此定義中,排除了boundary points).
  • aS\forall a \in S , aintS a \in \text{int} S. (開集合中所有的點都是內點)
    • Proof: Given pS p \in S, r>0r > 0 and qNr(p)q \in N_r(p).
    • Let h=d(p,q)h = d(p,q), then r1=rd>0r_1 = r-d > 0, thus Nr1(q)Nr(p)N_{r_1}(q) \subseteq N_r(p). (QED).
  • S is an open set \Leftrightarrow S=intSS = \text{int} S.
  • S in not an open set \Leftrightarrow aS\exists a \in S, r>0\forall r > 0, Nr(a)SN_r(a) \nsubseteq S.
  • 任意(無限)個開集點的聯集仍是開集合AnA_n is an open set, then n=1An\cup_{n=1}^{\infty}A_n is an open set.
  • 有限個開集點的交集仍是開集合。 AnA_n is an open set, then n=1NAn\cap_{n=1}^{N} A_n is an open set.
  • 任意(無限)個開集合的交集不一定為開集合
    • E.g. An=(1n,1n)A_n=\left(\frac{-1}{n}, \frac{1}{n} \right), then n=1An=0\cap_{n=1}^{\infty} A_n =0 is a point, not an open set.

極限點(Limit point)

  • SXS \subseteq X, and xXx \in X

  • 定義: xx is called a limit point of SS if

    • r>0\forall r > 0, Nr(x)S{x}ϕN_r(x) \cap S - \lbrace x \rbrace \neq \phi.

    • (反) r>0\exists r > 0 , Nr(x)S{x}=ϕN_r(x) \cap S - \lbrace x \rbrace = \phi.

    • xx的任意鄰域Nr(x)N_r(x)必定與集合SS中不等於xx的元素之交集不為空集合。

  • 可用limit point的概念得到以下序列

    • r1=1r_1 =1,可得x1Nr1(x)Sx1x_1 \in N_{r_1} (x) \cap S \neq x_1.
    • r2=12x1x>12r_2 = \frac{1}{2} \begin{vmatrix} x_1 - x\end{vmatrix} > \frac{1}{2}, 得x2Nr2(x)S{x}x_2 \in N_{r_2}(x) \cap S - \lbrace x \rbrace,則 x2x1,xx_2 \neq x_1, x
    • 以此類推,得相異點{x1,x2,}S\lbrace x_1, x_2, \cdots \rbrace \subset S, 且limnxn=x\lim_{ n \rightarrow \infty} x_n = x.
  • E.g. S={1n, nN}S=\lbrace \frac{1}{n},\ \forall n \in \mathbb{N}\rbrace has 00 as its limit point.

  • E.g. The set of rational numbers has every real number as a limit point.

附著點(Adherent point)

  • 定義:SXS \subseteq X, and xXx \in X, xx is adherent to SS if

    • r>0,Nr(x)Sϕ\forall r >0, N_r(x) \cap S \neq \phi.

    • 任意xx的鄰域Nr(x)N_r(x)均包含至少一個SS中的元素。

    • adherent point與limit point的唯一差異為與SS交集中的點不可為xx本身。

  • If xSx \in S, xx must adherent to SS (集合中所有點均為附著點)。

  • Theorem: if xx is a limit point of set SS, then r>0\forall r > 0, Nr(x)N_r(x) contains infinitely many point of SS.
    • proof: 假設Nr(x)N_r(x)只包含了s1,,sms_1,\cdots, s_m個元素,取r=min{xs1,,xsm}r=\min \lbrace \lVert x-s_1 \rVert, \cdots, \lVert x-s_m \rVert \rbrace,則Nr/2(x)N_{r/2}(x)不包含除了xx以外的任何元素(矛盾)。

孤立點 (Isolated point)

  • 定義: If xSx \in S, but xx is not a limit point of SS, then xx is an isolated point of SS.

    • r>0Nr(x)S{x}=ϕ \exists r > 0 \ni N_r(x) \cap S - \lbrace x \rbrace = \phi.
  • Let xSx \in S, if xx is not a limit point of SS, then Nr(x)S={x}N_r(x) \cap S = \lbrace x \rbrace.

  • E.g. S={1n nN} S = \lbrace \frac{1}{n} \vert \ n\in \mathbb{N} \rbrace, then every point in SS is an isolated point of SS.

閉集合(Closed set)

  • 定義 1:SXS \subseteq X is a closed set \Leftrightarrow X\SX\backslash S is an open set.
  • 定義 2:SXS \subset X is a closed set if every limit point of SS is a point of SS.
  • 有限個閉集合的聯集仍然是閉集合。
    • AnA_n is a closed set, then n=1An\cup_{n=1}^{\infty} A_n is a closed set.

    • 無限多個閉集合的聯集可能會變成開集合。E.g. An=(1/n,1/n)A_n = (-1/n, 1/n).

  • 任意(無限)閉集合的交集仍是閉集合
    • AnA_n is a closed set, then n=1An\cap_{n=1}^{\infty} A_n is a closed set.
  • If AA is an open set, BB is an close set, then ABA-B is an open set, and BAB-A is a closed set.
    • AB=A(X\B)\because A-B = A \cap (X\backslash B),而兩個open sets的有限個交集仍為open set。

    • BA=B(X\A)\because B-A= B \cap (X \backslash A),為兩個closed set的交集,仍為closed set。

  • SXS \subseteq X is a closed set \Leftrightarrow SS contains all its adherent(limit) points.
    • "\Rightarrow" 假設SS is closed set,且xxSS的adherent point,to prove xSx \in S

      • If xSx \notin S, then xX\Sx \in X\backslash S
      • X\S\because X\backslash S is open set, r>0Nr(x)X\S\therefore \exists r > 0 \ni N_r(x) \in X \backslash S, thus Nr(x)N_r(x) contains no points of SS (矛盾)。
    • "\Leftarrow" Assume xX\Sx \in X \backslash S, then xSx \notin S, xx is not adherent to SS, r>0Nr(x)S=ϕ\therefore \exists r > 0 \ni N_r(x) \cap S = \phi, then Nr(x)X\SN_r(x) \in X\backslash S, X\S\therefore X\backslash S is open set, S\therefore S is closed set.

閉包 (Closure)

  • 定義 1: The set of all adherent points of a set SS is called the closure of SS, denoted by S¯\bar{S}.
  • 定義 2: SXS \subseteq X and SS^{'} is the set of all limit points of SS in XX, the closure of SS is the set S¯=SS\bar{S} = S \cup S^{'}.
    • 依定義S¯\bar{S}為包含SS的最小閉集合。

    • SS¯ S \subseteq \bar{S} (Every point of SS is adherent to SS).

    • S¯S\bar{S} \subseteq S \Leftrightarrow SS is closed.

      • xx is limit point of SS or SS^{'}.
      • 如果xx is limit point of SS \Rightarrow xS¯x\in \bar{S}.
      • xx is limit of SS^{'},因SS^{'} is closed xSS¯\therefore x \in S^{'} \subseteq \bar{S}.
    • SS is closed \Leftrightarrow S=S¯S=\bar{S}.

    • SS is closed set \Leftrightarrow SS contains all its limit points.

    • Theorem: ϕSsubseteqX\phi \neq S subseteq X is bounded above. Let y=supS y = \sup S, then yS¯y \in \bar{S}.

完全集合(Perfect set)

  • 定義:SXS \subseteq X is perfect set if

    • SS is closed set and
    • every point of SS is a limit point of SS.
    • i.e. S=SS = S^{'}
  • E.g. [0,1][0,1], [0,)[0, \infty), (,0](-\infty, 0] are all perfect sets.

  • E.g. Nr(x)N_r(x), r>0r > 0 is not perfect set.
  • E.g. N\mathbb{N} is not perfect set.
  • E.g. Rn\mathbb{R}^n, nN\forall n \in \mathbb{N} are perfect sets,但是Rn\mathbb{R}^n中有限元素的子集合均不為perfect set。

稠密集(Dense set)

  • 定義:SXS \subseteq X is dense set if every point of XX is a limit point of SS or a point of EE.
  • E.g. The set of all rational numbers Q\mathbb{Q} is dense in R\mathbb{R}, and so is the set of all irrational numbers.

凸集(Convex set)

  • 定義:SXS \subseteq X is convex set if x,yS\forall x,y \in S and 0<λ<10 < \lambda < 1, λx+(1λ)yS \lambda x + (1-\lambda)y \in S,

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