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開集合

A subset U of the Euclidean n-space Rn is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in Rn whose Euclidean distance from x is smaller than ε, y also belongs to U. Equivalently, U is open if every point in U has a neighbourhood contained in U.

The Euclidean distance between points $P=(p_1,p_2,\dots,p_n)\,$ and $Q=(q_1,q_2,\dots,q_n)\,$, in Euclidean n-space, is defined as:

$\sqrt{(p_1-q_1)^2 + (p_2-q_2)^2 + \cdots + (p_n-q_n)^2} = \sqrt{\sum_{i=1}^n (p_i-q_i)^2}.$

A metric space is a tuple (M,d) where M is a set and d is a metric on M, that is, a function

such that

1. d(x, y) ≥ 0     (non-negativity)
2. d(x, y) = 0   if and only if   x = y     (identity of indiscernibles)
3. d(x, y) = d(y, x)     (symmetry)
4. d(x, z) ≤ d(x, y) + d(y, z)     (triangle inequality).

也因為如此，因為度量空間比歐氏空間還要寬鬆，所以只要一個空間是歐氏空間，必定是一個度量空間，但反之則不一定成立。

The most common way to define a topological space is as a set X together with a collection T of subsets of X satisfying the following axioms:

1. The empty set and X are in T.
2. The union of any collection of sets in T is also in T.
3. The intersection of any finite collection of sets in T is also in T.

The collection T is called a topology on X, and the elements of X are called points. Under this definition, the sets in T are the open sets, and their complements in X are the closed sets.

T這個集合的集合 (定義裡面又用Collection來代表T這個集合的集合)，就是把X集合裡面的元素，某些元素規定在一起，某些元素放在一起，用「集合」本身，來規範空間裡面元素之間的關係，用「集合」運算裡面的「屬於」和「不屬於」，來定義空間裡面元素的關係。正式的T的定義，規定T要有封閉性，也就是T裡面的集合聯集(union)無限多次，仍就會變成T裡面的某個集合，也就是還在T裡面。T裡面的集合，有限次的交集(intersection)之後，還是在T裡面。符合上面這些條件，X集合、和T這個集合的集合，一起就稱為一個拓樸空間。

下面開始圖示法，綠色點就是集合X裡面的元素，藍色橢圓，就是塑造陶土形狀的集合，這些塑造形狀的集合，在變成一個大T集合的元素，也就是紅色橢圓，就是一個集合的集合 (因為集合裡面的元素，就是一個集合，所以叫做集合的集合)。大T又稱為 Collection，大T和X在一起，就定義出了一個拓樸空間

不同空間的開集合定義

http://www.math.washington.edu/~lee/Books/manifolds.html

http://www.math.washington.edu/~lee/Books/Manifolds/c1.pdf

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