Free ebooks ==> www.Ebook777.com www.Ebook777.com ELEMENTS OF TOPOLOGY K13301_FM.indd 4/12/13 1:25 PM Free ebooks ==> www.Ebook777.com www.Ebook777.com K13301_FM.indd 4/12/13 1:25 PM ELEMENTS OF TOPOLOGY Tej Bahadur Singh K13301_FM.indd 4/12/13 1:25 PM Free ebooks ==> www.Ebook777.com CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2013 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 20130426 International Standard Book Number-13: 978-1-4822-1566-3 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com www.Ebook777.com Dedicated to my mother and grandchildren Amishi, Amil and Pradyumn Free ebooks ==> www.Ebook777.com www.Ebook777.com Contents Author Bio xi Preface xiii Suggested Course Outlines xvii Acknowledgements xix List of Symbols xxi TOPOLOGICAL SPACES 1.1 1.2 1.3 1.4 1.5 Metric Spaces Topologies Derived Concepts Bases Subspaces CONTINUITY AND PRODUCTS 2.1 2.2 35 Continuity Product Topology CONNECTEDNESS 3.1 3.2 3.3 3.4 Connected Spaces Components Path-connected Spaces Local Connectivity 35 45 63 CONVERGENCE 4.1 12 19 30 Sequences 63 72 77 82 93 93 vii Free ebooks ==> www.Ebook777.com viii 4.2 4.3 4.4 Nets 96 Filters 103 Hausdorff Spaces 106 COUNTABILITY AXIOMS 5.1 5.2 113 1st and 2nd Countable Spaces Separable and Lindelăof Spaces 113 119 COMPACTNESS 6.1 6.2 6.3 6.4 6.5 Compact Spaces Countably Compact Spaces Compact Metric Spaces Locally Compact Spaces Proper Maps 125 TOPOLOGICAL CONSTRUCTIONS 7.1 7.2 7.3 7.4 7.5 7.6 Quotient Spaces Identification Maps Cones, Suspensions and Joins Wedge Sums and Smash Products Adjunction Spaces Coinduced and Coherent Topologies 159 SEPARATION AXIOMS 8.1 8.2 8.3 8.4 Regular Spaces Normal Spaces Completely Regular Spaces ˇ Stone–Cech Compactification 159 173 180 188 195 202 211 PARACOMPACTNESS AND METRISABILITY 9.1 9.2 125 136 140 148 155 211 216 229 235 241 Paracompact Spaces 241 A Metrisation Theorem 252 www.Ebook777.com ix 10 COMPLETENESS 257 10.1 Complete Spaces 257 10.2 Completion 265 10.3 Baire Spaces 269 11 FUNCTION SPACES 275 11.1 Topology of Pointwise Convergence 275 11.2 Compact-Open Topology 283 11.3 Topology of Compact Convergence 301 12 TOPOLOGICAL GROUPS 12.1 12.2 12.3 12.4 Examples and Basic Properties Subgroups Isomorphisms Direct Products 313 13 TRANSFORMATION GROUPS 313 324 331 341 347 13.1 Group Actions 347 13.2 Orbit Spaces 365 14 THE FUNDAMENTAL GROUP 14.1 14.2 14.3 14.4 14.5 371 Homotopic Maps The Fundamental Group Fundamental Groups of Spheres Some Group Theory The Seifert–van Kampen Theorem 15 COVERING SPACES 15.1 15.2 15.3 15.4 15.5 Covering Maps The Lifting Problem The Universal Covering Space Deck Transformations The Existence of Covering Spaces 371 383 397 408 424 439 439 448 459 468 480 Set Theory 513 many elements For example, if ω ∪ {q} is the ordinal obtained from the ordinal ω by adjoining q to it as the last element, then ω ∪ {q} is not isomorphic to ω, by Proposition A.6.5 But ϕ : ω ∪ {q} → ω defined by ϕ (n) = n + 1, ϕ (q) = is a bijection By Theorem A.6.3, every set X can be well-ordered and, by Theorem A.7.6, has the cardinality of an ordinal number We define |X| to be the least ordinal number λ such that X is equipotent to λ Clearly, the object |X| is uniquely determined by X, and is called the cardinal number of X If X is finite and has n elements, then |X| is the ordinal number n The cardinal number |N| is denoted by ℵ0 , and the cardinal number |R| is denoted by ℵ1 and also by c, called the cardinality of the continuum The mapping x → x/(1 − |x|) is a bijection between the open interval (−1, 1) and R, and the mapping x → (2x − a − b)/(b − a) is a bijection between an open interval (a, b) and (−1, 1) Therefore the cardinality of any open interval (a, b) is c By Theorem A.5.14, we see that the cardinality of a closed interval or a half open interval is also c Given two sets X and Y , we have |X| < |Y | if and only if there exists an injection X → Y but there is no bijection between X and Y Accordingly, it is logical to say that X has fewer elements than Y when |X| < |Y | As seen in §A5, there is no surjection from N to I while there is an injection N → I, since R and I are equipotent So we have ℵ0 < c The sum of two cardinal numbers α and β, denoted by α + β, is the cardinality of the disjoint union of two sets A and B, where |A| = α and |B| = β Obviously, N = {1, 2, , n} ∪ {n + 1, n + 2, } and so n + ℵ0 = ℵ0 Since N is the union of disjoint sets {1, 3, 5, } and {2, 4, 6, }, we have ℵ0 + ℵ0 = ℵ0 Similarly, we obtain ℵ0 + · · · + ℵ0 = ℵ0 Considering the interval [0, 2) as the union of the intervals [0, 1) and [1, 2), we see that c + c = c Therefore, for any integer n ≥ 0, c ≤ n + c ≤ ℵ0 + c ≤ c + c = c which implies that n + c = ℵ0 + c = c + c = c If ∑ M is a set and for each m ∈ M , we have a cardinal number αm , then m∈M αm denotes the cardinal number of the union of pairwise disjoint sets Am , where |Am | = αm For example, the set N can be written as the disjoint union of countably many countable sets: Free ebooks ==> www.Ebook777.com 514 Elements of Topology ··· 10 24 · · · 12 20 28 · · · 24 40 56 · · · ··· So ℵ0 + ℵ0 + · · · = ℵ0 Thus, for all cardinal numbers ≤ αn ≤ ℵ0 , n = 1, 2, 3, , we have α1 + α2 + · · · = ℵ0 This is also immediate from Proposition A.5.11 The decomposition of the interval [0, ∞) into the intervals [n−1, n), where n = 1, 2, , shows that c + c + · · · = c If α is an infinite cardinal number, then α + ℵ0 = α To see this, we first observe that every infinite set X contains a countably infinite subset Choose an element x1 ∈ X Since X ̸= {x1 }, we can choose an element x2 ∈ X such that x2 ̸= x1 We still have X − {x1 , x2 } = ̸ ∅ So we can choose an element x3 ∈ X such that x3 ̸= x1 , x2 Assume that we have chosen n distinct elements x1 , x2 , , xn of X Then X ̸= {x1 , x2 , , xn }, since X is infinite Therefore we can find a xn+1 ∈ X − {x1 , x2 , , xn } Thus we have a sequence ⟨x1 , x2 , ⟩ of distinct points of X The set {x1 , x2 , } constructed in this way is obviously countably infinite Now, given an infinite cardinal number α, find a set X with α = |X| Let Y be a countably infinite subset X Then the equality X = Y ∪ (X − Y ) implies α = ℵ0 + β, where β = |X − Y | So α + ℵ0 = β + ℵ0 + ℵ0 = β + ℵ0 = α The product of two cardinal numbers α and β, denoted by αβ, is the cardinality of the cartesian product of two sets A and B, where |A| = α and |B| = β If µ is a cardinal number, then the µth power of α, denoted by αµ , is the cardinal number of the set AM , where M is a set with |M | = µ Suppose that ∑for every m ∈ M , there ∏ is the sameµ cardinal number αm = α Then m∈M αm = αµ and m∈M αm = α∪ For the first equation, we note that A × M = m∈M Am , where Am = A × {m} Obviously, Am is equivalent to∑A, and the family {A |Am | and we have ∑ m |m ∈ M } is pairwise disjoint So |A ì M | = = The second formula is immediate from the equality ∏ m∈M m M A = A , where A = A for all m ∈ M m m∈M m www.Ebook777.com Set Theory 515 Thus, we have 2ℵ0 = ℵ0 + ℵ0 = ℵ0 , nℵ0 = ℵ0 + · · · + ℵ0 = ℵ0 , ℵ0 ℵ0 = ℵ0 + ℵ0 + · · · = ℵ0 , nc = c + ··· + c = c, ℵ0 c = c + c + ··· = c, where n is any positive integer We also have Proposition A.8.1 c = 2ℵ0 = |2N | Proof As seen in §A5, every real number in the interval [0, 1] has one or at most two decimal representations as 0.a1 a2 · · · an · · · , where each an is one of the digits 0,1, ,9 Similarly, we can express each x in [0, 1] as x = 0.a1 a2 · · · an · · · , where each an is either or This is referred to as the binary or dyadic expansion of x For any cardinal number α, αℵ0 is the cardinality of all sequences ⟨a1 , a2 , ⟩ in a set A with cardinality α In particular, 2ℵ0 is the cardinality of the set of all sequences which have terms the digits and only (that is, the dyadic sequences) As each real number in [0, 1] has at least one and at most two binary expansions, we have c ≤ |2N | ≤ 2c = c ♢ Note that ℵ20 = ℵ0 ℵ0 = ℵ0 and ℵn0 = ℵ0 · · · ℵ0 = ℵ0 , by induction on n Similar results can be proved for any transfinite cardinal number α To this end, we first establish the following Proposition A.8.2 For any infinite cardinal number α, 2α = α + α = α Proof Let X be an infinite set with |X| = α Denote the two-point set {0, 1} by Then, for any set A, 2×A is the union of disjoint sets {0}×A and {1} × A and so we have 2|A| = |A| + |A| = |2 × A| Now, consider the family F of all pairs (A, f ) such that A ⊆ X and f : A → × A is a bijection By Theorem A.5.7, X contains a countably infinite set A and, by Propoposition A.5.11, the set × A is also countable So there is a bijection f between A and 2×A Obviously, the pair (A, f ) belongs to F, and thus F is nonempty Partially order F by (A, f ) ≤ (B, g) if A ⊆ B and f = g|B It is easily verified that every chain in F has an upper bound in F so that the Zorn’s lemma applies Let (M, h) Free ebooks ==> www.Ebook777.com 516 Elements of Topology be a maximal member in F Then |M | + |M | = |M | We show that |M | = |X| If X − M is infinite, then it contains a countably infinite set B Suppose that g is a bijection between B and 2×B Then we have a bijection k : B ∪ M → × (B ∪ M ) defined by k|B = g and k|M = h This contradicts the maximality of M Therefore X −M must be finite; accordingly, M is infinite If Y ⊆ M is a countably infinite set (which does exist), then |Y ∪ (X − M )| = |Y | + |X − M | = |Y | and we obtain |X| = |Y ∪ (X − M ) ∪ (M − Y )| = |Y ∪ (X − M )| + |M − Y | = |Y | + |M − Y | = |M | ♢ As an immediate consequence of this result, we see that if α is an infinite cardinal number, then, by induction, nα = α for every integer n > And, for a cardinal number β ≤ α, α + β = α, since α ≤ α + β ≤ 2α = α Proposition A.8.3 For any infinite cardinal number α, α2 = αα = α Proof Let X be an infinite set with |X| = α and let F bethe family of all pairs (A, f ) such that A ⊆ X and f : A → A × A is a bijection Then X contains a countably infinite set A and, by Lemma A.5.10, the set A × A is also countable So there is a bijection f between A and A × A Thus the pair (A, f ) belongs to F and F is nonempty Partially order F by (A, f ) ≤ (B, g)∪if A ⊆ B and f = g|B Given a chain C = {Ai , fi } in F, put B = Ai and define f : B → B × B by setting f (x) = fi (x) if x ∈ Ai Then f is clearly a bijection so that the pair (B, f ) belongs to F It is obvious that (B, f ) is an upper bound for the chain C By the Zorn’s lemma, we have a maximal member (M, h) in F Since h : M → M × M is a bijection, |M | = |M ||M | We observe that |X| = |M | As M ⊆ X, |M | ≤ |X| If |M | < |X|, then we find that |M | < |X −M | For, |X −M | ≤ |M | implies that |X| = |M |+|X −M | ≤ |M | + |M | = |M |, by the preceding proposition This contradicts our assumption that |M | < |X| Let j : M → X − M be an injection and put Y = j(M ) Then |Y | = |M | By the preceding proposition, we have 3|Y | = |Y | and, therefore, we can find three disjoint subsets A, B, C of Y such that Y = A ∪ B ∪ C and |Y | = |A| = |B| = |C| It follows that |A| = |M × Y |, |B| = |Y × M | and |C| = |Y × Y |; accordingly, there is a bijection k : Y → (M × Y ) ∪ (Y × M ) ∪ (Y × Y ) Since M ∩ Y ̸= ∅, www.Ebook777.com Set Theory 517 we have a bijection M ∪ Y → (M ∪ Y ) × (M ∪ Y ) which extends h (and k) This contradicts the maximality of M , and hence |M | = |X| ♢ It is now clear by induction on n that if α is an infinite cardinal number, then αn = α for every integer n > And, for a cardinal number β ≤ α, αβ = α, since α ≤ αβ ≤ αα = α Theorem A.8.4 Let X be an infinite set and F be the family of all finite subsets of X Then |F| = |X| Proof Since the mapping X → F, x → {x}, is an injection, we have |X| ≤ |F| To see the opposite inequality, for each integer n > 0, let Fn denote the family of all those subsets of X, which contain exactly n elements Now, for each set F ∈ Fn , choose an element ϕ(F ) in X × · · · × X = X n (n copies), which has all its coordinates in F Of course, there are several choices for ϕ(F ), we pick any one of these Then ϕ : Fn → X n is an∪injection, and so |Fn | ≤ |X n | = |X|n = |X| Obviously, we have F = n Fn and therefore ∑ ∑ |F| ≤ n |Fn | ≤ n |X| = ℵ0 |X| = |X| ♢ This completes the proof Let X be a set and A ⊆ X The function fA : X → {0, 1} defined by { fA (x) = for x ̸∈ A, and for x ∈ A is called the characteristic function of A The function which is zero everywhere is the characteristic function of the empty set, and the function which is identically on X is the characteristic function of X The set of all functions : X → {0, 1} is denoted by 2X Obviously, every element of 2X is a characteristic function on X Proposition A.8.5 For any set X, there is a bijection between the set P (X) of all its subsets and 2X Proof For any subset A ⊆ X, we have the function fA : X → {0, 1} defined by { for x ̸∈ A, and fA (x) = for x ∈ A So the mapping ϕ : 2X → P (X), f → f −1 (1), is surjective This is injective, too For, if f ̸= g are in 2X , then we have f (x) ̸= g (x) for Free ebooks ==> www.Ebook777.com 518 Elements of Topology some x ∈ X This implies that x belongs to one of the sets f −1 (1) and g −1 (1) , but not the other So ϕ (f ) ̸= ϕ (g), and ϕ is a bijection ♢ It follows that |P (X) | = 2|X| The function x → {x} is obviously an injection from X into the set P (X), but there is no bijection between these, by Theorem A.5.13 So |X| < |P (X) |, and we have established Proposition A.8.6 For any set X, |X| < 2|X| www.Ebook777.com Appendix B Fields R, C and H B.1 B.2 B.3 The Real Numbers The Complex Numbers The Quaternions 519 521 523 In this appendix, we will discuss the fundamental properties of the fields of real numbers, complex numbers and quaternions B.1 The Real Numbers We shall not concern ourselves here with the construction of the real number system on the basis of a more primitive concept such as the positive integers or the rational numbers Instead, we assume familiarity with the system R of real numbers as an ordered field which is complete (that is, it has the least upper bound property) We review and list the essential properties of R With the usual addition and multiplication, the set R has the following properties Theorem B.1.1 (a) R is an abelian group under addition, the number acts as the neutral element (b) R − {0} is an abelian group under multiplication, the number acts as the multiplicative identity (unit element) (c) For all a, b, c ∈ R, a (b + c) = ab + ac A field is a set F containing at least two elements ̸= together with two binary operations called addition and multiplication, denoted by + and · (or juxtaposition), respectively, which satisfy B.1.1 The element is the identity element for addition, and acts as the mul519 Free ebooks ==> www.Ebook777.com 520 Elements of Topology tiplicative identity A skew-field satisfies all field axioms except the commutativity of multiplication; this is also called a division ring With this terminology, the set R is a field under the usual addition and multiplication This field has an order relation < which satisfies (a) a + b < a + c if b < c, and (b) ab > if a > and b > A field which also has an order relation satisfying these two conditions is called an ordered field The set Q of all rational numbers is another example of an ordered field We call an element a in an ordered field positive if a > 0, and negative if a < Theorem B.1.2 The following statements are true in every ordered field (a) a > ⇒ −a < 0, and vice versa (b) a > and b < c ⇒ ab < ac, a < and b < c ⇒ ab > ac (c) a ̸= ⇒ a2 > In particular > (d) a > ⇒ a−1 > 0, a < ⇒ a−1 < (e ab > ⇒ either both a > and b > or both a < and b < (f) ab < ⇒ either both a < and b > or both a > and b < The ordered field R has the least upper bound property: If S ⊆ R is nonempty and bounded above, then sup S exists in R This is also called the completeness property of R Thus R is a complete ordered field Using this property, it can be shown that every nonempty set of real numbers with a lower bound has a infimum Another important consequence of the completeness property of R is the following Theorem B.1.3 If a real number x > 0, then given any real number y, there exists a positive integer n such that nx > y This is called the archimedean property of R Using this property of R, one can prove that, for any two real numbers x < y, there is a rational number r ∈ Q such that x < r < y This fact is usually stated by saying that Q is dense in R www.Ebook777.com Fields R, C and H 521 An ordered set (X, ≺) containing more than one point is called a linear continuum if it is order complete, and has no gaps (i.e., for any two points x ≺ y in X, there exists z ∈ X such that x ≺ z ≺ y) It follows that R is a linear continuum The absolute value of a real number x is defined by { x if x ≥ 0, |x| = -x if x < Notice that |x| ≥ for all x ∈ R Proposition B.1.4 For any x, y ∈ R, we have (a) |x| = ⇔ x = (b) | − x| = |x| (c) |xy| = |x||y| (d) for y ≥ 0, |x| ≤ y ⇔ −y ≤ x ≤ y (e) − |x| ≤ x ≤ |x| (f) |x + y| ≤ |x| + |y| (g) ||x| − |y|| ≤ |x − y| (h) |x − y| ≤ |x| + |y| By (c), |x| = B.2 √ x2 The Complex Numbers The set R2 of all ordered pairs (x, y) of real numbers turns into a field under the following addition and multiplication: (x, y) + (x′ , y ′ ) = (x + x′ , y + y ′ ), (x, y) (x′ , y ′ ) = (xx′ − yy ′ , xy ′ + yx′ ) The element (0, 0) acts as the neutral element for addition, and the element (1, 0) plays the role of multiplicative identity It is routine to check that R2 is a field under these definitions It is usually denoted by C, and its elements are referred to as the complex numbers It is readily verified that (x, 0) + (y, 0) = (x + y, 0), and (x, 0) (y, 0) = (xy, 0) This shows that the complex numbers of the form (x, 0) form a subfield of C, which is isomorphic to R under the correspondence x → (x, 0) We Free ebooks ==> www.Ebook777.com 522 Elements of Topology can therefore identify this subfield of C with the real field and regard R ⊂ C Writing ı = (0, 1), we have ı2 = −1 and (x, y) = (x, 0) + (y, 0) (0, 1) = x + yı, using the identification x ↔ (x, 0) Thus C = {x + yı|x, y ∈ R}, where ı2 = −1 If z = x + yı is a complex number, then we call x the real part of z (denoted by Re(z)), and y the imaginary part of z (denoted by Im(z)) If z = x + yı ∈ C, its conjugate is defined to be the complex number z¯ = x − yı For any complex numbers z and w, we have z + w = z¯ + w, ¯ zw = z¯w ¯ Observe that z z¯ is a positive real number unless z = √ The absolute value |z| of a complex number z is defined to be z z¯ (the nonnegative square root) It is immediate from the definition that |z| > except when z = 0, and |0| = Also, it is obvious that |Re (z) | ≤ |z| and |z| = |¯ z | If z and w are any two complex numbers, then it is easily checked that |zw| = |z||w| and |z + w| ≤ |z| + |w| B.2.1 (Schwarz Inequality) If z1 , , zn and w1 , , wn are complex numbers, then we have (∑n ) ( ∑n ) ∑n 2 | zj w¯j |2 ≤ |zj | |wj | ∑n ∑n ∑n Proof Put α = |zj |2 , β = |wj |2 and γ = zj w ¯j We need to show that αβ − |γ| ≥ If α = or β = 0, then the conclusion is trivial We therefore assume that α ̸= ̸= β We have ∑n ∑n = |βzj − γwj | (βzj − γwj ) (β z¯j − γwj ) ∑n ∑n = β γ zj w¯j − |zj | − β¯ ∑n ∑n βγ z¯j wj + |γ|2 |wj |2 = β α − β|γ|2 ( ) = β αβ − |γ|2 Obviously, the left-hand side is nonnegative and β > So αβ−|γ|2 ≥ 0, and the desired inequality holds ♢ www.Ebook777.com Fields R, C and H B.3 523 The Quaternions By using the usual scalar and (non-associative) vector product in R3 , Hamilton defined (in 1843) a multiplication in R4 , which together with componentwise addition makes it into a skew-field (that is, a division ring) This field has proven to be fundamental in several areas of mathematics and physics We discuss this field here The mapping R4 → R × R3 , (x0 , x1 , x2 , x3 ) → (x0 , (x1 , x2 , x3 )), is a bijection If we define the vector space structure on R × R3 over R componentwise: (a, x) + (b, y) = (a + b, x + y) and c (a, x) = (ca, cx), then this mapping becomes an isomorphism So we can identify R × R3 = H with R4 , and call its elements quaternions If q = (a, x), we refer to a as the real part of q and x as the vector part of q There are canonical monomorphisms R → H, a → (a, 0), and R3 → H, x → (0, x), of vector spaces Accordingly, we identify a with (a, 0), and x with (0, x), and write (a, x) = a + x Then, for any quaternions q = a+x, r = b+y and a real c, we have q+r = a+b+x+y, cq = ca+cx If x, y ∈ R3 ⊂ H, we first define xy = −x · y + x × y, where · is the usual scalar product, and × is the usual vector product in R3 Notice that xy is in general an element of H It is easily checked that this multiplication of vectors in H is associative As the multiplication in H ought to be distributive, for q = a + x and r = b + y, we set qr = ab + ay + bx + xy We leave it to the reader to verify the following conditions: q (cr) = c (qr) = (cq) r, q (r + s) = qr + qs, q (rs) = (qr) s, (r + s) q = rq + sq for all quaternions q, r, s and real c The element ∈ H acts as the identity: 1q = q = q1 for any q ∈ H To prove that H is a skew-field, it remains to verify that every nonzero quaternion q has a multiplicative inverse For this, we define the conjugate of q = a + x by q¯ = a − x Observe that q + r = q¯ + r¯, cq = c¯ q , qr = r¯q¯ for any quaternions q, r, s and real c Also, it is straightforward to see that q q¯ = q¯q = a2 + x · x √ (a real) We define the modulus of q to be |q| = q q¯ Thus |q| is Free ebooks ==> www.Ebook777.com 524 Elements of Topology the euclidean norm of q when it is considered as an element of R4 If q, r ∈ H, then we have |qr|2 = (qr) (qr) = qr¯ rq¯ = q|r|2 q¯ = |r|2 q q¯ = |r|2 |q|2 So |qr| (= |q||r| that ) Also, note ( ) q = ⇔ |q| = Obviously, if q ̸= 0, then q q¯/|q|2 = = q¯/|q|2 q Thus q¯/|q|2 is the inverse of q in H, usually denoted by q −1 Observe that x ∈ R3 is a unit vector ⇔ x · x = ⇔ x2 = −1, and two vectors x, y ∈ R3 are orthogonal ⇔ x · y = ⇔ xy = −yx A right-handed orthonormal system in R3 is an ordered triple ı, ȷ, k of vectors in R3 such that ı, ȷ, k are of unit length, mutually orthogonal and ı × ȷ = k So, if ı, ȷ, k form a right-handed orthonormal system, then ı2 = ȷ2 = k = −1 and ıȷk = −1 Conversely, these conditions imply that ı, ȷ, k are of unit length, and ıȷ = k whence ı · ȷ = and ı × ȷ = k Thus ı, ȷ, k form a right-handed orthonormal system Suppose now that ı, ȷ, k is a right-handed orthonormal system in R3 Then any vector x ∈ R3 can be written uniquely as x = x1 ı+x2 ȷ+x3 k, xi ∈ R; accordingly, any quaternion q can be expressed uniquely as q = q0 + q1 ı + q2 ȷ + q3 k, qi ∈ R Clearly, we have ıȷ = k = −ȷı, ȷk = ı = −kȷ, kı = ȷ = −ık Using these rules, we obtain the following formula for the product of two elements q = q0 + q1 ı + q2 ȷ + q3 k, q ′ = q0′ + q1′ ı + q2′ ȷ + q3′ k in H: qq ′ = (q0 q0′ − q1 q1′ − q2 q2′ − q3 q3′ ) + (q0 q1′ + q1 q0′ + q2 q3′ − q3 q2′ ) ı + (q0 q2′ + q2 q0′ + q3 q1′ − q1 q3′ ) ȷ + (q0 q3′ + q3 q0′ + q1 q2′ − q2 q1′ ) k √ We also note that q¯ = q0 −q1 ı−q2 ȷ−q3 k and |q| = (q02 + q12 + q22 + q32 ) For any unit vector x ∈ R3 , the set of quaternions a + bx, a, b ∈ R, is a subfield of H isomorphic to C under the mapping a + bx → a + bı In particular, the subfield of quaternions with no ȷ and k components is identified with C, and we regard C as a subfield of H Thus, we have field inclusions R ⊂ C ⊂ H It is obvious that any real number commutes with every element of H Conversely, if a quaternion q commutes with every element of H, then q ∈ R We emphasize, however, that the elements of C not commute with the elements of H www.Ebook777.com Bibliography [1] G.E Bredon Topology and Geometry Springer-Verlag, New York, NY, 1993 [2] R Brown Elements of Modern Topology McGraw-Hill, London, 1968 [3] J Dugundji Topology Allyn and Bacon, Boston, MA, 1965 [4] D.B Fuks and V.A Rokhlin Beginner’s Course in Topology Springer-Verlag, Berlin, Heidelberg, 1984 [5] I.M James Topological and Uniform Spaces Springer-Verlag, New York, NY, 1987 [6] J.L Kelley General Topology van Nostrand, New York, NY, 1955 [7] W.S Massey Algebraic Topology: An Introduction SpringerVerlag, New York, NY, 1967 [8] G McCarty Topology: An Introduction with Application to Topological Groups McGraw-Hill, New York, NY, 1967 [9] D Montgomery and L Zippin Topological Transformation Groups Interscience Publishers, New York, NY, 1955 [10] J.R Munkres Topology: A First Course Prentice-Hall, Englewood Cliffs, NJ, 1974 [11] L Pontriajagin Topological Groups Princeton University Press, Princeton, NJ, 1939 [12] I.M Singer and J.A Thorpe Lecture Notes on Elementary Topology and Geometry Scott, Foresman and Company, Glenview, IL, 1967 525 Free ebooks ==> www.Ebook777.com 526 Elements of Topology [13] E.H Spanier Algebraic Topology McGraw-Hill, New York, NY, 1967 [14] L.A Steen and J.A Seebach Counterexamples in Topology Springer-Verlag, New York, 1978 ˇ [15] R.C Walker The Stone-Cech Compactification Springer-Verlag, New York, 1974 [16] S Willard General Topology Addison-Wesley, Reading, MA, 1970 www.Ebook777.com ... family F of subsets of X satisfying the conditions in 1.2.4 In fact, the family of complements of the members of F is a topology for X such that F consists of precisely the closed subsets of X Thus... “Closeness” of elements of a set can be measured most conveniently as distance between the elements In any set endowed with a suitable notion of distance, one can define convergence of sequences... number of exercises of varying degree of difficulty at the end of each section These provide ample opportunity to consolidate the results in the body of text, and in some exercises, a line of development