Point topology 定義與性質2

開覆蓋(Open covering)

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

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

  • open cover之index set $I$可為uncountable。

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

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

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

緊緻集 (Compact set)

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

• $E \subseteq X$ is compact set if $E$ is open covered by finite many collection $G_i$.

  • i.e. $E \subseteq \cup_{i=1}^{n} G_i$.

• 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.

• $S \subseteq \mathbb{R}^n$，以下三個定義等價:

  • (a) $S$ is closed and bounded set.

  • (b) $S$ is compact set.

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

• 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)

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

• $A$ and $B$ are separated if $A \cap \bar{B} = \phi$ and $\bar{A} \cap B = \phi$.

• 上述定義即no points of $A$ lies in the closure of $B$ and no point of $B$ lies in the closure of $A$.

• Separated set $\Rightarrow$ disjoint set.

• 反之不成立。e.g. $A=[0,1]$, $B=(1,2)$, but $A\cap \bar{B}=1$.

連通集( Connected set)

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