Topology and Its Applications

By William F. Basener

Discover a unique and modern treatment of topology employing a cross-disciplinary approach

Implemented recently to understand diverse topics, such as cell biology, superconductors, and robot motion, topology has been transformed from a theoretical field that highlights mathematical theory to a subject that plays a growing role in nearly all fields of scientific investigation. Moving from the concrete to the abstract, Topology and Its Applications displays both the beauty and utility of topology, first presenting the essentials of topology followed by its emerging role within the new frontiers in research.

Filling a gap between the teaching of topology and its modern uses in real-world phenomena, Topology and Its Applications is organized around the mathematical theory of topology, a framework of rigorous theorems, and clear, elegant proofs.

This book is the first of its kind to present applications in computer graphics, economics, dynamical systems, condensed matter physics, biology, robotics, chemistry, cosmology, material science, computational topology, and population modeling, as well as other areas of science and engineering. Many of these applications are presented in optional sections, allowing an instructor to customize the presentation.

The author presents a diversity of topological areas, including point-set topology, geometric topology, differential topology, and algebraic/combinatorial topology. Topics within these areas include:

• Open sets
• Compactness
• Homotopy
• Surface classification
• Index theory on surfaces
• Manifolds and complexes
• Topological groups
• The fundamental group and homology

Special "core intuition" segments throughout the book briefly explain the basic intuition essential to understanding several topics. A generous number of figures and examples, many of which come from applications such as liquid crystals, space probe data, and computer graphics, are all available from the publisher's Web site.

• Language: English
• Publisher: Wiley-Interscience
• Release date: Jun 12, 2013
• ISBN: 9781118626221

Book preview

Preface

This is a textbook for a first course in either topology beginning in Chapter 1 or geometric topology beginning in Chapter 3. Our goal is to present the essentials of topology that underpin mathematics while quickly moving to the most interesting and useful topics. The framework of this text is rigorous theorems and proofs. We have the philosophy that a good proof should be clean and elegant, and that clear and complete logic elucidates the heart of a matter more than does a long intuitive discussion. However, we are generous with exposition outside of the proofs, and we introduce geometric examples and interesting applications as early as possible. We hope that the reader gains intuition early in the text and appreciates the beauty of topology as well as its importance to mathematics and science.

The range of topics is distributed among the topological subfields of point set topology, combinatorial topology, differential topology, geometric topology, and algebraic topology, while offering a broad variety of examples and applications. Choices in subject matter reflect the desire to present the elegant and complete theory of topology, with numerous examples and figures, while leaving time in a course for applications. Applied examples investigate the use of topology in physics, computer graphics, condensed matter, economics, chemistry, robotics, cosmology, dynamical systems, modeling, groups, and other mathematical and scientific fields. However, our presentation is planned around the theoretic framework of topology, and the applications are used to add intuition and utility to the subject.

Applications of topology are different from applications of other areas of mathematics. The utility of topology comes from its ability to categorize and count objects using qualitative approximate information as opposed to exact values. Our primary criteria in choosing applications is to look for questions from outside of topology whose solution involved topology and would have been either significantly more difficult or impossible without topology. (Farmers might use calculus to optimize their fence planning, but do not need the Jordan curve theorem to determine whether their chickens can escape from a fenced-in area!) This criterion was suggested informally by Jeff Weeks.

In most applications the topology is employed out of a need to handle the qualitative information. In condensed-matter physics, for example, a main goal is to determine the emergent behavior of a very large number of interacting molecules. Because the exact positions of all individual molecules cannot be determined practically, and because of the nature of the interactions, understanding the topological qualitative properties of the interactions is an essential part of determining the properties of materials such as superconductors (Section 5.7). A primary goal in cosmology is to determine the topology, or shape, of the universe as a 3–manifold. This shape of the universe determines, among other things, whether the universe is destined to eventually collapse in on itself in a big crunch. (Section 3.7) A primary goal in dynamical systems, discussed throughout this text, is to use qualitative statements about a model to make qualitative, although certain, predictions about the resulting behavior. Qualitative properties of interactions in game theory discussed in Section 4.7 result in Nash equilibria, which govern many important interactions in economics. The basic principle in dynamical systems and much of game theory is that governing laws, especially those involving social or biological interactions, can be known only approximately. Moreover, even when precise laws are known, chaotic interactions can make the resulting behavior too complicated for precise predictions to be useful. Topology enables us to handle qualitative laws and determine qualitative, but provable, resulting behavior.

Most of the applications appear in separate sections. This provides the reader (or instructor) with flexibility, choosing the applications that are most relevant. This format also provides ample room for background exposition with each application. Instructors may choose to cover any variety of the applications, or may assign them as reading for the students. One possible format, which has proved useful, is to have students read the applied sections and give presentations on applications, teaching each other.

Every scientific discipline has its own jargon, its own set of goals, and its own way of viewing the world. Thus, in each applied section there is a balancing act between presenting the material from the point of view of the applied field and presenting it in a manner consistent with the theory of topology. The result, due mostly to the background of the author, is a presentation of the applied topology from the perspective of a mathematician with all possible respect for the applied field. For a thorough treatment of the applied field the reader should consult the references cited in the sections.

One other unique feature of this book is the occasional core intuition segments. These short paragraphs explain the basic intuition for some of the topics. Hopefully, this will aid the reader encountering the theory of topology for the first time. One has to take great care, of course, to avoid depending too heavily on intuition. Like a magician in front of an audience, theory can play tricks on us when we look only for what we want to see.

A good student will learn to read the text with a pencil and paper in hand. Questions should be asked about all definitions: Can I think of examples? Can I create an equivalent formulation of the definition? Can I draw the picture of an example? What are each of the parts of the definition there for? Similar questions should be considered when encountering a theorem: Does the theorem make intuitive sense? Does it look similar to another theorem I know? How would I begin to prove it? Do I recall all terms used in the theorem? Can I think of an example? Can I think of a counterexample? (Probably not, but trying to beat the theorem often gives insights as to why it is true!) Can I draw a picture of it? Is it true if I remove some of the conditions? Can I generalize it or think of a specific simple case? A proof should be read not only step by step to see its logical progression, but as a whole. It is often helpful to try to summarize the proof in a single sentence.

The most important logical prerequisite is a standard sequence in calculus. Some of the material, particularly the sections on topological groups, the fundamental group, and homology, involves the algebra of groups. Chapter B provides the basic theory. One recurring theme is the demonstration of connections between topology and topics from mathematics and science. In most cases no previous experience is assumed. For example, Chapter 1 begins with coverage of the ε, δ definition of continuity and we prove that the open set definition of continuity is a generalization. No prior exposure to the ε, δ definition is assumed.

The chapters are organized to be covered in order. However, Chapter 6 does not rely on Chapter 5, with the exception of Section 6.7. So it is possible to skip some or all of Chapter 5. This allows an instructor to cover the basics both the fundamental group (Chapter 5) and the basics of Homology (Chapter 4) in a course with limited time.

The author is honored to thank a number of people who helped create this book. George Thurston was a great help at every stage of writing, suggesting many of the applications in quantum physics and thermodynamics. George also proofread much of the book and made numerous helpful suggestions about pedagogy. Tamas Wiandt read the book in its entirety and made too many good suggestions to count. Glenn R. Hall and Bob Devaney both assisted with the sections on the history and notions of chaos (Sections 1.7.1 and 1.7.3). Robert Ghrist provided guidance in the section on topology of robot coordination (Section 3.7.1). Jeff Weeks was a great help with the section of topology in cosmology (Section 3.7.3), as was the NASA WMAP team. Nicolas Ray was very helpful with the section on index theory in computer graphics (Section 4.9). James Sethna greatly improved the section on condensed matter (Section 5.7.1). Afra Zomorodian assisted with the section of computational topology (Section 6.9). Denis Blackmore also provided help with the section on computational topology. Bernie Brooks, Matthew Coppenbarger, Doug Meadows, and Joel Zablow each read significant portions of the book and provided helpful feedback.

The author would like to thank the National Science Foundation, and John Haddock, for their support for the project. The Rochester Institute of Technology, especially Sophia Maggelakis and Ian Gatley, provided abundant support in both time and encouragement. Everyone at John Wiley deserves thanks for their efforts in making this work possible.

I would also like to thank several people on a personal level. Richard McGovern, my undergraduate advisor, gave much to introduce me to the beauty and power of math. My graduate advisor, Glenn R. Hall, showed me the power of topology in dynamical systems and continues to be a valued personal mentor. I would also like to thank my parents, Richard and Carol, who have always believed in me. Most of all, I am blessed to thank my wife, Amber Basener, whose friendship and encouragement are invaluable.

Of course, all errors are the responsibility of the author alone. All comments and suggestions about this work are encouraged, and can be emailed to the author.

WILLIAM F. BASENER

Rochester, New York

[e-mail]

Introduction

One may ask Why study topology? A good answer is that topology is both highly beautiful and highly useful. Its beauty comes from both its wondrous geometric constructions and its logical rigor. Its utility comes from the fundamental importance of continuous functions and geometric objects in mathematics and science. Topology gives insights into practically all other branches of mathematics, including algebra, real analysis, complex analysis, functional analysis, graph theory, number theory, modeling, dynamical systems, and differential equations.

A natural question to ask when beginning a course in topology is What is topology? There are many different answers to this question: rubber sheet geometry, the study of shape without reference to distances. Perhaps the most accurate answer is the study of continuous functions, but this is probably also the most boring answer. We endeavor to give an intuitive, albeit incomplete, answer to this question in this introduction.

In topology, one studies continuous functions f : X → Y and studies properties of the spaces X and Y that make the functions continuous. (The notation "f : X → Y" means that f is a function whose domain is X and range is Y.) One of the great strengths of the subject is that it does not rely on other properties, such as distances, angles, and derivatives. This means that when one uses a topological theorem, whether it is dealing with functions, sets, or differential equations, one only needs to know continuous type of information instead of distance-angle-derivative type of information. In applications, one often does not need an exact formulation of the information, but only needs appropriately correct information. So a topological theorem about circles also applies to ellipses, ovals, and all other circle-shaped curves. In the language of topology, the theorem applies to all curves that are homeomorphic to a circle.

Another strength of topology is that the simple axioms for continuity do not rely on much background. In fact, one of the pioneers of topology, N. E. Steenrod, coauthored with W. G. Chinn a text [1] on topology for high-school students covering topics including homotopies of curves, indices of vector fields, the fundamental theorem of algebra, and the celebrated ham sandwich and pancake theorems. (The pancake theorem is mentioned in the list below.)

One way to answer the question What is topology? is to describe what topology can do. After all, if one asks an inventor what her new invention is, one probably wants to know what the invention does, not the dimensions and materials list. The following is a list of questions that can be answered using topology and which are discussed later in the book. (We take some liberty with intuition here to avoid long definitions.)

1. Does every nonconstant polynomial in the complex numbers have a root? (Answer: Yes.)

2. If a student stirs his coffee gently, is there always a drop of coffee that ends up in the same location where it began? (Answer: Yes.)

3. Is it possible that the velocity of the wind on Earth is nonzero at every point on the planet? (Answer: No, but this would be possible if the Earth were shaped like a huge doughnut.)

4. If a rocket flies straight up from the north pole and never stops flying, will it eventually fly around the universe and crash into the south pole or will it fly away from the Earth forever? (Answer: Nobody knows, but Section 3.7.3 explains how topology is being used to investigate this question.)

5. The derivative operator can be regarded as a function whose domain is the set of continuously differentiable functions and whose range is a space of continuous functions. In other words, this operator takes the function f(x) to the function . Is this operator a continuous operator? (And what would continuous mean?) (Answer: It depends, but you need topology just to ask the question correctly.)

6. Let G be a group and for some element ∈ G define the left multiplication function and define the inverse function by Inv . What are the algebraic consequences on G if and Inv are continuous for every ∈ G? (Answer: There are many consequences, but topology is needed to describe them.)

7. Are there always two points on the equator that are 180° apart in longitude and have the same temperature? (Answer: Yes.)

8. Can you comb the hair on a tennis ball so that there is no cowlick? (Answer: No.)

9. You can link two circles in three dimensions. Can you link two spheres in four dimensions? (Answer: Yes.)

10. If two irregularly shaped pancakes are placed on a platter, is it always possible to cut both pancakes in half with one cut of the knife? (Answer: Yes.)

One of our goals in this text is to illustrate applications of topology to other mathematical and scientific fields. In the logical structure of mathematics as a whole, no field stands alone. The more one learns about a specific field the more one learns how deeply various mathematical fields rely on each other. Topology is particularly interdependent with other fields because many if not most of the important questions in topology were motivated by other mathematical fields.

Another way to answer What is topology is through its history. Historically, topology began in part out of necessity in the field of analysis, but has developed into a subject of its own that is very geometric in flavor. The problems that forced the development of topology were of a nature that an exact answer was too difficult for analysis but indirect topological means were sufficient to provide an ample answer. For example, given a continuous function f : , real numbers a < b, and a real number N between f(a) and f(b), it is often extremely difficult to find an exact value for a number c such that f(c) = N. However, the intermediate value theorem of calculus (which is really a topological theorem) tells us that such a number c must exist and this fact alone is sufficient for many purposes.

Many different mathematicians throughout history worked on topology, and there is no one single source from which topology originates. To give a brief summary of the history of topology we take a quote from the introduction to Chinn and Steenrod’s book [1] from 1966:

The beginnings of topology can be found in the works of Karl Weierstrass during the 1860’s in which he analyzed the concept of a limit of a function (as used in calculus). In this endeavor, he reconstructed the real number system and revealed certain properties now called topological. Then came George Cantor’s bold construction of the theory of point sets (1874–1895); it provided a foundation on which topology eventually built its own house. A second aspect of topology, called combinatorial or algebraic topology, was initiated in the 1890’s by the remarkable work of Henri Poincaré dealing with the theory of integral calculus in higher dimensions. The first aspect, called set-theoretical topology, was placed on a firm foundation by F. Hausdorff and others during the period 1900–1910. A union of combinatorial and set-theoretic aspects of topology was achieved first by L. E. Brouwer in his investigation (1908–1912) of the concept of dimension. The unified theory was given a solid development in the period 1915–1930 by J. W. Alexander, P. L. Alexandrov, S. Lefschetz and others. Until 1930 topology was called analysis situs. It was Lefschetz who first used and popularized the name topology by publishing a book with this title in 1930.

Of course, any description of the history of topology is immediately open to debate because it depends on what results one considers to be part of topology. For example, Chinn and Steenrod’complement equals the set of alls excellent description of the history of topology above makes no mention of the classification of closed surfaces, which was done by Möbius in 1861 at the level of rigor required in his day. It also makes no mention of Leonhard Euler’s work on polyhedron, including his famous formula υ – e + f = 2 (see Theorem 43.) Perhaps this is because this classification result is more part of what is known today as geometric topology, a field that continues to increase in importance and has numerous applications of topology in dynamical systems.

The only way to truly answer the question what is topology is to read on, think deeply about the simple things, and enjoy the beauty of the subject.

I.1 PRELIMINARIES

We set some basic terminology and set theory in this preliminary section. By a set, we mean a collection of things (without being too formal about what these things might be) which we call the elements or points of the set. If X is a set then we write x ∈ X to mean that x is an element of X. The expression x ∈ X is read "x is in X." If X and Y are sets then A ⊆ X means that every element of A is also an element of X. The expression A ⊆ X is read "A is a subset of X." We use x ∉ A to mean x is not in A. If A is a subset of a set X then we define the complement of A to be the set

This expression is read as "A complement equals the set of all x in X such that x is not in A." We denote the union of sets A and B by A ∪ B = {x | x ∈ A or x ∈ B} and the intersection of A and B by A ∩ B = {x | x ∈ A and x ∈ B}. Two important rules for working with sets are DeMorgan’s laws, which are

and

The most common sets we consider at the real numbers , the integers , the natural numbers , and the rational numbers . We will also consider the nonnegative reals and the complex numbers = {x + iy | i² = −1}. The cartesian product of sets A and B is the set A × B consisting of all pairs of points (a, b) with a ∈ A and b ∈ B,

A function is a rule for assigning to each element of a set X a unique element of some set Y. If x ∈ X then the point that a function f assigns to x is written f(x). The set X is the domain of f and Y is the range of f. In general we will write a function f whose domain domain is X and range is Y by f : X → Y, which is read "f is a function from X to Y."

The function from a set A to itself that takes each x to itself is called the identity function on A and is denoted by IdA : A → A, or just Id : A → A. (So Id(x) = x for all x ∈A.) A function f : A → B is one-to-one, or injective, if for x and y in A, x ≠ y implies that f(x) ≠ f(y). We can write this as x ≠ y ⇒ f(x) ≠ f(y), and the arrow ⇒ is read as implies. A function f : A → B is onto, or surjective, if for each y ∈ B there exists a x ∈ A such that f(x) = y. A function is said to be a bijection if it is one-to-one and onto. If f : A → B is a bijection then there exists a function f−1 : B → A such that

for all x ∈ A and y ∈ B. This condition can also be written as f o f−1 = IdB and f−1 o f = IdA, where o denotes the composition of functions. The function f−1 is called the inverse of f.

The image of a function f : A → B is a the set

This is equivalent to the set

If X ⊂ A the the image of X under f is the set

The inverse image of a point y ∈ B under f : A → B is the set

Of course, if f is a bijection then f−l(y) is both the inverse image of y under f and the image of y under f−l.

We define the empty set, denoted Ø, to be the set which contains no elements. We say that a set is nonempty if it is not the empty set. We say that two sets are disjoint if their intersection is empty.

A basic property of inverse images is:

LEMMA 1. f−1(O) = (f−1(Oc))c

Proof. We give this proof using the useful notation, which is read if and only if.

Formally, if X is a set than a relation on X is a collection of pairs of points in X. If R is a relation on X then we write xRy to mean that (x, y) is in R. The expression xRy is read as "x relates to y by R." The most important type of relation for us will be equivalence relations, which act like a form of equality.

DEFINITION 1. An equivalence relation on a set X is a relation ~ such that

1. x ~ x for all x ∈ X.

2. x ~ y implies y ~ x.

3. x ~ y and y ~ z implies that x ~ z.

DEFINITION 2. A partition on a set X is a collection of disjoint subsets of X whose union is X.

If ~ is an equivalence relation on X then we define the equivalence class of a point x ∈ X to be the set

Clearly, the equivalence classes forms a partition of X.

If A is a nonempty subset of that is bounded above we define the supremum of A to be the real number α, written

such that

1. For all a ∈ A, a ≤ α.(That is, α is an upper bound for A.)

2. If α′ is any upper bound for A then α ≤ α′. (So α is the least upper bound for A.)

The supremum of a set is also called the least upper bound of the set. It is an axiom of , called the completeness axiom, that every nonempty subset of that is bounded above has a supremum. Observe that if x < sup A then there exists a point a ∈ A such that x < a < sup A, for if no such a existed then x would be an upper bound of A violating part (1) of the definition above. This property is useful so we make it a lemma.

LEMMA 2. If x < sup A then there exists a point a ∈ A such that x < a ≤ sup A.

If A is a nonempty subset of that is bounded below we define the infimum of A to be the real number β, written

such that

1. For all a ∈ A, a ≤ β. (That is, β is a lower bound for A.)

2. If β′ is any lower bound for A then β ≥ β′. (So β is the greatest lower bound for A.)

By the completeness axiom, every nonempty subset of that is bounded below has an infimum.

LEMMA 3. If x > inf A then there exists a point a ∈ A such that x > a ≥ inf A.

Problems

0.1 Prove DeMorgan’s Laws.

0.2 Prove Lemma 2.

0.3 Prove Lemma 3.

I.2 CARDINALITY

The basic idea in cardinality is to determine when two sets have the same number of elements. If the sets are finite then we simply count the elements; if we assign numbers 1 though n to the elements of the set then there are n elements. If the sets are infinite we need more sophisticated construction.

The basic idea is that two sets A and B have the same cardinality if there is a bijection from A to B. Since the domain and range of the bijection is not relevant here, we often refer to a bijection from A to B as a bijection between the sets, or a one-to-one correspondence between the elements of the sets.

Let Jn denote the set {1, 2,…, n}. We say that a set A is finite if there is a bijection from A to some Jn. If a bijection f : A → Jn exists then we say that A has n elements. If a set is not finite then we say it is infinite.

We say that a set A is countable if there is a bijection from A to a subset of . A set is countably infinite if there is a bijection from A to itself. If a set is not countable then we say that the set is uncountable.

Example: The set of even numbers A = {2, 4, 6, 8,…} is countably infinite. A bijection f : A → is given by f(n) = n/2 − 1.

Example: The set of integers is countably infinite. A bijection f : is given by the function

If a set is countable then it is often convientent to write the set as a list. Specifically, if A is a set and is a bijection then we can write A as

Conversely, if we can write a set in this form, as an infinite list of distinct terms, then the set is countable. Accordingly, a countable set is sometimes said to be listable.

PROPOSITION 1. An infinite subset of a countable set is countably infinite.

Proof. Let A = {a0, a1, a2,…} be a countably infinite set and let B be an infinite subset of A. Let

Let b0 = ano. Inductively define

and let bk = . Then B = {b0, b1, b2, …} and B is countably infinite.

PROPOSITION 2. If A and B are countably infinite then A × B is countably infinite.

Proof. Write A = {a0, a1, a2,….} and B = {b0, b1, b2, …}. The we can write A × B = . The following diagram shows a path which lists all of the elements of A × B.

It is clear that by following the path we would write down each of the elements of A × B exactly once, creating a bijection between and A × B.

PROPOSITION 3. The set is countably infinite.

Proof. Define the map f : by

where p/q is written in reduced form. This map is a one-to-one correspondence between and a subset of a countable set, and hence is countable. Since contans , it is infinite. (See Problem 0.7.) Hence is countably infinite.

PROPOSITION 4. The set is uncountable.

Proof. Suppose, to obtain a contradiction, that is countable. Then the interval (0, 1) is countable. So (0, 1) = {a0, a1, a2, …}. Let x ∈ (0, 1) be the number whose digit in the kth decimal place is given by (See Figure 1.1.)

Then x is an element of (0, 1) which is not equal to any of the ai. This is a contradiction.

The process we used in the proof of Proposition 4 is called Cantor’s diagonalization argument. (See Figure 1.1.) This process can also be used to prove the following.

Fig. 1.1 Cantor’s diagonalization argument.

PROPOSITION 5. For each n ∈ suppose that An is a countable set. Define the cartesian product of the An by

Then is uncountable.

Problems

0.4 Prove, by writing down a bijection, the the union of a countable set with a set consisting of a single element is countable.

0.5 Is there a bijection from to ? Prove your answer.

0.6 Prove that the union of two countable sets is countable. [HINT: Modify the construction we used to show that the integers is countable using the idea that the integers is the union of the the positive integers with the negative ones.]

0.7 Prove that if A is an infinite set and A ⊆ B then B is infinite.

0.8 Prove that the union of an uncountable set with a countable one is uncountable.

0.9 Prove that the union of countably many countable sets is countable. [HINT: Modify the proof from Proposition 2.]

0.10 Prove that the product of countably many countable sets is uncountable, Proposition 5. [HINT: Modify the proof from Proposition 4.]

1

Continuity

1.1 CONTINUITY AND OPEN SETS IN

Continuity is the foundation on which topology is built. We show in this chapter that continuity relies solely on open sets. Focus on these most basic notions, continuity and open sets, lends elegance and generality to the theory of topology. Moreover, because topology is built on continuity, applications of topology often depend only on continuous type of information and not necessarily distance-angle-derivative type of information, which makes topology a particularly powerful tool. Such is the power of good abstraction; it allows us to focus on the essential core that applies to a spectrum of situations. In this first section we study continuity in the familiar setting of with the goal of extracting the essential core properties of continuity. In Section 1.2 we will use this core to motivate our axioms for topology and begin to build the general theory.

The usual distance between points in is

We will sometimes consider with a different formula for the distance between points or without any notion of distance, but unless otherwise noted, we will assume that means with its usual distance function. The distance between x and y in is d(x, y ) = |x − y|.

Let us begin with perhaps the earliest definition of continuity that the reader has encountered. For a function f: [0, 1] → [0, 1], the grade school definition of continuity says that f is continuous if you can draw its graph without lifting your pencil. In other words, the function is not continuous if its graph has a tear, or makes a jump.

Figure 1.1 shows the graph of a continuous function f and the graph of a function g that is not continuous at the point c. In calculus one defines a function f to be continuous at a point c if the limit of f(x) as x approaches c is equal to f(c). From the graph of f we see that limx→c f(x) = f(c). In contrast, limx→c g(x) ≠ g(c) because the left and righthand limits of g are not equal at x, and hence limx→c g(x) is undefined.

Fig. 1.1 The graph of a function f that is continuous and a function g that is not continuous.

Both of these notions of continuity have the same core idea:

CORE INTUITION 1. For a continuous function f, if y is close to x then f (y) is close to f(x).

This is the fundamental idea in continuity. This description of continuity raises two immediate questions: How close is close? and How can you define close without reference to distances? The first question is answered by the − δ definition of continuity from calculus or real analysis:

DEFINITION 3. A function f: is continuous at a point x ∈ if for every > 0 there exists a δ > 0 such that

The function f is said to be continuous if it is continuous at every point in its domain.

This definition declares how close x and y need to be so that f(x) and f(y) are close. This definition can be summarized by saying that f(x) and f(y) can be made arbitrarily close if x and y are chosen close enough.

Example:

It is proven in a calculus or analysis course that the following types of functions are continuous on their domain of definition:

Polynomial functions: P(x) = an xn + an−1xn−1··· + a1x + a0.

Rational functions: f(x) = P(x)/Q(x) where P(x) and Q(x) are polynomial functions.

Exponential and logarithmic functions: f(x) = ex and g(x) = ln(x).

Trigonometric functions: sin x, cos x, tan x, and so on.

Sums, products, quotients and compositions of the functions listed above.

Also, a function f: , written

is continuous if and only if each of the functions fi are continuous.

In general topology we need to deal with continuity even if there is no distance function. Hence, we want to define continuity in a way that does not depend on distance. This seems contradictory at first, since Definition 3 relies in an apparently fundamental way on the distance function. However, this definition can be loosely summarized as saying that all points near x are sent by f to points near f(x). So continuity rests on the meaning of "points near f(x) and points near x." Toward this end, we make the following definitions.

DEFINITION 4. For a point x ∈ , the open ball of radius r around x is the set Br(x) = {y ∈ : d(x, y ) < r}. The closed ball of radius r around x is the set

Note that open balls and closed balls are just higher dimensional generalizations of open intervals and a closed intervals. We think of the open ball around x as containing all of the points near x. We can reformulate Definition 3 in terms of open balls as follows. (The reader should verify that Definition 3 is equivalent to Definition 5.)

DEFINITIONS 5. A function f: is continuous at the point x ∈ if for every > 0 there exists a δ > 0 such that

More succinctly, we say that f is continuous at x if for every > 0 there exists a δ > 0 such that

(see Figure 1.2). The function f is said to be continuous if it is continuous at every point in its domain.

Using our intuition that an open ball around x contains all the points near x, Definition 5 says that if y is near x, then f(y) is near f(x).

We now generalize the concept of "points near x" with the definition of open sets.

DEFINITION 6. A subset O ⊆ is said to be open if for every x ∈ 0 there exists an r > 0 such that Br(x)⊆ O.

CORE INTUITION 2. If O is open and x ∈ 0, then O contains all points near x.

Figure 1.2. The definition of continuity using balls.

Suppose that f is a continuous function. Intuitively, if y is near x, then f(y) is near f(x). So if O is any set containing all points near f(x), then the preimage of O must contain all points near x. Using our intuition for open sets, for a continuous function f, if O is an open set containing f(x), then f−1 (O) is an open set containing x. Since f(x) ∈ O trivially implies that x ∈ f−1 (O), it is sufficient to say that for a continuous function f, if O is an open set, then f−1(0) is an open set. This is precisely the statement of Proposition 6.

PROPOSITION 6. A function f: is continuous if and only if f−1(O) is open for every open set O ⊆ .

Proof. Suppose that f is continuous as in Definition 5 and let O ⊆ be an open set (see Figure 1.3.). Let x be any point in f−1(O). It suffices to show that there is an open ball containing x and contained in f−1 (O). Since O is open and f(x) ∈ O, there exists an > O such that B (f(x)) ⊆ O. Since f is continuous, by Definition 5 there exists a δ > 0 such that f(B (x)) ⊆ B (f(x)). Therefore, Bδ(x) ⊆ f−1(B (f(x))) ⊆ f−1(0) and hence f−1(0) is open.

Suppose that f−1 (O) is open for every open set O ⊆ and choose an arbitrary point x ∈ and > 0. Because B (f(x)) is open, the set f−1(B (f(x))) is open (see Figure 1.3.). Observe that x ∈ f−1(B (f(x))). By Definition 6 there exists a δ > 0 such that Bδ(x) ⊆ f−1(B (f(x))). Then, f(Bδ(x)) ⊆ (B (f(x)), and f is continuous according to Definition 5.

To illustrate Proposition 6, in Figure 1.4. we show the graphs of f and g from Figure 1.1. with an open interval whose preimage under the continuous function f is open but whose preimage under g is not open.

Problems

1.1 Consider the function g shown in Figure 1.4.. Define a function h that is equal to g for all x other than c. (Where c is the point of discontinuity of g and h is undefined at c.) Is h continuous?

Fig. 1.3 The open balls from the proof of Proposition 6.

Fig. 1.4 The function f is continuous and the function g is not continuous

1.2 Define the function f: → by f(x) = 0 if x < 0 and f(x) = 1 if x ≥ 0. Sketch the graph of

Reviews for Topology and Its Applications

[cpg_7 · Rating: 3 out of 5 stars]

Needs corrections?A student came by my office yesterday with a question about an exercise from this book that asked the reader to show that the product topology and the standard metric topology on R^2 are equivalent. The student thought he had a counterexample (a disc containing part of its boundary), and he was right as far Basener's faulty definition of the product topology was concerned. Basener defines a set to be open in the product topology if and only if its image under each projection is open.I scoured the Internet for mention of this error and came across the Zentralblatt review (which is omitted from Amazon's list of editorial reviews for this book). That review lists this error and others and states in summary: "The book is absolutely terrible."That's obviously a strong assertion, and, not having read the book, I'm in no position to confirm or deny it. Still, I felt it was important to post this note to warn potential purchasers/readers of problems with this text, and to encourage the author and publisher to fix things. Neither the publisher website nor the (apparently broken) author website nor a Google search yielded a list of errata.[8/19/2008 Update: The author's website has been fixed, and it has a list of errata. Furthermore, the author assures me that the error in the definition of the product topology has been fixed in printings after the first.]