Point topology 定義與性質2

開覆蓋(Open covering)

  • (X,d)(X, d) is metric space, and Gi, iIG_i, \ i\in I are open sets.

  • EX E \subseteq X is open covered by iIGi\cup_{i\in I} G_i if EiIGiE \subseteq \cup_{i \in I} G_i.

    • open cover之index set II可為uncountable。

    • E.g. (countable covering) Gi=1/i<x<2/iG_i=1/i < x < 2/i, i=2,3,4,i=2,3,4,\cdots, then (0,1)iGi(0,1)\subset \cup_i G_i.

    • E.g. (uncountable covering) R\mathbb{R} is open covered by the collection of all open intervals (a,b)(a,b).

    • E.g. R\mathbb{R} is open covered by the collection of (n,n+1)(n, n+1).

緊緻集 (Compact set)

  • (X,d)(X, d) is metric space, and Gi, iIG_i, \ i\in I are open sets.

  • EX E \subseteq X is compact set if EE is open covered by finite many collection GiG_i.

    • i.e. Ei=1nGiE \subseteq \cup_{i=1}^{n} G_i.
  • Every finite set is compact. (因為finite set必定為有限個open sets聯集的子集合)。

  • Compact set \Leftrightarrow closed set.
  • SRnS \subseteq \mathbb{R}^n,以下三個定義等價:

    • (a) SS is closed and bounded set.

    • (b) SS is compact set.

    • (c) Every infinite subset of SS has a limit point in SS.

  • 而在一般的metric space時

    • (b) \Rightarrow (c) \Rightarrow (a) \neq (b).
    • (b) \Leftarrow (c) \neq (a).

分離集(Separated set)

  • (X,d)(X,d) metric space, and A,BXA, B \subseteq X.

  • AA and BB are separated if AB¯=ϕA \cap \bar{B} = \phi and A¯B=ϕ\bar{A} \cap B = \phi.

  • 上述定義即no points of AA lies in the closure of BB and no point of BB lies in the closure of AA.
  • Separated set \Rightarrow disjoint set.
  • 反之不成立。e.g. A=[0,1]A=[0,1], B=(1,2)B=(1,2), but AB¯=1A\cap \bar{B}=1.

連通集( Connected set)

  • EXE \subseteq X is connected set if EE is not a union of two nonempty separated sets.

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