点集拓扑学基础（I）

点集拓扑学基础（I）
我们的目的是学懂、学透微积分。但是，只懂得欧几里得平面基本知识还不够，中间缺少欧几里得平面点集拓扑的概念。
本文附件（英文）阐述了欧几里得平面点集拓扑的基本概念（I），值得一读。
袁萌  陈启清  3月23日
附件：
Basic Concepts of Point Set Topology
Notes for U course Math 4853 Spring 2011
A. Miller
1. Introduction.
The definitions of ‘metric space’ and ’topological space’ were developed in the early 1900’s, largely through the work of Maurice Frechet (for metric spaces) and Felix Hausdorff (for topological spaces). The main impetus for this work was to provide a framework in which to discuss continuous functions, with the goal of examining their attributes more thoroughly and extending the concept beyond the realm of calculus. At the time, these mathematicians were particularly interested in understanding and generalizing the Extreme Value Theorem and the Intermediate Value Theorem. Here are statements of these important theorems, which are well-known to us from calculus: Theorem 1.1 (Extreme Value Theorem). Every continuous function f : [a,b] → R achieves a maximum value and a minimum value. Theorem 1.2 (Intermediate Value Theorem). If f : [a,b] → R is a continuous function and y is a real number between f(a) and f(b) then there is a real number x in the interval [a,b] such that f(x) = y.
In these statements we recall the following basic terminology. For real numbers a and b, [a,b] denotes the closed finite interval [a,b] = {x ∈R| a ≤ x ≤ b} . If c is a real number in the interval [a,b] such that f(c) ≥ f(x) for every x ∈ [a,b] then f(c) is a maximum value for f on [a,b], and if d is a real number in the interval [a,b] such that f(d) ≤ f(x) for every x ∈ [a,b] then f(d) is a minimum value for f on [a,b]. The function f : [a,b] →R is continuous provided that for each x0 ∈ [a,b] and each  > 0 there is a positive real number δ such that if x ∈ [a,b] and |x−x0| < δ then |f(x)−f(x0)| < . As we proceed to develop the ideas of point set topology, we will review and examine these notations more thoroughly. Notice that in the statement of these two theorems the fact that the domain of the function f is a closed finite interval [a,b] is crucial. Here are two examples that should convince you that this hypothesis is needed in both statements.
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Example 1.3. Consider the function f : (0,2] →R given by f(x) = 1/x. From our calculus experience we know that f is a continuous function (for example f is a rational function whose denominator does not have any roots inside the interval (0,2]). This function has a minimum value of f(2) = 1/2 but it has no maximum value because limx→0+ f(x) = +∞. So the conclusion of the Extreme Value Theorem fails, but of course the ‘half-open interval’ (0,2] = {x ∈ R | 0 < x ≤ 2} is not a closed finite interval—so the example does not violate the statement of the Extreme Value Theorem.
Example 1.4. Consider the function g : [−1,0) ∪ (0,2] → R given by g(x) = 1/x (whose domain is the set [−1,0)∪(0,2] = {x ∈ R | −1 ≤ x ≤ 2 and x 6= 0}). Again from calculus experience we know that g is a continuous function. This function satisfies g(−1) = −1 and g(2) = 1/2 however there is no real number x with g(x) = 0 even though g(−1) < 0 < g(2). So the conclusion of the Intermediate Value Theorem fails, but again the set [−1,0)∪(0,2] is not a closed finite interval so the example does not contradict the statement of the Intermediate Value Theorem. [Sketch the graphs of the functions in these two examples so that you can better visualize what goes wrong.]
Mathematicians in the early twentieth century, such as Frechet and Hausdorff, came to realize that the validity of the Extreme Value Theorem and the Intermediate Value Theorem particularly relied on the fact that any closed finite interval [a,b] satisfies two key ‘topological properties’ known as ‘compactness’ and ‘connectedness’. Our goal in the point-set topology portion of this course is to introduce the language of topology to the extent that we can talk about continuous functions and the properties of compactness and connectedness, and to use these ideas to establish more general versions of the Extreme Value and Intermediate Value Theorems. We will end with a discussion of the difference in perspective between the Frechet and Hausdorff definitions, and how attempts to bridge these definitions contributed to the early development of the subject of point set topology.
2. Review of Set Theory.
A set A is a collection of objects or elements. In order to avoid certain paradoxes of set theory we will assume that an ‘object’ is always chosen from some universal set. If x is an element and A is a set then ‘x ∈ A’ means that x is an element of A, and ‘x / ∈ A’ means x is not an element in A. Sets are usually described by either listing all of their elements or by using ‘set-builder notation’.
Example 2.1. The set A consisting of three elements labeled by 1, 2 and 3 might be described in one of the following three ways: A = {1,2,3} = {x | x is a positive integer and x ≤ 3} = {x | x is a positive integer and 0 < x2 < 10} The first description lists the elements of the set while the last two descriptions illustrate set-builder notation.
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Examples. Here are some important special examples of sets that we will refer to frequently:
• The empty set ∅ is the set which contains no elements. This means that x /∈ ∅ for every element x in the universal set. • Z+ = {1,2,3,...} is the set of positive integers. • Z = {...,−2,−1,0,1,2,...} is the set of integers. • Q = {m/n | m,n ∈ Z and n 6= 0} is the set of rational numbers. Recall that every rational number can be expressed as m/n where the greatest common divisor of m and n equals 1—when this happens we say that m/n is in reduced form. (For example, the reduced form for the rational number 39/52 would be 3/4.) • R is the set of real numbers. • R−Q = {x ∈R| x 6= Q} is the set of irrational numbers. • R+ = {x ∈R| x > 0} is the set of positive real numbers. • If a and b are real numbers then we can define various ‘intervals from a to b’: [a,b] = {x ∈R| a ≤ x ≤ b} called a closed interval (a,b) = {x ∈R| a < x < b} called a closed interval [a,b) = {x ∈R| a ≤ x < b} (a,b] = {x ∈R| a < x ≤ b} (a,∞] = {x ∈R| a < x} (−∞,b] = {x ∈Rx ≤ b}
Notice that the ‘interval’ [a,b] is the empty set if a > b, and that [a,a] = {a}. Definition 2.2. We list some of the most important basic definitions in set theory. In the following let A and B be sets.
1. A is a subset of B (written A ⊆ B) provided that if x ∈ A then x ∈ B. 2. A is equal to B (written A = B) if and only if A is a subset of B and B is a subset of A. This is equivalent to saying that x is an element of A iff x is an element of B. (Notation: ‘iff’ is short for ‘if and only if’.) 3. A is a proper subset of B (written A ( B or A ⊂ B) if A is a subset of B but A is not equal to B. 4. The union of A and B is the set A∪B = {x | x ∈ A or x ∈ B}. 5. The intersection of A and B is the set A∩B = {x | x ∈ A and x ∈ B}. 6. The set difference of A and B is the set A−B = {x | x ∈ A and x / ∈ B}. Notice that typically A−B and B−A are two different sets.
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Definition 2.3. In addition to defining the union and intersection of two sets we can define the union and intersection of any finite collection of sets as follows. Let n be a positive integer (that is, n ∈Z+), and let A1,A2,...,An be sets. Then define A1 ∪A2 ∪•••∪An = {x | x ∈ Ai for some i ∈{1,2,...n}} A1 ∩A2 ∩•••∩An = {x | x ∈ Ai for every i ∈{1,2,...n}}
With a little more background we can also define unions and intersections of arbitrary families of sets. Let J be a non-empty set, and let Aj be a set for each j ∈ J. In this situation we say that {Aj | j ∈ J} is a family (or collection) of sets indexed by J, and we refer to J as the index set for this family. The union and intersection of this family are respectively defined by the following: [ j∈J Aj = [{Aj | j ∈ J} = {x | x ∈ Aj for some j ∈ J} \ j∈J Aj = \{Aj | j ∈ J} = {x | x ∈ Aj for every j ∈ J} In particular notice that a finite union A1 ∪ A2 ∪•••∪ An is the same asSj∈J Aj where J = {1,2,...,n}, and similarly for intersections. Here are two examples of infinite unions and intersections.
Example 2.4. For each n ∈ Z+ let An = [0,1/n] (which is a closed interval in R). Then {An | n ∈ Z+} is a family of sets indexed by the positive integers, with A1 = [0,1], A2 = [0,1/2], A3 = [0,1/3] and so on. Notice that each set An in this family contains its successor An+1, when this happens we say the family is a decreasing family of nested sets. Here we have [n ∈Z+ An = [ n∈Z+ [0,1/n] = [0,1] and \ n∈Z+ An = \ n∈Z+ [0,1/n] = {0}. Since the determination of infinite unions and/or intersections like these can sometimes be difficult, let’s give a formal proof of the second of these statements. As is often the case, being able to write down a formal proof is tantamount to being able to systematically determine the result in the first place. Proof thatTn∈Z+ An = {0}: First suppose that x is an element of Tn∈Z+ An. By definition of intersection this means that x ∈ An = [0,1/n] for every positive integer n. In particular, x is an element of A1 = [0,1] and so we must have x ≥ 0. Suppose first that x > 0. Then by the Archimedean Principle1 there is a positive integer N such that 1/N < x. It follows that x / ∈ [0,1/N] = AN which contradicts our choice of x as being an element ofTn∈Z+ An.1 The Archimedean Principle asserts that for every positive real number x there is a positive integer N such that 1/N < x. This is equivalent to the statement that the limit of the sequence (1/n)n∈Z+ is 0.
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Thus our supposition that x > 0 must be false (using ‘proof by contradiction’). Since we have already observed that x ≥ 0 it must be that x = 0. So we have that x ∈ {0}, and since x was an arbitrary element ofTn∈Z+ An, this shows thatTn∈Z+ An ⊆{0} (by the definition of subset). Next, let’s suppose that x ∈{0}. Then x = 0 (since {0} is a set with only one element). For each positive integer n, x = 0 is an element of An = [0,1/n]. Therefore x ∈Tn∈Z+ An by the definition of intersection, and this shows that {0} is a subset ofTn∈Z+ An. We have shown that each of the sets {0} andTn∈Z+ An is a subset of the other, and therefore the two sets are equal by the definition of set equality.
[See if you can write down a formal proof of the first statement above, and of the two claims in the next example.]
Example 2.5. For each n ∈Z+ let Bn = [0,1−1/n] (which is a closed interval in R). This is a family of sets indexed by the set of positive integers, with B1 = [0,0] = {0}, B2 = [0,1/2], B3 = [0,2/3] and so on. Notice that each set Bn in this family is a subset of its successor Bn+1, when this happens we say the family is an increasing family of nested sets. Here we have [ n∈Z+ Bn = [ n∈Z+ [0,1−1/n] = [0,1) and \ n∈Z+ Bn = \ n∈Z+ [0,1−1/n] = {0}.
The basic definitions of set theory satisfy a variety of well-known laws. For example, the ‘associative law for union’ says that for any three sets A, B and C, (A∪B)∪C = A∪(B∪C). And the ‘commutative law for intersection’ says that for all sets A and B, A∩B = B ∩A. Of particular interest are DeMorgan’s Laws which can be expressed as follows: Let A be a set and let {Bj | j ∈ J} be a family of sets then A−[ j∈J Bj = \ j∈J (A−Bj) A−\ j∈J Bj = [ j∈J (A−Bj)
We will not give a comprehensive list of all of the elementary laws of set theory here, but any book on set theory or discrete mathematics will contain such lists. In particular, the theory of sets can be completely described by certain lists of laws, one well-known such list is referred to as the ‘Zermelo-Fraenkel Axioms for Set Theory’. Another definition involving sets that is very important and useful is the Cartesian product. Let n be a positive integer and let A1,A2,...,An be a family of n sets. Then the Cartesian product of this family is the set A1 ×A2 ו••×An = {(a1,a2,...,an) | ai ∈ Ai for each i ∈{1,2,...,n}}.
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The formal symbol (a1,a2,...,an) is called an ordered n-tuple, and we refer to ai as the ith coordinate of the n-tuple. Notice that in this definition we don’t require that the sets Ai all be distinct from each other. In fact, in the extreme case where all of the Ai’s are the same set A we obtain An = A×Aו••×A = {(a1,a2,...,an) | ai ∈ A for each i ∈{1,2,...,n}}. For example, Rn denotes the set of all ordered n-tuples of real numbers. One can also define infinite Cartesian products but we will not need them in this course. If X is a set then the power set of X is the set of all subsets of X
P(X) = {A | A ⊆ X}. (Some authors denote the power set P(X) by 2X.) Note that the empty set ∅ and the set X itself are always elements of P(X). If X is a finite set with n elements then P(X) is a set with 2n elements.
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.
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(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 + ) .
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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.
4. Review of Functions.
Let X and Y be sets. A function f from X to Y is a rule that associates each element of X with exactly one element of Y . For each element x ∈ X, the unique element of Y associated with x is denoted by f(x) and called the image of x under f. We write f : X → Y to signify that f is a function from X to Y . Given a function f : X → T, the set X is referred to as the domain of f, and the set Y is called the codomain of f. The set {y ∈ Y | y = f(x) for some x ∈ X} is called the range of the function f. Notice that by definition the range of a function is a subset of its co-domain. A function f : X → Y is said to be one–to–one (or injective) provided that if x1 and x2 are elements of X and f(x1) = f(x2) then x1 = x2. Thus a function is one–to–one iff each element in the range of the function is associated with only one element of the domain. A function f : X → Y is onto (or surjective) provided that for each y ∈ Y there is at least one x ∈ X such that f(x) = y. In other words, f : X → Y is onto iff the range of f equals the codomain of f. A function f : X → Y which is both one–to–one and onto is called a one–to–one correspondence (or a bijection). If f : X → Y and g : Y → Z are functions (where the codomain of f equals the domain of g) then we can form the composition of f with g to be the function g◦f : X → Z given by g ◦f(x) = g(f(x)) for each x ∈ X. Note that the domain of g ◦f equals the domain of f, while the codomain of g ◦f equals the codomain of g. If X is a set and A is a subset of X then there is function i : A → X called the inclusion function from A to X defined by i(a) = a for each a ∈ A. The inclusion function from a set X to itself is called the identity function on X. Definition 4.1 (Images and Inverse Images of Sets). Let f : X → Y be a function with domain X and codomain Y . If A ⊂ X then the image of A under f is the set f(A) = {y ∈ Y | y = f(a) for some a ∈ A} . If B is a subset of Y then the inverse image of B under f is the set f−1(B) = {x ∈ X | f(x) ∈ B} . These definitions can be summarized by saying that each function f : X → Y induces two functions on power sets. The first fromP(X) toP(Y ) given by A 7→ f(A) for each A ∈P(X), and the second from P(Y ) to P(X) given by B 7→ f−1(B) for each B ∈P(Y ).
Example 4.2. Let f : R → R be the function defined by f(x) = x2 + 1 for all x ∈ R. Let A be the half-open interval A = (1,3], which in this example is a subset of both the domain
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and codomain of f. Then f(A) = f((1,3]) = {y ∈R| y = x2 + 1 for some x ∈ (1,3]} = {x2 + 1 | 1 < x ≤ 3} = (2,10] and f−1(A) = f−1((1,3]) = {x ∈R| x2 + 1 ∈ (1,3]} = {x | 0 < x2 ≤ 3} = {x | 0 < x ≤ √3 or −√3 ≤ x < 0} = [−√3,0)∪(0,√3].
Example 4.3. Let X be a set and let A be a subset of X. If i : A → X is the inclusion function and B ⊂ X then i−1(B) = {a ∈ A | i(a) = a ∈ B} = {aA | a ∈ A and a ∈ B} = A∩B .
5. Definition of Topology and Basic Terminology.
Let’s start by repeating the definition of a topology on a set X: Definition 5.1. Let X be a set and let T be a family of subsets of T 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 each j ∈ J thenSj∈J Uj is in T. (T4) If U1 and U2 are in T then U1 ∩U2 is in T. then we say that T is a topology on the set X. Here the axiom (T3) is referred to by saying T is closed under arbitrary unions and axiom (T4) is referred by saying T is closed under pairwise intersection.
A set X together with a topology T on that set is called a topological space, more formally we will refer to this topological space as (X,T). If (X,T) is a topological space then the elements of T are called open sets in X. Theorem 5.2 (T is closed under finite intersections.). Let T be a topology on a set X, and let U1,U2,...,Un be open sets in this topology where n is a positive integer. Then the intersection U1 ∩U2 ∩•••∩Un is an open set.
Proof. We prove this statement by induction on n. If n = 1 then the collection of open sets has only one set U1 and the intersection is just U1 which is an open set. This shows that the statement is true when n = 1. Now suppose the statement is true for a positive integer n (this is the induction hypothesis). To complete the proof by induction we need to show that the statement is true for n+1. So suppose that U1,U2,...,Un+1 are open sets. Then we can write U1 ∩U2 ∩•••∩Un+1 = V ∩Un+1 where V = U1 ∩U2 ∩•••∩Un. Then V is an open set by the induction hypothesis, and V ∩Un+1 is open by axiom (T4). This shows that the statement is true for n+1 (that is, U1 ∩U2 ∩•••∩Un+1 is open) and completes the proof by induction.
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If x is an element of X and U is an open set which contains x then we say that U is a neighborhood of x. Theorem 5.3. A subset V ⊆ X is an open set if and only if every element x ∈ V has a neighborhood that is contained in V .
Proof. (⇒) Suppose that V is an open set in X. Then for every element x ∈ V , V is a neighborhood of x and V ⊆ V . (⇐) Suppose that V is a subset of X and each element x ∈ V has a neighborhood Ux such that Ux ⊆ V . Then V =Sx∈V Ux and since each Ux is open then the union is open by axiom (T3). This shows that V is an open set and completes the proof. [In the second part of the previous proof, carefully explain the equality V =Sx∈V Ux?] If T1 and T2 are two topologies on a set X and T2 ⊆ T1, then we say that T1 is finer than T2, or that T2 is coarser than T1. If T2 ( T1 then T1 is strictly finer than T2. If neither of the topologiesT1 andT2 are finer than the other then we say that the two topologies are incomparable. We end this section by describing a number of examples of topological spaces. Example 5.4. For any set X let Tdiscrete = P(X) = {A | A ⊆ X}. Clearly axioms (T1) and (T2) hold since ∅ and X are subsets of X. The union and the intersection of any collection of subsets of X is again a subset of X and this shows that axioms (T3) and (T4) hold as well. Therefore Tdiscrete forms a topology on X and this is called the discrete topology on X. So in the discrete topology, every subset of X is an open set. Therefore Tdiscrete is the largest possible topology on X, which is to say that the discrete topology is finer than every topology on X.
Example 5.5. For any set X let Ttrivial = {∅,X}. It is easily verified that Ttrivial is a topology on X and this is called the trivial topology on X. This topology contains only two open sets (assuming that X is a nonempty set). Therefore Ttrivial is the smallest possible topology on X, which is to say that every topology on X is finer than the trivial topology.
Example 5.6. Just a reminder that for each positive integer n, Teuclid is a topology on Rn called the Euclidean topology. In particular, taking n = 1 gives the Euclidean topology Teuclid on the real line R, referred to briefly as the Euclidean line. Here Teuclid = {U ⊂R| for each x ∈ U there is > 0 such that (x−,x + ) ⊆ U}. With this definition it is not hard to show that every open interval in R is an open set in the Euclidean topology. [Be sure you can write out an explanation for this.] But note carefully that there are open sets in the Euclidean topology that are not open intervals. For example, the union of two or more disjoint intervals (such as (−1,1) ∪ (√2,π)) will be an open set which is not an interval.
Example 5.7. Consider the collection of subsets T` of the real line R given by T` = {U ⊆R| for each x ∈ U there is  > 0 so that [x,x + ) ⊆ U}.
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This set forms a topology T` on R called the lower limit topology on R. It is not hard to show that (1) each finite half-open interval of the form [a,b) where a < b is an open set in the lower limit topology but not an open set in the Euclidean topology, while (2) every open set in the Euclidean topology on R is open in the lower limit topology. Thus the lower limit topology on R is strictly finer than the Euclidean topology on R (that is Teuclid (T`).
Example 5.8. Let X be a set and let x0 be an element of X. Define T to be the collection of subsets of X consisting of X itself and all subsets of X which do not contain x0. Thus T = {X}∪{U ⊆ X | x0 / ∈ U} . This forms a topology on X which is called the excluded point topology.
Example 5.9. Let X be any set and define Tcofinite = {∅}∪{U ⊆ X | X−U is a finite set } = {∅}∪{X−F | F is a finite subset of X}. Then it is not hard to show thatTcofinite forms a topology on X, called the cofinite topology on X. (We will explain this using closed sets in the next section.) Note that if the set X itself is finite then every subset of X will be finite, and so the cofinite topology on a finite set X is the same as the discrete topology on X.
6. Closed Sets and the Closure Operation.
Throughout this section let (X,T) be a topological space. A subset C ⊆ X is said to be a closed set in X provided that its complement X −C is an open set (that is, X −C ∈T). Notice that taking complements in X defines a bijection between the collection of open sets in a topological space and the collection of closed sets. Basic properties of closed sets are described by the next theorem, whose proof depends principally on DeMorgan’s Laws. Theorem 6.1. The collection C of all closed sets in X satisfies the properties (1) The empty set ∅ is an element of C. (2) The set X is an element of C. (3) If Uj ∈C for each j ∈ J thenTj∈J Uj is in C. (4) If U1 and U2 are in C then U1 ∪U2 is in C. Moreover, for any collection C of subsets of X which satisfies properties (1)–(4) there is a unique topology on X for which C is the collection of closed sets. Example 6.2. For any set X consider the collection of subsets given by C = {X}∪{F ⊆ X | F is a finite set } . Then it is easy to check that C satisfies the four conditions in theorem 6.1 (for example, the intersection of any collection of finite sets is finite, and the union of two finite sets is finite). Therefore C forms the collection of closed sets for some topology on X, and this is the cofinite topology Tcofinite = {∅}∪{U ⊆ X | X −U is a finite set } as described in example 5.9.
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If A is an arbitrary subset of X then the closure of A is the intersection cl(A) of all closed sets in X which contain A. The next theorem is easily proved using this definition. The first two parts of this theorem can be summarized by saying that the closure cl(A) of a subset A is the smallest closed set which contains A. Theorem 6.3. For any subset A ⊆ X the closure cl(A) is a closed set. If C is a closed set containing A then cl(A) ⊆ C. The subset A is a closed set if and only if cl(A) = A. Example 6.4. Consider the trivial topology Ttrivial = {∅,X} on a set X. The closed sets in this topology are X−∅ = X and X−X = ∅. Then the closure of a subset A of X is described by cl(A) = ∅, if A = ∅ X, if A 6= ∅. To explain: If A is the empty set then both closed sets∅and X contain A, so cl(A) = ∅∩X = ∅. If A is a nonempty subset of X then the only closed set containing A is X and so the intersection of all the closed sets containing A is X.
Example 6.5. Consider the discrete topologyTdiscrete = P(X) on a set X. Then every subset of X is closed, and so here the closure operation is described by cl(A) = A (by theorem 6.3).
Definition 6.6. Let A be a subset of X. An element x ∈ X is called a limit point of A provided that every neighborhood U of x contains an element of A other than x itself. The set of all limit points of A is called the derived set of A and denoted by A0.
The next theorem describes a relationship between the set of limit points of a set and the closure of a set. The proof of this theorem is quite typical of proofs in point set topology. Theorem 6.7. For any subset A ⊆ X, cl(A) = A∪A0. Proof. The proof is broken into two parts where first we will show that cl(A) ⊆ A∪A0 and then that A∪A0 ⊆ cl(A). cl(A) ⊆ A∪A0: Suppose x ∈ cl(A). Then we must show that x is an element of A∪A0. For contradiction, let us assume that x / ∈ A∪A0. Then x / ∈ A and x / ∈ A0 (by the definition of union). Since x / ∈ A0, x is not a limit point of A and this means that there is a neighborhood U of x which contains no element of A other than x. In fact, because x / ∈ A, U cannot contain any elements of A. Thus A is a subset of X −U. Since X −U is a closed set (as U is open) and cl(A) is the smallest closed set containing A (by theorem 6.3), cl(A) must be a subset of X −U. On the other hand, x is an element of cl(A) (by supposition) but x / ∈ X −U (since U is a neighborhood of x). This yields a contradiction and shows that x ∈ A∪A0, as desired. A∪A0 ⊆ cl(A): Suppose x ∈ A∪A0. For contradiction, let’s assume that x / ∈ cl(A). This means that x / ∈ A (since A is a subset of cl(A)) and that x ∈ A0 (because x ∈ (A∪A0)−A). Observe that X −cl(A) is an open set (being the complement of the closed set cl(A)), and that X −cl(A) is a neighborhood of x (x ∈ X −cl(A) since x / ∈ cl(A)). As x is a limit point of A, this neighborhood X −cl(A) must contain an element of A. But since A is a subset of
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cl(A) it can’t contain any elements of X−cl(A). This contradiction leads us to conclude that x ∈ cl(A), showing that A∪A0 ⊆ cl(A).
Theorem 6.7 is often helpful for determining the closure cl(A) of a given subset A in a topological space. However, it is not so useful for determining the derived set A0, in part this is because there is no general relationship between the sets A and A0 (in particular, these two sets need not be disjoint). Example 6.8. In the Euclidean line (R,Teuclid) consider the subsets A1 = (0,3), A2 = (0,3)∪{5} and A3 = {1/n | n ∈Z+} . Then it is not difficult to determine the derived sets to be
A0 1 = [0,3], A0 2 = [0,3] and A0 3 = {0} , and from this we find the closures cl(A1) = [0,3], cl(A2) = [0,3]∪{5} and cl(A3) = {0}∪{1/n | n ∈Z+} . Notice that of these three sets A it’s only the last one where A∩A0 = ∅.
7. Continuous Functions and Subspaces.
Let (X,T1) and (Y,T2) be topological spaces. A function f : X → Y is said to be continuous provided that f−1(U) ∈T1 whenever U ∈T2. To summarize this situation we will sometimes write that f : (X,T1) → (Y,T2) is a continuous function. Example 7.1. Consider the function f : R → R given by f(x) = x2, and the lower limit topology T` on R. The half-open interval [2,3) is open in the lower limit topology but f−1([2,∞)) = {x ∈R| x2 ∈ [2,∞)} = {x ∈R| 2 ≤ x2} = (−∞,−√2]∪[√2,∞) is not in T` (because −√2 ∈ f−1([2,∞)) but [−√2,−√2 + ) is not a subset of f−1([2,∞)) for any  > 0.) This shows that f : (R,T`) → (R,T`) is not continuous. Notice that since f−1([2,3)) is not open in the Euclidean topology, this also shows that f : (R,Teuclid) → (R,T`) is not continuous. But f : (R,Teuclid) → (R,Teuclid) is continuous (see theorem 7.4 below), and f : (R,T`) → (R,Teuclid) is also continuous.
Theorem 7.2. Let (X,T1) and (Y,T2) be topological spaces. A function f : X → Y is continuous if and only if f−1(C) is a closed set in X whenever C is a closed set in Y .
Proof. Suppose that f : X → Y is continuous and that C is a closed set in Y . Then Y −C is an open set in Y , and so f−1(Y −C) is an open set in X by the definition of continuity. Then X −f−1(Y −C) = {x ∈ X | x / ∈ f−1(Y −C)} = {x ∈ X | f(x) / ∈ Y −C} = {x ∈ X | f(x) ∈ C} = f−1(C) .
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Since X −f−1(Y −C) is a closed set (because f−1(Y −C) is open), this shows that f−1(C) is a closed set in X. The other direction of the if-and-only-if statement is proved in a similar fashion. Example 7.3. Again consider the function f : R→R given by f(x) = x2 but this time with respect to the cofinite topology Tcofinite on R. In this topology a closed set is either R or a finite subset F ⊂R. Since f−1(R) = R and f−1(F) is finite [Can you explain this? How many elements does f−1(F) have?], the function f : (R,Tcofinite) → (R,Tcofinite) is continuous by the previous theorem.
Theorem 7.4. Let f : R→R be a function. Then f : (R,Teuclid) → (R,Teuclid) is continuous if and only if f : R→R is continuous with the δ–definition of continuity. Proof. Let f : R → R be a function. Before starting on the proof let’s recall that the δ– definition of continuity for f means: for each x0 ∈ R and each  > 0 there is a δ > 0 such that |f(x)−f(x0)| <  whenever |x−x0| < δ. First assume that f : R → R is continuous with the δ–definition. Suppose that U ∈ Teuclid and that x0 ∈ f−1(U). By the definition of the Euclidean topology on R, there is an  > 0 such that B(f(x0,)) = (f(x0) − ,f(x0) + ) ⊆ U. By the δ–definition, there is a δ > 0 so that if |x − x0| < δ then |f(x) − f(x0)| < . Thus if |x − x0| < δ then f(x) ∈ (f(x0)−,f(x0) + ). It follows that f−1(U) ⊇ f−1((f(x0)−,f(x0) + )) ⊇{x ∈R||x−x0| < δ} = (x0 −δ,x0 + δ) . So we have shown that if x0 ∈ f−1(U) then there exists δ > 0 with B(x0,δ) ⊆ f−1(U) and this implies that f−1(U) is open in the Euclidean topology on R. Since U was an arbitrary open set we have established that f is continuous. Now assume that f is continuous (with the topology definition). Let x0 ∈ R and let be a positive real number. The open interval U = (f(x0)−,f(x0) + ) is an open set in the Euclidean topology. Therefore f−1(U) is an open set as well (applying the continuity of f). Since x0 ∈ f−1(U) (because f(x0) ∈ U) there is positive real number δ with B(x0,δ) ⊆ f−1(U). If x is a real number with |x−x0| < δ then x ∈ B(x0,δ) and therefore x ∈ f−1(U). This means that f(x) is an element of U = (f(x0)−,f(x0)+), in other words|f(x)−f(x0)| < . Thus we have shown that f is continuous with the δ–definition of continuity, and that completes the proof. Consequence: Every function f : R→R known to be continuous in Calculus I is continuous with respect to the Euclidean topology on R. For example, polynomial functions are continuous as is f(x) = cos(x), and etc. This consequence even extends to functions f : X → R where X is a subset of R (here we take the topology on X to be the ‘subspace topology’ coming from the Euclidean topology on R as described below). Thus for example, rational functions and the natural logarithm function are continuous when we take the domain of the function to be the ‘domain of definition’ (that is, the set of all real numbers for which the equation makes sense).
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Let (X,T) be a topological space, and let A be a subset of X. We define a collection TA of subsets of A by: TA = {V ⊂ A | V = U ∩A for some U ∈T} = {U ∩A | U ∈T} . Theorem 7.5. Let (X,T) be a topological space, and let A be a subset of X. Then TA forms a topology on A. This is called the subspace topology on A.
Proof. I leave it to you to verify the four axioms (T1)–(T4) for a topology. Theorem 7.6. Let (X,T) be a topological space and let (A,TA) be a subspace. Then the inclusion map i : A → X is continuous. Proof. If U is an open set in X then i−1(U) = U ∩A and U ∩A ∈TA. Example 7.7. Consider the Euclidean line (R,Teuclid), and the subsets A1 = [0,4], A2 = [0,1]∪[3,4] and A3 = Z . We describe some aspects of the subspace topology on A1, A2 and A3. Subspace topology on A1: Notice that the interval [0,1) ⊂ A1 is open in the subspace topology on A1 since [0,1) = (−1,1)∩A1 and (−1,1) (being an open interval) is an open set in the Euclidean topology. On the other hand [0,1) is not an open set in the Euclidean topology on R. Subspace topology on A2: Since [0,1] = (−1,2)∩A2 and [3,4] = (2,5)∩A2, the sets [1,2] and [3,4] are open sets in the subspace topology on A2. Moreover, observe that A2 −[1,2] = [3,4] and it follows that [0,1] is a closed set in the subspace topology on A2. Subspace topology on A3: Let n an element of A3 = Z. Then the open interval (n− 1/2,n + 1/2) is an open set in the Euclidean line. Thus (n−1/2,n + 1/2)∩Z = {n} is an open set in the subspace topology on Z. This shows that every singleton subset {n} of Z is open in the subspace topology. If V is any subset of Z then V = ∪n∈V{n} so V is an open set by axiom (T3). This shows that the subspace topology on Z is the discrete topology.
When A is a subset of Euclidean n-space Rn then we refer to the subspace topology as the Euclidean topology on A. Theorem 7.8. Let X and Y be topological spaces and let f : X → Y be a continuous function. Let A be a subset of X. Then the function g : A → f(A) defined by g(a) = f(a) for each a ∈ A is continuous when A ⊆ X and f(A) ⊆ Y have the subspace topologies. Proof. Let f : X → Y be a continuous, let A ⊆ X and let g : A → f(A) be as defined. Suppose U is an open subset of f(A) with the subspace topology. Then there is an open set U0 in Y so that U = U0∩f(A). Then g−1(U) = {a ∈ A | g(a) ∈ U} = {a ∈ A | g(a) ∈ U0} = {x ∈ X | f(x) ∈ U0}∩A = f−1(U0)∩A . Since f−1(U0) is open in X by the continuity of f, it follows that g−1(U) is open in the subspace topology on A. Therefore g : A → f(A) is continuous.