欧几里德平面中的开集合
我们研究的对象“点集”是在欧几里德平面“里面”（in it）,还是在其“上面”（on it），这是需要首先高清楚的问题。正确答案是：点集在欧几里德平面的的内部，而不在其上面。
   根据希尔伯特《几何基础》 。欧几里德平面上可以建立笛卡尔直角坐标系XoY，在其上，点P与实数序偶（x，y）一一对应，x与y分别是点P的横坐标与纵坐标。
    例如，函数y=f（x）的几何图像F，是嵌入欧几里德平面中的曲线。（点集），而不是“画”（帖）上去的。学微积分，不懂得这个道理就是“糊涂虫”。
   根据以上所述，在欧几里德平面任意两点之间可以定义距离，从而定义给定圆心与半径的“开圆” （去除圆周）。假定点集U中的每一个点都在U中至少存在一个“开圆”把该点包含在其中，则称U是欧几里德平面中的开开集合。
   开集合是点集拓扑的基本概哟用处很大。
袁萌  陈启清  3月25日
附件：
3. Open Sets in the Euclidean Plane.
A very familiar example of a Cartesian product is the set of ordered pairs of real numbers R2 = {(x1,x2) | x1 ∈R and x2 ∈R} which can be identified with the set of points in a plane using Cartesian coordinates. Writing x = (x1,x2) and y = (y1,y2), the Pythagorean Theorem leads to the formula d(x,y) = p(x1 −y1)2 + (x2 −y2)2 measuring the distance between the two points x and y in the plane. This distance function allows us to examine the standard Euclidean geometry of the plane, and for this reason we refer to R2 in conjunction with the distance formula d as the Euclidean plane. Given x = (x1,x2) ∈ R2 and a positive real number   > 0 we define the open disk of radius   centered at x to be the set B(x, ) = {y ∈R2 | d(x,y) <  } (which is also sometimes called an open ball and denoted by B (x)). Observe that B(x, ) coincides with the set of all points inside (but not on) the circle of radius   centered at x. We say that a subset U ⊆R2 of the Euclidean plane is an open set provided that for each element x ∈ U there is a real number   > 0 so that B(x, ) ⊆ U. This can be loosely paraphrased by saying U ⊆ R2 is an open set iff for each element x ∈ U all nearby points are contained in U. However note that the term ”nearby” must be interpreted in relative terms (which is equivalent to observing that different values of   may be required for different points x ∈ U). Example 3.1. The set {(x1,x2) ∈R2 | x1 > 3}, which consists of all points lying strictly to the right of the vertical line x1 = 3 in the x1x2-plane, is an open set in R2. However the set {(x1,x2) ∈R2 | x1 ≥ 3}, which consists of points on or to the right of x1 = 3 is not an open set. Also each open disk B(x, ) can be shown to be an open set in the Euclidean plane.
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The next theorem provides an important description of the open sets in R2. Theorem 3.2. The collection of open sets in the Euclidean plane R2 satisfies the following properties: (1) The empty set is an open set. (2) The entire plane R2 is an open set. (3) If Uj is an open set for each j ∈ J thenSj∈J Uj is an open set. (4) If U1 and U2 are open sets then U1 ∩U2 is an open set. Proof. Part (1) follows immediately from our definition of open set since the empty set does not contain any elements. To prove (2): suppose x ∈ R2 then B(x, ) is contained in R2 for any choice of   > 0. Now consider (3). Let Uj be an open set for each j ∈ J and let x be an element of the unionSj∈J Uj. Then x ∈ Uj0 for some j0 ∈ J (definition of union). Since Uj0 is an open set, there is a real number   > 0 so that B(x, ) ⊆ Uj0. Since Uj0 ⊆Sj∈J Uj, it follows that B(x, ) ⊆Sj∈J Uj, and this shows thatSj∈J Uj is an open set. Finally, consider (4). Let U1 and U2 be open sets, and let x ∈ U1∩U2 (which means that x ∈ U1 and x ∈ U2). Then there are real numbers  1 > 0 such that B(x, 1) ⊆ U1, and  2 > 0 such that B(x, 2) ⊆ U2. Let   be the smaller of the two numbers  1 and  2. Then   > 0, B(x, ) ⊆ B(x, 1) ⊆ U1 and B(x, ) ⊆ B(x, 2) ⊆ U2. Therefore, B(x, ) ⊆ U1 ∩U2 and it follows that U1 ∩U2 is an open set. Note that the key observation in the final paragraph of the proof of the Theorem is that if   ≤  1 then B(x, ) ⊆ B(x, 1) which follows immediately from the definitions of B(x, ) and B(x, 1). This Theorem suggests the following general definition which is the central focus for ‘point set topology’ (this is essentially Hausdorff’s definition): Definition 3.3. Let X be a set and let T be a family of subsets of X which satisfies the following four axioms: (T1) The empty set ∅ is an element of T. (T2) The set X is an element of T. (T3) If Uj ∈T for every j ∈ J then Sj∈J Uj is an element of T. (T4) If U1 and U2 are elements of T then the intersection U1 ∩U2 is an element of T. then we say that T is a topology on the set X. Here the axiom (T3) is called closure of T under arbitrary unions and axiom (T4) is called closure of T under pairwise intersections.
Comments:
(i) Notice that the definition of topology seems unbalanced in the sense that unions and intersections are treated differently—we only require the intersection of two elements of T to be in T, but we require arbitrary unions of elements of T to be in T. However it’s really this imbalance which contributes to the definition leading to a rich and useful theory.
7
(ii) WARNING: Every set X that has at least two elements will have more than one different collection of subsets that form a topology. That is, one set X will generally have MANY different possible topologies on it. So to specify a topology we have to specify both the set X and the collection of subsets T. Example 3.4. By theorem 3.2, the collection Teuclid = {U ⊂R2 | U is an open set in R2} forms a topology on the Euclidean plane. This is called the Euclidean topology on R2. Notice that for each positive integer n the subset Un ⊆R2 defined by Un = {(x1,x2) ∈R2 | x1 > 3−1/n} is an open set in R2 , however the intersection \n ∈Z+ Un = {(x1,x2) ∈R2 | x1 ≥ 3} is not an open set. This shows that part (4) of Theorem 3.2 will not be true for arbitrary intersections, and justifies the imbalance mentioned in Comment (i) above.
To end this section, we will describe how the example of the Euclidean topology of the plane can be generalized to ‘Euclidean n-space’. Example 3.5. For any positive integer n, we can endow the set Rn of ordered n-tuples of real numbers with a distance function d(x,y) = p(x1 −y1)2 + (x2 −y2)2 +•••+ (xn −yn)2where x = (x1,x2,...,xn) and y = (y1,y2,...,yn). The set Rn together with the distance formula d is referred to as Euclidean n-space. For each x = (x1,x2,...,xn) ∈Rn and each real number   > 0, let B(x, ) = {y ∈Rn | d(x,y) <  }. Then a subset U ∈ Rn is said to be an open set in Euclidean n-space provided that for each x ∈ U there is a real number   > 0 such that B(x, ) ⊆ U. With this definition theorem 3.2 extends easily to describe the collection of open sets in Euclidean n-space (just replace R2 with Rn in the statement and proof of that theorem). As a consequence, it follows that the collection Teuclid of open sets in Euclidean n-space forms a topology on Rn. This is called the Euclidean topology on Rn.
Example 3.6. When n = 1, Euclidean n-space is called the Euclidean line R1 = R. Here the distance function is given by d(x1,y1) = p(x1 −y1)2 = |x1 −y1| ,for x1,y1 ∈R, and the open disks are open intervals: B(x, ) = {y ∈R||x−y| <  } = (x− ,x +  ) .
8
Thus a subset U of the Euclidean line R is an open set iff for each x ∈ U there is an   > 0 such that (x− ,x +  ) ⊆ U. In other words, U ⊂R is an open set iff for each x ∈ U there is an   > 0 such that if y is a real number with |x−y| <   then y ∈ U. This collection of open sets forms the Euclidean topology （Teuclid） on the real line R

欧几里德平面“里面”（in it）,还是在其“上面”（on it），这是需要首先高清楚的问题。正确答案是：点集在欧几里德平面的的内部，而不在其上面。
   根据希尔伯特《几何基础》 。欧几里德平面上可以建立笛卡尔直角坐标系XoY，在其上，点P与实数序偶（x，y）一一对应，x与y分别是点P的横坐标与纵坐标。
    例如，函数y=f（x）的几何图像F，是嵌入欧几里德平面中的曲线。（点集），而不是“画”（帖）上去的。学微积分，不懂得这个道理就是“糊涂虫”。
   根据以上所述，在欧几里德平面任意两点之间可以定义距离，从而定义给定圆心与半径的“开圆” （去除圆周）。假定点集U中的每一个点都在U中至少存在一个“开圆”把该点包含在其中，则称U是欧几里德平面中的开开集合。
   开集合是点集拓扑的基本概哟用处很大。
袁萌  陈启清  3月25日
附件：
3. Open Sets in the Euclidean Plane.
A very familiar example of a Cartesian product is the set of ordered pairs of real numbers R2 = {(x1,x2) | x1 ∈R and x2 ∈R} which can be identified with the set of points in a plane using Cartesian coordinates. Writing x = (x1,x2) and y = (y1,y2), the Pythagorean Theorem leads to the formula d(x,y) = p(x1 −y1)2 + (x2 −y2)2 measuring the distance between the two points x and y in the plane. This distance function allows us to examine the standard Euclidean geometry of the plane, and for this reason we refer to R2 in conjunction with the distance formula d as the Euclidean plane. Given x = (x1,x2) ∈ R2 and a positive real number   > 0 we define the open disk of radius   centered at x to be the set B(x, ) = {y ∈R2 | d(x,y) <  } (which is also sometimes called an open ball and denoted by B (x)). Observe that B(x, ) coincides with the set of all points inside (but not on) the circle of radius   centered at x. We say that a subset U ⊆R2 of the Euclidean plane is an open set provided that for each element x ∈ U there is a real number   > 0 so that B(x, ) ⊆ U. This can be loosely paraphrased by saying U ⊆ R2 is an open set iff for each element x ∈ U all nearby points are contained in U. However note that the term ”nearby” must be interpreted in relative terms (which is equivalent to observing that different values of   may be required for different points x ∈ U). Example 3.1. The set {(x1,x2) ∈R2 | x1 > 3}, which consists of all points lying strictly to the right of the vertical line x1 = 3 in the x1x2-plane, is an open set in R2. However the set {(x1,x2) ∈R2 | x1 ≥ 3}, which consists of points on or to the right of x1 = 3 is not an open set. Also each open disk B(x, ) can be shown to be an open set in the Euclidean plane.
6
The next theorem provides an important description of the open sets in R2. Theorem 3.2. The collection of open sets in the Euclidean plane R2 satisfies the following properties: (1) The empty set is an open set. (2) The entire plane R2 is an open set. (3) If Uj is an open set for each j ∈ J thenSj∈J Uj is an open set. (4) If U1 and U2 are open sets then U1 ∩U2 is an open set. Proof. Part (1) follows immediately from our definition of open set since the empty set does not contain any elements. To prove (2): suppose x ∈ R2 then B(x, ) is contained in R2 for any choice of   > 0. Now consider (3). Let Uj be an open set for each j ∈ J and let x be an element of the unionSj∈J Uj. Then x ∈ Uj0 for some j0 ∈ J (definition of union). Since Uj0 is an open set, there is a real number   > 0 so that B(x, ) ⊆ Uj0. Since Uj0 ⊆Sj∈J Uj, it follows that B(x, ) ⊆Sj∈J Uj, and this shows thatSj∈J Uj is an open set. Finally, consider (4). Let U1 and U2 be open sets, and let x ∈ U1∩U2 (which means that x ∈ U1 and x ∈ U2). Then there are real numbers  1 > 0 such that B(x, 1) ⊆ U1, and  2 > 0 such that B(x, 2) ⊆ U2. Let   be the smaller of the two numbers  1 and  2. Then   > 0, B(x, ) ⊆ B(x, 1) ⊆ U1 and B(x, ) ⊆ B(x, 2) ⊆ U2. Therefore, B(x, ) ⊆ U1 ∩U2 and it follows that U1 ∩U2 is an open set. Note that the key observation in the final paragraph of the proof of the Theorem is that if   ≤  1 then B(x, ) ⊆ B(x, 1) which follows immediately from the definitions of B(x, ) and B(x, 1). This Theorem suggests the following general definition which is the central focus for ‘point set topology’ (this is essentially Hausdorff’s definition): Definition 3.3. Let X be a set and let T be a family of subsets of X which satisfies the following four axioms: (T1) The empty set ∅ is an element of T. (T2) The set X is an element of T. (T3) If Uj ∈T for every j ∈ J then Sj∈J Uj is an element of T. (T4) If U1 and U2 are elements of T then the intersection U1 ∩U2 is an element of T. then we say that T is a topology on the set X. Here the axiom (T3) is called closure of T under arbitrary unions and axiom (T4) is called closure of T under pairwise intersections.
Comments:
(i) Notice that the definition of topology seems unbalanced in the sense that unions and intersections are treated differently—we only require the intersection of two elements of T to be in T, but we require arbitrary unions of elements of T to be in T. However it’s really this imbalance which contributes to the definition leading to a rich and useful theory.
7
(ii) WARNING: Every set X that has at least two elements will have more than one different collection of subsets that form a topology. That is, one set X will generally have MANY different possible topologies on it. So to specify a topology we have to specify both the set X and the collection of subsets T. Example 3.4. By theorem 3.2, the collection Teuclid = {U ⊂R2 | U is an open set in R2} forms a topology on the Euclidean plane. This is called the Euclidean topology on R2. Notice that for each positive integer n the subset Un ⊆R2 defined by Un = {(x1,x2) ∈R2 | x1 > 3−1/n} is an open set in R2 , however the intersection \n ∈Z+ Un = {(x1,x2) ∈R2 | x1 ≥ 3} is not an open set. This shows that part (4) of Theorem 3.2 will not be true for arbitrary intersections, and justifies the imbalance mentioned in Comment (i) above.
To end this section, we will describe how the example of the Euclidean topology of the plane can be generalized to ‘Euclidean n-space’. Example 3.5. For any positive integer n, we can endow the set Rn of ordered n-tuples of real numbers with a distance function d(x,y) = p(x1 −y1)2 + (x2 −y2)2 +•••+ (xn −yn)2where x = (x1,x2,...,xn) and y = (y1,y2,...,yn). The set Rn together with the distance formula d is referred to as Euclidean n-space. For each x = (x1,x2,...,xn) ∈Rn and each real number   > 0, let B(x, ) = {y ∈Rn | d(x,y) <  }. Then a subset U ∈ Rn is said to be an open set in Euclidean n-space provided that for each x ∈ U there is a real number   > 0 such that B(x, ) ⊆ U. With this definition theorem 3.2 extends easily to describe the collection of open sets in Euclidean n-space (just replace R2 with Rn in the statement and proof of that theorem). As a consequence, it follows that the collection Teuclid of open sets in Euclidean n-space forms a topology on Rn. This is called the Euclidean topology on Rn.
Example 3.6. When n = 1, Euclidean n-space is called the Euclidean line R1 = R. Here the distance function is given by d(x1,y1) = p(x1 −y1)2 = |x1 −y1| ,for x1,y1 ∈R, and the open disks are open intervals: B(x, ) = {y ∈R||x−y| <  } = (x− ,x +  ) .
8
Thus a subset U of the Euclidean line R is an open set iff for each x ∈ U there is an   > 0 such that (x− ,x +  ) ⊆ U. In other words, U ⊂R is an open set iff for each x ∈ U there is an   > 0 such that if y is a real number with |x−y| <   then y ∈ U. This collection of open sets forms the Euclidean topology （Teuclid） on the real line R

我们研究的对象“点集”是在欧几里德平面“里面”（in it）,还是在其“上面”（on it），这是需要首先高清楚的问题。正确答案是：点集在欧几里德平面的的内部，而不在其上面。
   根据希尔伯特《几何基础》 。欧几里德平面上可以建立笛卡尔直角坐标系XoY，在其上，点P与实数序偶（x，y）一一对应，x与y分别是点P的横坐标与纵坐标。
    例如，函数y=f（x）的几何图像F，是嵌入欧几里德平面中的曲线。（点集），而不是“画”（帖）上去的。学微积分，不懂得这个道理就是“糊涂虫”。
   根据以上所述，在欧几里德平面任意两点之间可以定义距离，从而定义给定圆心与半径的“开圆” （去除圆周）。假定点集U中的每一个点都在U中至少存在一个“开圆”把该点包含在其中，则称U是欧几里德平面中的开开集合。
   开集合是点集拓扑的基本概哟用处很大。
袁萌  陈启清  3月25日
附件：
3. Open Sets in the Euclidean Plane.
A very familiar example of a Cartesian product is the set of ordered pairs of real numbers R2 = {(x1,x2) | x1 ∈R and x2 ∈R} which can be identified with the set of points in a plane using Cartesian coordinates. Writing x = (x1,x2) and y = (y1,y2), the Pythagorean Theorem leads to the formula d(x,y) = p(x1 −y1)2 + (x2 −y2)2 measuring the distance between the two points x and y in the plane. This distance function allows us to examine the standard Euclidean geometry of the plane, and for this reason we refer to R2 in conjunction with the distance formula d as the Euclidean plane. Given x = (x1,x2) ∈ R2 and a positive real number   > 0 we define the open disk of radius   centered at x to be the set B(x, ) = {y ∈R2 | d(x,y) <  } (which is also sometimes called an open ball and denoted by B (x)). Observe that B(x, ) coincides with the set of all points inside (but not on) the circle of radius   centered at x. We say that a subset U ⊆R2 of the Euclidean plane is an open set provided that for each element x ∈ U there is a real number   > 0 so that B(x, ) ⊆ U. This can be loosely paraphrased by saying U ⊆ R2 is an open set iff for each element x ∈ U all nearby points are contained in U. However note that the term ”nearby” must be interpreted in relative terms (which is equivalent to observing that different values of   may be required for different points x ∈ U). Example 3.1. The set {(x1,x2) ∈R2 | x1 > 3}, which consists of all points lying strictly to the right of the vertical line x1 = 3 in the x1x2-plane, is an open set in R2. However the set {(x1,x2) ∈R2 | x1 ≥ 3}, which consists of points on or to the right of x1 = 3 is not an open set. Also each open disk B(x, ) can be shown to be an open set in the Euclidean plane.
6
The next theorem provides an important description of the open sets in R2. Theorem 3.2. The collection of open sets in the Euclidean plane R2 satisfies the following properties: (1) The empty set is an open set. (2) The entire plane R2 is an open set. (3) If Uj is an open set for each j ∈ J thenSj∈J Uj is an open set. (4) If U1 and U2 are open sets then U1 ∩U2 is an open set. Proof. Part (1) follows immediately from our definition of open set since the empty set does not contain any elements. To prove (2): suppose x ∈ R2 then B(x, ) is contained in R2 for any choice of   > 0. Now consider (3). Let Uj be an open set for each j ∈ J and let x be an element of the unionSj∈J Uj. Then x ∈ Uj0 for some j0 ∈ J (definition of union). Since Uj0 is an open set, there is a real number   > 0 so that B(x, ) ⊆ Uj0. Since Uj0 ⊆Sj∈J Uj, it follows that B(x, ) ⊆Sj∈J Uj, and this shows thatSj∈J Uj is an open set. Finally, consider (4). Let U1 and U2 be open sets, and let x ∈ U1∩U2 (which means that x ∈ U1 and x ∈ U2). Then there are real numbers  1 > 0 such that B(x, 1) ⊆ U1, and  2 > 0 such that B(x, 2) ⊆ U2. Let   be the smaller of the two numbers  1 and  2. Then   > 0, B(x, ) ⊆ B(x, 1) ⊆ U1 and B(x, ) ⊆ B(x, 2) ⊆ U2. Therefore, B(x, ) ⊆ U1 ∩U2 and it follows that U1 ∩U2 is an open set. Note that the key observation in the final paragraph of the proof of the Theorem is that if   ≤  1 then B(x, ) ⊆ B(x, 1) which follows immediately from the definitions of B(x, ) and B(x, 1). This Theorem suggests the following general definition which is the central focus for ‘point set topology’ (this is essentially Hausdorff’s definition): Definition 3.3. Let X be a set and let T be a family of subsets of X which satisfies the following four axioms: (T1) The empty set ∅ is an element of T. (T2) The set X is an element of T. (T3) If Uj ∈T for every j ∈ J then Sj∈J Uj is an element of T. (T4) If U1 and U2 are elements of T then the intersection U1 ∩U2 is an element of T. then we say that T is a topology on the set X. Here the axiom (T3) is called closure of T under arbitrary unions and axiom (T4) is called closure of T under pairwise intersections.
Comments:
(i) Notice that the definition of topology seems unbalanced in the sense that unions and intersections are treated differently—we only require the intersection of two elements of T to be in T, but we require arbitrary unions of elements of T to be in T. However it’s really this imbalance which contributes to the definition leading to a rich and useful theory.
7
(ii) WARNING: Every set X that has at least two elements will have more than one different collection of subsets that form a topology. That is, one set X will generally have MANY different possible topologies on it. So to specify a topology we have to specify both the set X and the collection of subsets T. Example 3.4. By theorem 3.2, the collection Teuclid = {U ⊂R2 | U is an open set in R2} forms a topology on the Euclidean plane. This is called the Euclidean topology on R2. Notice that for each positive integer n the subset Un ⊆R2 defined by Un = {(x1,x2) ∈R2 | x1 > 3−1/n} is an open set in R2 , however the intersection \n ∈Z+ Un = {(x1,x2) ∈R2 | x1 ≥ 3} is not an open set. This shows that part (4) of Theorem 3.2 will not be true for arbitrary intersections, and justifies the imbalance mentioned in Comment (i) above.
To end this section, we will describe how the example of the Euclidean topology of the plane can be generalized to ‘Euclidean n-space’. Example 3.5. For any positive integer n, we can endow the set Rn of ordered n-tuples of real numbers with a distance function d(x,y) = p(x1 −y1)2 + (x2 −y2)2 +•••+ (xn −yn)2where x = (x1,x2,...,xn) and y = (y1,y2,...,yn). The set Rn together with the distance formula d is referred to as Euclidean n-space. For each x = (x1,x2,...,xn) ∈Rn and each real number   > 0, let B(x, ) = {y ∈Rn | d(x,y) <  }. Then a subset U ∈ Rn is said to be an open set in Euclidean n-space provided that for each x ∈ U there is a real number   > 0 such that B(x, ) ⊆ U. With this definition theorem 3.2 extends easily to describe the collection of open sets in Euclidean n-space (just replace R2 with Rn in the statement and proof of that theorem). As a consequence, it follows that the collection Teuclid of open sets in Euclidean n-space forms a topology on Rn. This is called the Euclidean topology on Rn.
Example 3.6. When n = 1, Euclidean n-space is called the Euclidean line R1 = R. Here the distance function is given by d(x1,y1) = p(x1 −y1)2 = |x1 −y1| ,for x1,y1 ∈R, and the open disks are open intervals: B(x, ) = {y ∈R||x−y| <  } = (x− ,x +  ) .
8
Thus a subset U of the Euclidean line R is an open set iff for each x ∈ U there is an   > 0 such that (x− ,x +  ) ⊆ U. In other words, U ⊂R is an open set iff for each x ∈ U there is an   > 0 such that if y is a real number with |x−y| <   then y ∈ U. This collection of open sets forms the Euclidean topology （Teuclid） on the real line R