Point topology 定義與性質2
開覆蓋(Open covering)
is metric space, and are open sets.
is open covered by if .
open cover之index set 可為uncountable。
E.g. (countable covering) , , then .
E.g. (uncountable covering) is open covered by the collection of all open intervals .
E.g. is open covered by the collection of .
緊緻集 (Compact set)
is metric space, and are open sets.
is compact set if is open covered by finite many collection .
- i.e. .
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 closed set.
,以下三個定義等價:
(a) is closed and bounded set.
(b) is compact set.
(c) Every infinite subset of has a limit point in .
而在一般的metric space時
- (b) (c) (a) (b).
- (b) (c) (a).
分離集(Separated set)
metric space, and .
and are separated if and .
- 上述定義即no points of lies in the closure of and no point of lies in the closure of .
- Separated set disjoint set.
- 反之不成立。e.g. , , but .
連通集( Connected set)
- is connected set if is not a union of two nonempty separated sets.