1. Open sets
Definition:
A subset U of a metric space (M, d) is called open if for any point x in U, there exists a real number ε > 0 such that any point y satisfying d(x, y) < ε belongs to U.
Equivalently, U is open if every point in U has a neighborhood contained in U.
可以理解為，對於開集合 U 當中的每一個元素 x ，與 x 夠靠近的點也都落在 U 裡面。
也就是，以任意元素 x 為圓心，都可以畫出一個小圓圈(半徑可以是任意小的正數)，
這個小圓圈整個落在 U 裡面。
用此定義來檢查我們熟悉的開/閉區間:
(0,1) 開區間是一個開集合，[0,1] 閉區間卻不是一個開集合。
對於 [0,1] 裡的元素 1 ，無論選擇的半徑有多小，畫出來的 neighborhood 都不可能整個落在 [0,1] 裡面
(右半邊一定超出 [0,1] )
要注意的是，「不是開集合」不代表一定是閉集合。
2. Closed sets
Definition1 :
A closed set is a set that contains all its limit points.
Definition2 :
A closed set is a set whose complement is an open set.
Note: A point x is a limit point (cluster point/accumulation point) of the set S if every neighborhood of x contains at least one point of S different from x itself.
3. Clopen sets
A clopen set is both open and closed.
Example: The empty set; the whole space X
4. Some sets are neither open nor closed.
Example: The half-open interval [0,1) in the real numbers.
