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so that (6) may be used, we have F (a) has been given as a Riemann sum for f plus some error term R. But it appears now − R | | ≤ < n F (xi) Xi=1 n ε(xi − Xi=1 F (xi − 1) f (ξi)(xi − − xi − − 1) = ε(b xi − − a). 1) ClassicalRealAnalysis.comThomson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008) 532 The I... |
defines a function f on D by the equation fn} { ∈ be a sequence of functions defined on a common domain D. If limn fn(x) converges pointwise on D. This limit →∞ D, we say that the sequence fn} { We write limn fn = f or fn → f . For the infinite sum, the simplest idea is to extend the definition of finite sum using our famil... |
k . 9.3 Uniform Limits Pointwise limits do not allow the interchange of limit operations. In many situations, uniform limits will. To see how the definition of a uniform limit needs to be formulated, let us return to the sequence of Example 9.4. That sequence illustrated the fact that a pointwise limit of continuous fun... |
N , ≥ for all x ∈ convergent on D. D. Thus the sequence Sn} { Sm(x) + Mn. Sn(x) | ∞k=0 Mk converges by hypothesis, there exists an integer N such that if Mm+1 + | ≤ · · · − Mm+1 + · · · + Mn < ε. Sn(x) Sm(x) < ε | − | is uniformly convergent on D; that is, the series ∞0 fk is uniformly P Example 9.17: Consider again t... |
1, 1). Show that this series does converge uniformly ∞ k=1 kxk−1 converges pointwise but not uniformly on ( on every closed interval [a, b] contained in ( P − 1, 1). − f (x + 1/n) , −∞ ∞ f (x) lim n sup x∈D | fn(x) f (x) | − = 0. { xk} fnk (xk) | f (xk) ε0. | ≥ − ClassicalRealAnalysis.comThomson*Bruckner*BrucknerElemen... |
es pointwise to a continuous function f on [0, Why does this not contradict Theorem 9.24? { ∞ on the interval [0, ) that is monotonic decreasing ) but for which the convergence is not uniform. ∞ 9.4.6 Let fn} { f be a sequence of continuous nondecreasing functions defined on an interval [a, b]. Suppose fn → pointwise on... |
but now fn(1/(2n)) = 1 fn (x) dx 0. → 0 Z These functions form a uniformly bounded sequence of functions : that is, there exists a constant M (M = 1 in this case) such that [0, 1]. A theorem (whose proof is fn} beyond the scope of this chapter) asserts that if a uniformly bounded sequence of integrable functions conve... |
b], x ≥ ∈ fm fn(x0) fm(x0) < ε | − | From this we deduce that fn(x) fm(x) − − [fn(x0) − fm(x0)] = (x x0)[f ′n(ξ) f ′m(ξ)]. − − (17) fn(x) | − fm(x) | ≤ | fn(x0) fm(x0) − < ε(1 + (b x0)(f ′n(ξ) f ′m(ξ) | − − (x + | a)) | − ≥ N . Since this N depends only on ε this assertion is true for all x for any n, m verified that th... |
zero. Similarly, Exercise 7.4.2 provides an example of a differentiable function g whose derivative g′ is discontinuous at every point of a Cantor set that does not have measure zero. We might ask the question, “Can the derivative of a differentiable function be discontinuous everywhere?” We shall see that the answer is ... |
empting this chapter. The notion of a radius of convergence depends naturally on these concepts. 10.1 Introduction One of the simplest and, arguably, the most important type of series of functions is the power series. This is a series of the form or, more generally, ∞ It represents the notion of an “infinitely long” pol... |
10 If the coefficients ak} convergence is at least 1. { Xk=0 of a power series Xk=1 ∞ 0 akxk form a bounded sequence show that the radius of P ∞ 0 akxk has a radius of convergence Ra and the power series ak| ≤ | | P ∞ 0 akxk has a radius of convergence R, what must be the radius of convergence of the for all k sufficiently... |
ion of Power Series If a function is represented by a power series, is it possible to differentiate that function by differentiating the power series term by term? Note that for continuity and integration we were able to prove Theorems 10.11 and 10.13 immediately from general theorems on uniform convergence. To prove a t... |
nverges to f on I? Or even that the series converges at all on I. The answer to both questions is “no.” Example 10.22: Consider, for example, the function f (x) = 1/(1 + x2). This function is infinitely differentiable on all of the real line. Its Taylor series about x = 0 is, as we have seen in Example 10.21, 1 − x2 + x4... |
s)n ds − n!f (b) bn+1 ≤ 623 10.5.4 Let f (x) = 0, e−1/x2 , if x = 0 = 0. if x Prove that f is infinitely differentiable on the real line. Show that f (k)(0) = 0 for all k IN. Explain why the Taylor series for f about x = 0 does not represent f in any neighborhood of zero. Is f analytic at x = c for = 0? c See Note 253 ∈ ... |
known as Fourier series. The aj and bj are called the Fourier coefficients of f . 10.8.1 Uniform Convergence of Trigonometric Series " Enrichment section. May be omitted. For a first taste of this theory we prove an interesting theorem that justifies some of Fourier’s original intuitions. We show that if a trigonometric se... |
ect. 10.8.4 Weierstrass Approximation Theorem " Enrichment section. May be omitted. Fej´er’s theorem allows us to prove the famous Weierstrass approximation theorem. Note that a consequence of Fej´er’s theorem is that continuous, 2π-periodic functions can be uniformly approximated by trigonometric polynomials. The Weie... |
*BrucknerElementary Real Analysis, 2nd Edition (2008) Section 11.2. The Metric Structure of Rn 647 (a) Show that “ ” satisfies the conditions of Theorem 11.2. · (b) Show that the set of functions form an orthogonal set of functions in , that is, f C · g = 0 if f, g ∈ T = sin nx, cos mx : (n = 1, 2, . . . m = 1, 2, . . .... |
bsets of Rn that are both open and closed. 11.3.4 Prove the following analogues of Theorems 4.17 and 4.18 in the setting of Rn. ∅ and Rn are both open and closed. (a) The sets (b) Any intersection of a finite number of open sets is open. (c) Any union of an arbitrary collection of open sets is open. (d) The complement o... |
k2 (d) xk = ( ln k k2+1 , ( − (e) xk = ( sin k k , k sin 1 k ) √k, 7) (f) xk = (√k + 1 1)k) − 11.4.8 Which of the sequences in Exercise 11.4.7 have convergent subsequences? 11.4.9 Prove that if , then euclidean norm is replaced by the norms xk} → x as k → ∞ { xkk → k k k·k1 or k → ∞ k·k∞ of Exercise 11.4.1? x as k . D... |
ssicalRealAnalysis.comThomson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008) 668 The Euclidean Spaces Rn Chapter 11 (a) Describe the sets in the xy-plane that H maps onto horizontal lines in the uv-plane. Do the same for vertical lines. (b) Determine a set S ⊂ R2 such that H(S) = [1, 2] [4, 5]. × 11.6 Li... |
⊂ then let x0 be an accumulation point of E and let f, g : E x lim x0 → f (x) = y0 and lim x0 → x g(x) = z0, Rm. If → x lim x0 → (f (x) g(x)) = y0 · · z0. Proof. We apply Lemma 11.30. Let xk → vk → Then uk → y0 and vk → z0, so uk · = x0. For k z0 by Theorem 11.16(iii). E, xk 6 x0, xk ∈ y0 · IN, let uk = f (xk), vk = g(... |
nce with E such that f (xk) = yk. Since E is compact, the sequence { E. Since f is continuous at x0, yk} { that converges to the point f (x0) ∈ f (E). Corollary 11.47: Let E maximum and absolute minimum on E. ⊂ Rn be compact and let f : E R be continuous. Then f achieves an absolute → Proof. The set f (E) is a compact ... |
n in Section 11.10. 267Exercise 11.3.8. The point of this problem is that since the open sets are exactly the same, so too will be all the other concepts whose definitions can be given entirely in terms of open sets. The same will be true when in later sections we consider convergence of sequences or limits, continuity,... |
and second order and h11 + h22 = 0 at all points of D. This equation is called Laplace’s equation. Verify that each of the following is harmonic on all of R2 and, for each, verify that f12 = f21 and f112 = f121 = f211. (a) ex cos y (b) y2 3x2y − ClassicalRealAnalysis.comThomson*Bruckner*BrucknerElementary Real Analysis... |
the sense defined later in Section 12.4. f221 = (f2)21 = (f2)12 = f212 = (f21)2 = (f12)2 = f122. ClassicalRealAnalysis.comThomson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008) Section 12.2. Partial and Directional Derivatives 701 Partial Derivatives of f : Rn R Theorem 12.5 also extends to functions of m... |
at the tangent line is a close approximation to a differentiable function. 12.4.1 Approximation by Linear Functions Let’s see what is involved by looking at the one variable situation more carefully. Suppose f : R x0 ∈ R, and L(x) = a0x + a1 R, → ClassicalRealAnalysis.comThomson*Bruckner*BrucknerElementary Real Analysis... |
*BrucknerElementary Real Analysis, 2nd Edition (2008)6 716 Differentiation on Rn Chapter 12 Then f is differentiable at x if there exists a linear function L : Rn R such that when ε is defined by f (x1 + h1, . . . , xn + hn) n L(x)hi + ε → f (x1, . . . , xn) = − 2 + h1| ( | + 2) hn| | · · · p + 2) hn| | · · · → 0. Prove t... |
all tangent lines at a point exist and do lie in the same plane, then that plane must be the tangent plane.” Is the second statement correct? That is, if f : R2 R is continuous on R2 with f (0, 0) = 0 and every directional derivative at (0, 0) is zero, then f is differentiable at (0, 0) and the xy-plane is the tangent ... |
informal discussion simple, we avoid technicalities such as the domains of definition of the functions and the precise hypotheses needed for the resulting chain rules. Example 12.26: Let u = u(x, y) and let z = F (u). We can view z as a function G of x and y via the intermediate variable u. Thus z = F (u) = F (u(x, y))... |
t t0 and ∈ G′(t0) = F1(x0, y0)f ′(t0) + F2(x0, y0)g′(t0). ClassicalRealAnalysis.comThomson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008) Section 12.5. Chain Rules Proof. Write x = f (t), y = g(t). Then Since F is differentiable at (x0, y0), we can write G(t) − G(t0) = F (f (t), g(t)) = F (x, y) − F (x0, ... |
itutions u = x + ct, z = g(u) we arrive at the following schema, which is equivalent to schema (43) of Example 12.26. f (x, t) = g(x + ct), x H t HH HHj * u - z Thus the appropriate chain rule becomes f1(x, t) = g′(u) f2(x, t) = g′(u) ∂u ∂x ∂u ∂t = g′(x + ct) = cg′(x + ct) ClassicalRealAnalysis.comThomson*Bruckner*Bruc... |
nuous function of (x, y) in that neighborhood. p ≤ ± − ≤ − 9 x2 y2 (84) ◭ Theorem 12.44 that follows is the analogue of Theorem 12.40 when we deal with one equation in more than two variables. We state the theorem for n + 1 variables. ClassicalRealAnalysis.comThomson*Bruckner*BrucknerElementary Real Analysis, 2nd Editi... |
pose | such that J | ∂F ∂u ∂G ∂u is not zero at p0. Then there are neighborhoods I0 and J0 of (x0, y0) and (u0, v0), respectively, ∂(F, G) ∂(u, v) = = J | | . ∂F ∂v ∂G ∂v (i) To each (x, y) I0 there corresponds a unique (u, v) (x, y, u, v). This correspondence defines u and v as functions on I0 by ∈ ∈ J0 such that equat... |
s, we can write this in the form f (t0 + h) T (t0) = T (h) − − f (t0) h − T (h) k = 0. (95) k lim 0 h → Let’s look at the familiar setting, m = 2, that we used as an introduction to this section: x = x(t) y = y(t) (a t ≤ ≤ b), where x and y are continuous functions on [a, b]. As we mentioned, these equations define a fu... |
with partial derivatives.) . (100) Our task is to approximate a function given by (100) using a linear transformation given by (99). Specifically, we seek a linear transformation T from Rn to Rm, with T = T x depending on x, such that k lim h k→ k 0 f (x + h) f (x) h k − k − T (h) k = 0. ClassicalRealAnalysis.comThomso... |
f at x) and β is the magnification factor of B (and f at x). This suggests that the linear of g at f (x)), then βα is the magnification factor of BA (and of g ◦ ClassicalRealAnalysis.comThomson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008) 786 Differentiation on Rn Chapter 12 transformation BA is the deri... |
axioms, that for a partially ordered set. (Exercises 13.1.2 and 13.1.3 make these last two statements precise.) There are also several notions of “distance” that can be considered between pairs of continuous functions. We shall see some of them in this chapter. We shall develop an abstract structure on sets of objects... |
) for all x, y, z X ∈ As a question of mathematical taste, would you prefer to use these conditions rather than the four in Definition 13.1 for the definition of this term? 13.2.5 Let X = x1, x2, x3, . . . , xn} { entry is d(xi, xj). What properties must such a matrix have? be a finite set and let d be a metric on X. Cons... |
0, 1] furnished with the metric 1 Show that d is a metric on C d(f, g) = f (t) g(t) dt. 0 | Z [0, 1] different from the one in Example 13.10. − | 13.3.5 Let [0, 1] consist of all integrable functions on [0, 1] (not necessarily continuous). Let R d(f, g) = 1 f (t) 0 | Z g(t) dt. | − Show that d is not a metric on See Not... |
e the family of nonempty closed subsets of [0, 1] furnished with the Hausdorff metric of Exercise 13.3.9. Determine whether the following sequences converge and if so to what they converge. (a) An = [0, 1/n] ClassicalRealAnalysis.comThomson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008) 816 Metric Spaces ... |
rbitrary intersection of closed sets is closed. ClassicalRealAnalysis.comThomson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008) Section 13.5. Sets in a Metric Space 823 13.5.11 Let X denote the set of points in R furnished with the usual real metric. Answer the following questions: 0 { } ∪ { 1/k : k = 1,... |
again at Example 13.21. Here x1x2 1 + x2 x2 2 Let us check continuity at (0, 0). Since f (0, 0) = 0 (by definition), f will be continuous at (0,0) if and only if f (x1, x2) = f (0, 0) = 0. for every sequence of points (un, vn) that f is not continuous there. For example, observe that f (1/n, 1/n) f (0, 0) to be 1/2; in... |
the metric d2 the ball will be the inside of a circle of radius r, and for d1 the ball will be the inside of a square of side length r√2 with sides parallel to the lines x2 = x1. ∞ Let us denote the balls in the three spaces ± by B1(x, r), B2(x, r) and B r > 0 ∞ (R2, d1), (R2, d2), and (R2, d ) ∞ (x, r), respectively. ... |
that (X, e) is topologically equivalent to (X, d), where d is the discrete metric if and only if any one of the following properties holds: (a) Any intersection of a family of open sets is open. (b) For any open set G the closure G is also open. (c) Every point x in X is isolated. ClassicalRealAnalysis.comThomson*Bruck... |
le. For a countable dense set take the rationals ◭ Q. (There are many other countable dense sets in R.) Q of irrational numbers is separable. For a countable dense set take, for Example 13.48: The space R example, the set of all numbers of the form m√2/n, where m and n are integers. (Note that we cannot ◭ take Q this t... |
instead as a consequence of the completeness of M [a, b].) Recall that the metric here is the sup metric and convergence reduces to what we called uniform convergence in Chapter 9. Let sequence in M [a, b]. fk} { be a Cauchy d(f, g) = sup a t ≤ ≤ f (t) , g(t) | − b | Step 1. We wish to find a natural candidate for the ... |
function mapping X onto Y . (a) If X is separable, must Y be separable? (b) If X is complete, must Y be complete? (c) Is separability a topological property? Is completeness? (d) Do the answers to (a) and (b) change if f is an isometry? See Note 346 13.8.17 Let (X1, d1) and (X2, d2) be complete metric spaces. Is the pr... |
= max [0, 1 x 2 ] | g1(x) g2(x) | − x ∈ ∈ = max [0, 1 x [f1(t) − f2(t)] dt max [0, 1 t ∈ 2 ] | f1(t) f2(t) | dt − f1(t) f2(t) | − max [0, 1 x ∈ max [0(f1, f2). [0, 1/2] such that A(f ) = f , ∈ C Thus A is a contraction map. From Theorem 13.75 we can conclude that there is a unique function f that is, such that x f (x)... |
tioned in this section. Exercises 13.10.1 What conditions on the values of a and b in Example 13.78 will force the function A(x) = (1 be a contraction on R? Examine the sequence of iterates in this case. See Note 351 a)x + b to − 13.10.2 Apply the method of Example 13.79 to solve the system of two linear equations a11x... |
of. As we indicated in Example 13.82, we formulate our problem in terms of the integral equation δ, x0 + δ] for which φ(x0) = y0. φ(x) = y0 + f (t, φ(t)) dt, x (26) δ, x0 + δ]. Since D is open, there exists a closed sphere S centered which is to be valid on the interval [x0 − at (x0, y0) and contained entirely inside t... |
mple also illustrates. Consider the real line R but furnished with a different metric from the usual, the metric of Exercise 13.2.1(c) or (d). Since that metric is equivalent to the usual metric, every closed subset of R has all the properties of Theorem 13.90; the only difference here is that under the new metric all se... |
e ∈ 13.12.32 Use Exercise 13.12.31 to prove that if X is compact and f : X Y is continuous, then f is uniformly → ∈ C continuous. 13.12.33 Show that the property of Exercise 13.12.31 is not equivalent to compactness in R (i.e., find a noncompact R with this property). set K See Note 362 ⊂ 13.12.34 Show that a metric spa... |
that K is totally bounded in [a, b]. Then K is bounded in uniformly bounded family of functions. We show that K is equicontinuous. Let ε > 0, and let f1, f2, . . . , fn be an (ε/3)-net in K. Let f K. There exists j C C ∈ ≤ f (z) n such that < 1 fj(z) 3 ε. − | max z ∈ [a,b] | (28) (29) . Then, for x, y [a, b], ∈ f (x) f... |
be a closed rectangle contained in D having sides parallel to the coordinate axes and having (x0, y0) as center. Let M W = (x, y) R : { ∈ 1 be an upper bound for y0| ≤ x0|} M − x , | ≥ − y | on R. Let f | | and let [a, b] be the projection of W onto the x-axis, as in Figure 13.8. We next obtain a family K of functions... |
n A1 ∪ · · · ∪ dense in X. An is also nowhere 13.13.4 Is it true in an arbitrary metric space that every finite set is nowhere dense? See Note 374 13.13.5 Describe what property a metric space must have in order that every finite set is nowhere dense. See Note 375 13.13.6 (a) Show that a set A in a metric space X is nowh... |
Analysis.comThomson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008) 916 Metric Spaces Chapter 13 f g Figure 13.9. Graphs of f (x) = x | | p sin(1/x) and g(x) = f (x) . The asymptotes y = | ± | x | | are also shown. p Example 13.115: It is easy to give examples of continuous functions that “wiggle” so much... |
, 2nd Edition (2008) 922 Metric Spaces Chapter 13 s1 s1+δ m s1−δ s2 I Figure 13.11. The interval I in the proof that most members of are nowhere dense. K Suppose that we, like the nineteenth-century mathematicians, had not heard of Cantor sets, but, unlike nineteenth-century mathematicians, did know the Baire category ... |
a fixed x0 ∈ h(x) for the function (h(x)) (y) = d(x, y) X and define a mapping h : X M (X) by writing → (a) Show that each h(x) is a bounded function on X for each x (b) Show that h is an isometry of X to a subspace of M (X). ∈ d(y, x0) (y − X). ∈ X. (c) Show that every metric space is isometric to a subspace of some co... |
bilateral point. 330Exercise 13.6.47. Give an example of two subsets A and B of R, each of which is isometric to a subset of the other but that are not themselves isometric. Use and A = { 2, 3, 4, 3, 4, . . . } ∪ { . } 331Exercise 13.6.52. Note that (0, 1) ⊂ X would map onto a set of diameter 1. 332Exercise 13.7.3. Th... |
omson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008) 940 NOTES Show that AI,J is nowhere dense. and x2 ∈ J such that f (x1) = f (x2) } . 377Exercise 13.15.1. If E is not a countable union of members of , then E is closed. A 378Exercise 13.15.2. Hint for (b): Define in an appropriate manner and obtain n ∈ ... |
838 of the expression “mathematical induction” and the first to give a rigorous account of it. He has one interesting claim to fame, in addition to his “laws:” He was the tutor of Lady Ada Lovelace, who some say is the world’s first computer programmer. A puzzle of his survives: He claims that he “was x years old in the ... |
] and [n, n + 1]. n=1 \ A.2.4 Do you accept any of the following as an adequate definition of the function f ? (The domain is not specified but it is assumed that you will try to find a domain that might work.) (a) f (x) = 1/√1 (b) f (x) = x if x is rational and f (x) = x. − x if x is irrational. − (c) f (x) = 1 if x cont... |
e reading a proof: Try to remember what it is that has to be proved. Before reading the proof decide what it is that must be proved exactly. Ask yourself, “What would I have to show to prove that?” How to Write a Proof Practice! We learn to write proofs by writing proofs. Start by just copying nearly word for word a pr... |
lRealAnalysis.comThomson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008)6 Section A.8. Induction For example, if the formula is valid, then M + 1) = M (M + 1) 2 M (M + 1) 2 + (M + 1) = M (M + 1) + 2(M + 1) 2 = (M + 1)(M + 2) 2 , A-21 which is indeed the correct formula for n = M + 1. Thus there never can ... |
sis, 2nd Edition (2008) Subject Index in Rn, 652 algebra of functions, 795 algebraic number, 41 algebraic properties of sequence limits, 53 alternating harmonic series, 125 analytic function, 482, 620 annuity, 111 antiderivative, 485 archimedean property, 16 arithmetic progression, 33 arithmetic-geometric mean inequali... |
d, 664 function space, 806 functions series, 538 uniformly bounded, 361 fundamental theorem of calculus, 499, 525 generalized Riemann integral, 532 geometric progression, 34, 72 formula for sum, 108 geometric series, 124 Gordon, R. A., 532 gradient, 718 greatest integer function, 14 greatest lower bound, 13 harmonic fu... |
tegrable, 510 sums, 489 right continuous, 319 right-hand derivative, 399 right-hand limit, 274 rigid motion, 843 Rolle’s theorem, 427 Russell’s paradox, A-26 Russell, B., A-26 Saks, S., 473 second category set, 359, 912 second derivative, 401 second partial derivative, 691 separable space, 846 separate a set, 684 separ... |
alan). After having read part of the book from the non- printable pdf file, I have concluded that this is the book that I want to read to learn topology.” Long N., USA “I have never seen any book so clear on such a difficult subject”; Renato O., Chile: “Congratulations for your great book. I went through the first chapters... |
h the definition of a topology and is then devoted to some simple examples: finite topological spaces, discrete spaces, indiscrete spaces, and spaces with the finite-closed topology. Topology, like other branches of pure mathematics such as group theory, is an axiomatic subject. We start with a set of axioms and we use th... |
.] 1.1. TOPOLOGY 31 5. Let R be the set of all real numbers. Prove that each of the following collections of subsets of R is a topology. (i) τ 1 consists of R, Ø, and every interval (−n, n), for n any positive integer, where (−n, n) denotes the set {x ∈ R : −n < x < n}; (ii) τ 2 consists of R, Ø, and every interval [−n... |
uch Ui plus the set Y . Verify that τ i is indeed a topology on Y . Deduce that for each topology on X, there are at least n distinct topologies on Y .] (iii) If X is any infinite set of cardinality ℵ, prove that there are at least 2ℵ distinct topologies on X. Deduce that every infinite set has an uncountable number of d... |
the Sierpinski Space 5. A topological space (X, τ ) is said to be a T0-space if for each pair of distinct points a, b in X, either2 there exists an open set containing a and not b, or there exists an open set containing b and not a. (i) Prove that every T1-space is a T0-space. (ii) Which of (i)–(vi) in Exercise 3 abov... |
on R. sets in this topology and all closed intervals are closed sets. In particular, we shall see that all open intervals are indeed open (ii) Let r, s ∈ R with r < s. In the euclidean topology τ on R, the open interval (r, s) does indeed belong to τ and so is an open set. Proof. We are given the open interval (r, s).... |
le. Let X = {a, b, c} and B = {{a}, {c}, {a, b}, {b, c}}. Then B is not a basis for any topology on X. To see this, suppose that B is a basis for a topology τ . Then τ consists of all unions of sets in B; that is, τ = {X, Ø, {a}, {c}, {a, c}, {a, b}, {b, c}}. (Once again we use the fact that Ø is an empty union of memb... |
a union of members of B. So let V be any open set. Then for each x ∈ V , there is a Bx ∈ B such that x ∈ Bx ⊆ V . Clearly V = x∈V Bx. (Check this!) Thus V is a union of members of B. Let B be a basis for a topology τ on a set X. Then 2.3.3 Proposition. a subset U of X is open if and only if for each x ∈ U there exists ... |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... |
cal space we do not have a “distance function”, so we must proceed differently. We shall define the notion of limit point without resorting to distances. Even with our new definition of limit point, the point 0 will still be a limit point of (0, 1] . The introduction of the notion of limit point will lead us to a much bet... |
Using (iii) show that R has an uncountable number of distinct dense subsets. [Hint: Uncountable sets are discussed in Appendix 1.] (v)* Again using (iii), prove that R has an uncountable number of distinct countable dense subsets and 2c distinct uncountable dense subsets. [Hint: Note that c is discussed in Appendix 1.... |
st upper bound, if they exist. (i) S = R. (ii) S = Z = the set of all integers. (iii) S = [9, 10). (iv) S = the set of all real numbers of the form 1 − 3 n2 , where n is a positive integer. (v) S = (−∞, 3]. 1Most books use this property to define connectedness. 88 CHAPTER 3. LIMIT POINTS 3. Let (X, τ ) be any topologica... |
here A is closed in the euclidean topology on R and T is any subset of S. The complements of these closed sets form a topology τ on R which is Hausdorff but not regular. 96 CHAPTER 4. HOMEOMORPHISMS 4.2 Homeomorphisms We now turn to the notion of equivalent topological spaces. We begin by considering an example: and X =... |
homeomorphisms” amongst the exercises: (i) T0-space; (ii) T1-space; (iii) T2-space or Hausdorff space; (iv) regular space; (v) T3-space; (vi) satisfying the second axiom of countability; (vii) separable space. [See Exercises 4.2 #7.] There are also others: (viii) discrete space; (ix) indiscrete space; (x) finite-closed t... |
oduced the notion of group of homeomorphisms, which is an interesting and important topic in its own right. Chapter 5 Continuous Mappings Introduction In most branches of pure mathematics we study what in category theory are called “objects” and “arrows”. In linear algebra the objects are vector spaces and the arrows a... |
n indiscrete space, prove that f is continuous. 8. Let (X, τ ) and (Y, τ 1) be topological spaces and f : (X, τ ) → (Y, τ 1) a continuous mapping. Let A be a subset of X, τ 2 the induced topology on A, B = f (A), τ 3 the induced topology on B and g : (A, τ 2) → (B, τ 3) the restriction of f to A. Prove that g is contin... |
that an open subset of R2 is connected if and only if it is path-connected. 12.* Let A and B be subsets of a topological space (X, τ ). If A and B are both open or both closed, and A ∪ B and A ∩ B are both connected, show that A and B are connected. Zero-Dimensional Spaces 13. A topological space (X, τ ) is said to be ... |
ach a ∈ U there exists an ε > 0 such that the open ball Bε(a) ⊆ U . Assume that U ∈ τ . Then, by Propositions 2.3.2 and 6.1.17, for any Proof. a ∈ U there exists a point b ∈ X and a δ > 0 such that a ∈ Bδ(b) ⊆ U. Let ε = δ − d(a, b). Then it is readily seen that a ∈ Bε(a) ⊆ U. Conversely, assume that U is a subset of X... |
ve that every non-trivial interval (a, b), a, b ∈ R, is locally euclidean. (ii) Let S1 be the subspace of R2 consisting of all x ∈ R2 such that d(x, 0) = 1, where d is the Euclidean metric. Show that the space S1 is locally euclidean. (iii) Show that every topological space locally homeomorphic to Rn, for any positive ... |
ntially closed. Prove that if (X, τ ) is a metrizable space, then every sequentially closed set is closed and every sequentially open set is open. (ii)∗ Find an example of a (nonmetrizable) topological space in which not every sequentially closed subset is closed. (iii) A topological space (X, τ ) is said to be a seque... |
P of all irrational numbers with its induced topology is completely metrizable. Also as (0, 1) is a completely metrizable subspace of R which is not a closed subset, we see that Proposition 6.3.13(ii) would not be true if complete metric were replaced by completely metrizable. 6.3.17 Definition. A topological space is ... |
e metric subspace of a metric space is closed. 6. Prove that for each positive integer n, Rn is a Polish space. 160 CHAPTER 6. METRIC SPACES 7. Let a, b ∈ R, with a < b. Prove that each discrete subspace of R and each of the spaces [a, b], (a, b), [a, b), (a, b], (−∞, a), (−∞, a], (a, ∞), [a, ∞), and {a}, with the topo... |
This theorem is a consequence of the Baire Category Theorem 6.5.4. 6.5. BAIRE SPACES 167 6.5.11 Proposition. then the interior of Y is empty. If Y is a first category subset of a Baire space (X, τ ), Proof. As Y is first category, Y = dense. Let U ∈ τ be such that U ⊆ Y . Then U ⊆ ∞ n=1 Yn ⊆ ∞ n=1 Yn. ∞ n=1 Yn, where eac... |
trics on the same set can give rise to the same topology. Such metrics are called equivalent metrics. We were introduced to the study of function spaces, and in particular, C[0, 1]. En route we met normed vector spaces, a central topic in functional analysis. Not all topological spaces arise from metric spaces. We saw ... |
(X, τ ). Let Ui ∈ τ , i ∈ I, Proof. be any open covering of A. Then X ⊆ ( i∈I Ui) ∪ (X \ A); that is, Ui, i ∈ I, together with the open set X \ A is an open covering of X. , X \ A. [If X \ A is not Therefore there exists a finite subcovering Ui1, Ui2, . . . , Uik in the finite subcovering then we can include it and stil... |
if and only if it is compact. (iii) A topological space (X, τ ) is said to be locally compact if each point x ∈ X has at least one neighbourhood which is compact. Find an example of a locally compact Hausdorff space which is not countably compact. (iv) Show that every continuous image of a countably compact space is cou... |
f any finite number of indiscrete spaces is an indiscrete space. 5. Prove that the product of any finite number of Hausdorff spaces is Hausdorff. 6. Let (X, τ ) be a topological space and D = {(x, x) : x ∈ X} the diagonal in the product space (X, τ ) × (X, τ ) = (X × X, τ 1). Prove that (X, τ ) is a Hausdorff space if and o... |
y ∈ τ 1, W (x, y) ∈ τ 2 and x, y ∈ V (x, y) × W (x, y) ⊆ Ui. As x, y ranges over all points of X × Y we obtain an open covering V (x, y) × W (x, y), x ∈ X, y ∈ Y , of X × Y such that each V (x, y) × W (x, y) is a subset of some Ui, i ∈ I. Thus to prove (X, τ 1) × (Y, τ 2) is compact it suffices to find a finite subcoverin... |
is a locally convex space. (x) Show that a metrizable topological vector space V is separable if and only if it has a compact subset K such that any vector space containing K is dense in V . (xi)(a) Let G be a topological group. Prove that G is separable if and only if it has subgroups G1 ⊆ G2 ⊆ · · · ⊆ Gn ⊆ . . . such... |
Q(tw)| → 0 So there exists a real number t0 with 0 < t0 < 1 such that as t → 0. t0 |wk+1Q(t0w)| < |b0| So, by (5), P (t0w) = b0 + bk(t0w)k + (t0w)k+1Q(t0w) = b0 + bk t0 k −b0 bk + (t0w)k+1Q(t0w) = b0(1 − t0 k) + (t0w)k+1Q(t0w) Therefore |P (t0w)| (1 − t0 < (1 − t0 = |b0| = |P (0)| k)|b0| + t0 k) |b0| + t0 k+1|wk+1Q(t0w... |
continuous i=1 Yi, τ ) i=1 Xi, τ ) → (∞ that is, h(x1, x2, . . . , xn, . . . ) = i=1 hi(xi); It suffices to show that if O is a basic open set in (∞ i=1 Yi, τ ), then i=1 Xi, τ ). Consider the basic open set U1 × U2 × . . . Un × Proof. h−1(O) is open in (∞ Yn+1Yn+2 × . . . where Ui ∈ τ h−1(U1×· · ·×Un×Yn+1×Yn+2×. . . ) =... |
untably infinite product of topological spaces each homeomorphic to the discrete space N. What is much more surprising is the fact, as mentioned in Chapter 6, that N∞ is homeomorphic to P, the topological irrational numbers with the euclidean topology. See Engelking space of all [130] Exercise 4.3.G and Exercise 6.2.A. ... |
That i=1(Xi, τ i) is compact follows from Corollary 9.4.12 and Exercises 9.3 #9 (ii). Our next task is to verify the converse of Urysohn’s Theorem. To do this we introduce a new concept. (See Exercises 2.2 #4.) A topological space (X, τ ) is said to satisfy the second 9.4.14 Definition. axiom of countability (or to be ... |
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