Point topology 定義與性質
- 以下內容都是在 metric space 中討論。
- and is a metric space.
(集合)直徑 (Diameter)
- 定義: The diameter of is .
- 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: is bounded if and .
開球(鄰域)(Open ball (Neighborhood))
- 定義:Let and , the open ball .
- 包含了以為圓心,半徑小於的所有屬於集合中的點,但不含圓周上的點。
內點 (Interior point)
- 定義:Let , is called an interior point of .
內點會如此定義的原因是因為若點恰好在集合的邊界上中,會找不到符合上述定義的半徑 ;反之若點在集合內,則一定可以找到滿足上述定義的半徑 。
中所有interior points所形成的集合記為.
開集合 (Open set)
- 定義: is an open set if all its points are interior points (此定義中,排除了boundary points).
- , . (開集合中所有的點都是內點)
- Proof: Given , and .
- Let , then , thus . (QED).
- S is an open set .
- S in not an open set , , .
- 任意(無限)個開集點的聯集仍是開集合。 is an open set, then is an open set.
- 有限個開集點的交集仍是開集合。 is an open set, then is an open set.
- 任意(無限)個開集合的交集不一定為開集合。
- E.g. , then is a point, not an open set.
極限點(Limit point)
, and
定義: is called a limit point of if
, .
(反) , .
的任意鄰域必定與集合中不等於的元素之交集不為空集合。
可用limit point的概念得到以下序列
- 取,可得.
- 取, 得,則 。
- 以此類推,得相異點, 且.
E.g. has 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:, and , is adherent to if
.
任意的鄰域均包含至少一個中的元素。
adherent point與limit point的唯一差異為與交集中的點不可為本身。
If , must adherent to (集合中所有點均為附著點)。
- Theorem: if is a limit point of set , then , contains infinitely many point of .
- proof: 假設只包含了個元素,取,則不包含除了以外的任何元素(矛盾)。
孤立點 (Isolated point)
定義: If , but is not a limit point of , then is an isolated point of .
- .
Let , if is not a limit point of , then .
E.g. , then every point in is an isolated point of .
閉集合(Closed set)
- 定義 1: is a closed set is an open set.
- 定義 2: is a closed set if every limit point of is a point of .
- 有限個閉集合的聯集仍然是閉集合。
is a closed set, then is a closed set.
無限多個閉集合的聯集可能會變成開集合。E.g. .
- 任意(無限)閉集合的交集仍是閉集合
- is a closed set, then is a closed set.
- If is an open set, is an close set, then is an open set, and is a closed set.
,而兩個open sets的有限個交集仍為open set。
,為兩個closed set的交集,仍為closed set。
- is a closed set contains all its adherent(limit) points.
"" 假設 is closed set,且為的adherent point,to prove 。
- If , then 。
- is open set, , thus contains no points of (矛盾)。
"" Assume , then , is not adherent to , , then , is open set, is closed set.
閉包 (Closure)
- 定義 1: The set of all adherent points of a set is called the closure of , denoted by .
- 定義 2: and is the set of all limit points of in , the closure of is the set .
依定義為包含的最小閉集合。
(Every point of is adherent to ).
is closed.
- 令 is limit point of or .
- 如果 is limit point of .
- 若 is limit of ,因 is closed .
is closed .
is closed set contains all its limit points.
Theorem: is bounded above. Let , then .
完全集合(Perfect set)
定義: is perfect set if
- is closed set and
- every point of is a limit point of .
- i.e.
E.g. , , are all perfect sets.
- E.g. , is not perfect set.
- E.g. is not perfect set.
- E.g. , are perfect sets,但是中有限元素的子集合均不為perfect set。
稠密集(Dense set)
- 定義: is dense set if every point of is a limit point of or a point of .
E.g. The set of all rational numbers is dense in , and so is the set of all irrational numbers.
凸集(Convex set)
- 定義: is convex set if and , ,