book for topology for m.sc

7/29/2019 book for topology for m.sc

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TOPOLOGY

A READING GUIDEfor

MATH 513at

Binghamton University

by

Matthew G. Brin

1-st Edition

Department of Mathematical SciencesBinghamton University

State University of New York

c2012 Matthew G. Brin

All rights reserved.


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ii


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On the text

The text for this course is Topology, second edition, by James R. Munkres,published 2000 by Prentice Hall, NJ.

These notes are a guide to the reading. They will be added to as the semestergoes on.

Most of the mathematics in this course will be learned by reading the book.That is a very deliberate choice on the part of the instructor (me). Much ofmy work in these notes and in class will be helping you learn how to learnmathematics from a book.

My intention as I write this preface is to write chapters and sections of thesenotes that correspond to the chapters and sections of the book. The notes willcontain what to look for, and perhaps will indentify parts to emphasize, partsthat might be more difficult, possibly parts to ignore, and parts that mightbe of particular importance. There may be additional material introduced inthese notes of varying amounts. There may also be alternative points of viewthan those given in the book, and there will sometimes be either alternative oradditional notation.

There will be material in these notes that may not at first appear to bemathematical, and perhaps never appear to be mathematical.After writing a few pages, I find that I refer to the Munkres book either with

the word book or the word text. This guide will never be referred to with eitherof those words and their meaning will always be the Munkres book. If I wishto refer to this guide it will be with the word guide or the word notes.

iii


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iv ON THE TEXT


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A note to the reader

The books A note to the reader is on Pages xv and xvi in the text. Readit (the note) carefully. Do not worry if the four examples on Page xvi areincomprehensible. You are not expected to be familiar with either the words ornotation used in any of the examples. The four examples will have been definedand discussed to some extent by Page 118 in the text.

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vi A NOTE TO THE READER


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Part I

General Topology

1


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Chapter 1

Set theory and logic

On the material before Section 1. The important sentence starts four linesfrom the bottom of Page 3, starting with In this book, . . . Since this isnot a course in set theory, you will have to learn how to manipulate sets bywatching rather than having all the rules laid out for you at the beginning.Since the manipulations will be few, this is not too much of a burden, but it isa burden nonetheless. You will need to be attentative.

1.1 Fundamental concepts

As you read this section, you will need to focus on two concepts that are notstrictly mathematical content. They are language and interactions. Westart with language. Interactions will be discussed later in the section.

1.1.1 Language

Part of learning mathematics is learning how to communicate it. Saying youknow how to solve a problem but dont know how to explain the solution meansthat you have not learned enough. Missing from your learning would be math-ematical language adequate to explain the solution.

The language consists of words and symbols. Symbols are usually reservedfor very elementary and frequently used concepts, so I will treat them differently.Words, or terms, are usually defined. Definitions have three parts.

1. A general idea. Example: A derivative tells you how fast the value ofa function changes in comparison to the change of the argument of the

function.

2. The technical definition. Example:

f(x0) = limxx0

f(x) f(x0)

x x0.

3


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4 CHAPTER 1. SET THEORY AND LOGIC

3. The explanation of how the technical definition expresses the general idea.Example: From the definition of limit (not given), for any given > 0 there

is a > 0 so that if |x x0| is less than , thenf(x) f(x0)x x0 f(x0) <

which leads to

f(x0) 7 which is either true or falsedepending on the value of x. Thus for every x A, statement P(x) holdsbecomes

x A, P(x)


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and for at least one x A, statement Q(x) holds becomes

x A, Q(x).

1.1.3 Relations and operations

All of , = and are relations. Whatever they are given to work with, theresult takes on one of two values: true or false. On the other hand, and are operations. Whatever they are given, the result is a mathematical object.With these particular operations the result is a set. Operations such as + and generally give a result that is a number, although a is defined on sets inPage 10 whose result is a set. You should look back through the material ofSection 1 up to the start of Rules of Set Theory on Page 10 and identify allthe relations and operations that you can find.

This leads us to the topic of interactions.

1.1.4 Interactions

Mathematics takes place where concepts meet.We have barely enough symbols to see how they interact. To shorten state-

ments even more, we invent for not as in (not Q) is written as (Q). Therule at the bottom of Page 9 becaomes

(x A, P(x)) x A, (P(x)).

This is an example of an interaction. It shows how interacts with . Whenevertwo operators (each can be either a relation or an operation) are introducedin the same subject, it is often useful to see how they interact. The material onPages 10 and 11 on the Rules of Set Theory discuss the interactions of ,

and as operations on sets.As the operations and relations accumulate, you should take as a worthwhile

exercise to see how they interact. This is an open ended assignment. You willnever finish. But it is good to keep pecking away at it. Many of the exercisesat the end of the section on Pages 14 and 15 explore such interactions.

More on language

The formula(x A, P(x)) x A, (P(x)) (1.1)

illustrates the power of the symbols to shorten statements. What formula similarto (1.1) expresses the content of the top few lines on Page 10?

If P(x, y) is a statement that depends on two values (consider x > y asan example), then we can look at

x, (y, P(x, y)) (1.2)

andy, (x, P(x, y)). (1.3)


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1.1. FUNDAMENTAL CONCEPTS 7

The first says that for each x you can find a y that cooperates with theparticular x to make P(x, y) true. Thus one can choose a different y for each

x. The second says that there is one y that works for every single x to makeP(x, y) true. In the second statement, the burden on the one y is much greaterthan on any single y used in the first statement.

This distinction is crucial. Which statement is true if P(x, y) is take to bex > y?

As an exercise, negate each of (1.2) and (1.3). This exercise is supposed toconvince you of the value of the use of symbols and formulas such as (1.1) andits counterpars. If you are not convinced, try to find an efficient algorithm tonegate combinations of and using the formulas.

How would you negate

x, (y, (z, R(x,y,z)))?

1.1.5 Back to Section 1

The topic Collections of Sets starting near the bottom of Page 11 containsthe notion of a set of sets. This concept and the union and intersection of a setof sets in Arbitrary Unions and Intersections go together. In Collections ofSets is the definition of the power set of a set. This will enter many discussionsshortly.

Cartesian products come next and are based on ordered pairs. Grammat-ically, an ordered pair is a construct (a, b) using the five symbols ( a ,b ) with behavior defined by saying that two such pairs, (a, b) and (c, d) areequal if and only if a = c and b = d. The small type inset on Page 13 showsthat ordered pairs can be defined using only the notion of set. To make goodon this claim one has to prove that the statement

{{a}, {a, b}} = {{c}, {c, d}}

is true if and only if a = c and b = d. This looks obvious until it is pointed outthat a = b is not an assumption. The proof of the if and only if can be takenas an exercise, but you should be warned that it is annoying and when finished,only mildly satisfying.

The discussion near the bottom of Page 13 leaves out one more use of (a, b).If a and b are integers, then (a, b) is often used to denote the greatest commondivisor (GCD) of a and b. It is very common in mathematics to have onenotation used for several concepts.

1.1.6 The exercises on Page 1415

You should do them all, but you might nto regard that as practical. There isalso the problem of what is meant by check in Problem 1, or determine inProblems 2, 5 and 10, and prove in problem 9. Problems 3, 4, 6, 7 and 8 aremore straightforward and should be done first. The word show in Problem 8should give no difficulty.


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Proving statements will be dealt with piecemeal over the next few sections.For Problems 1, 2, 5, 9 and 10, do the best you can. Do as many parts of

Problem 2 as you can stomach. The more the better.

More work

See how many interactions you can analyze. You have learned that distributesover and vice versa. Does distribute over and? Does distribute overor? How about ? Not all answers are yes.

Note: There are symbols for and (it is ) and or (it is ). I rarely usethem (the words are already short enough) and the book does not use them atall. They are used heavily in books on logic.

1.2 Functions

The book comes close to messing up the definition of a function, but saves itselfin the end. It is definitely guilty of unnecessary generalization.

By Page 16, you know that a function f : A B must have a value f(x) forevery x A. This is so important a property for a function that it should nothave been delayed full page after the definition started. The rule of assignmentdefined on Page 15 will have no use in the rest of the course.

The other part of the definition of a function appears as the top line ofPage 16. The two parts of the definition of a function are often combined intostatements of existence and uniqueness. If f : A B is a function, then for

every x A, the exists a value for f(x) and this value is unique.As an exercise see if you can dig out the existence part from the definition

of a function in the book.

The words range and image are treated differently in different books. Youwill have to learn how this book uses them and get used to it. The booksinterpretation of these two words agrees with many other books. I am not boldenough to say most.

The technical definitions of injective and surjective could use quantifiers.What should the be and where should they go?

The only other point to make about this section is that it could be cleareron the double burden that the notation f1 carries. When f is a bijection, thenf1 is a function. However, if f is not a bijection, then f1 still has meaning.

Technically if f : A B is a function, then f1

is a function from P(B), thepower set of B, to P(A), the power set of A. Less technically, if Y is a subsetof B, then f1(Y) is a subset of A as defined in the middle of Page 19.

The rest of the section should be read carefully, omitting nothing. An im-portant concept is introduced in Problem 5 at the end of the section and isdiscussed below.


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1.2. FUNCTIONS 9

1.2.1 The exercises on Pages 2021

There are now more things to interact. There are f and f1

available to in-teract with each other and with the various constructs in Chapter 1. Not allinteractions are as nice as some others and the exercises work through them. Itis extremely important to know which work out nicely and which do not.

The word show starts to be more important here, making this a good timeto start discussions of proofs.

Some not very hard and fast rules are as follows.

1. Be guided by the conclusion. It is often a mistake to start with the hy-pothesis and try to find your way to the conclusion.

2. To prove P Q, assume P and then prove Q.

3. To prove that P and Q is true, first prove P and then prove Q.

4. To prove that P or Q is true, assume one (say P, for example) is falseand use that assumption to prove the other (Q in the example).

5. To prove that P and Q imply R just assume both P and Q and thenprove R.

6. To prove that (P or Q) implies R you have to do twice the work. Youhave to prove P implies R and you have to prove Q implies R. Thisis known as proof by cases.

7. To prove that x, P(x) is true, let x be arbitrary and prove P(x) for thatarbitrary x. It is against the rules to make x less than arbitrary. Thus itis not valid to say Let x be arbitrary. Now assume x = 7. Now prove

P(x). It is valid to say Let x be arbitrary. Now x > 7 or x 7 is true.We will prove that x > 7 implies P(x) and we will prove x 7 impliesP(x). Explain how this example combines two of the points made above.

8. To use x, P(x) in a hypothesis, just assume P(x) is true every time an xcomes up in the proof.

9. To prove x, P(x) find an x that works.

10. To use x, P(x) in a hypothesis, let x be arbitrary in every way exceptthat you assume that P(x) is true about that x. It is against the rulesto make x any less arbitrary. Thus it is not valid to say Let x be suchthat P(x) is true. Now assume x = 7. Now prove the desired conclusion.This comes up in using an assumed twice. Example: Assume f and g

are both continuous and an > 0 and an x are given. The definition ofcontinuity guarantees a value of > 0 that has some nice properties thatwe need not state. It is wrong to say Let > 0 be such that somethingnice happens around f(x), and let > 0 be such that something nicehappens around g(x). The second use of the assumed is wrong becausethe same letter is used for g that was used for f. That makes the


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picked for g not arbitrary. It has been assumed to be the same thatwas used for f. It is correct to say Let 1 > 0 be such that something

nice happens around f(x), and let 2 > 0 be such that something nicehappens around g(x).

If these rules are not familiar to you, then you will have to refer back tothem from time to time.

One difficulty in applying these rules is deciding at each state of a proof whatthe general shape is of what you are trying to prove. This will be discussed inclass for a while.

Some of the above rules can be validated by an exercise with truth tables.For 4, show that (P or Q) (P Q). Find a similar statement thatvalidates 6.

I assume no one will have trouble with applying (A B and B C)implies (A C) or the simpler (A and (A B)) implies B. The first is

referred to as the transitive property of , and the second is called modusponens.

Now that you have been given way too much information about proofs, tryall the problems on Pages 20 and 21.

Several concepts are introduced in Problem 5 on Page 21. This makes theproblem an important one. The two parts of the definition of a function and thetwo concepts one-toone and onto have nice symmetries. This problem isan exploration of those symmetries. What part of the definition of a function islike the definition of one-to-one and how does it differ? Answer the questionwith one-to-one replaced by onto.

1.3 Relations

Every function is a relation but not every relation is a function. Thus thedefinitions of a function and a relation can use much of the same words andsymbols. The book does this and a function in the book has been defined asthe set of all ordered pairs (a, f(a)) in A B as a runs over A. In standardnotation, this is the set

{(a, b) A B | b = f(a)}.

As just mentioned, this is just the graph of f.In words, this is just the set of all (a, b) in AB that are related by having

b be the image of a.The same approach is taken with relations. The definition makes a relation

the set of all related pairs. The main difference between function and relationis that a function has restrictions and a relation has none. Any set of orderedpairs makes a relation. This is not at all true of functions.

The book only defines relations on one set, so the pairs live in A A. Onecan have a relation between two different sets. The relation was born in is arelation from the set of people to the set of countries. One can have a relation


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1.3. RELATIONS 11

between elements of more sets. The relation was born in during is a relationbetween the set of people, the set of countries and the set of years. You can all

think of difficulties with these examples (people born at sea?) which is why wetend to stick to mathematics most of the time.

Example 1 in the book is another collection of non-mathematical examples.Hopefully equivalence relations are well known to you. If not, commit all

definitions to deep memory. Lemma 3.1 should be gone over until its key pointshave become obvious.

The connection between equivalence relations and partitions is complete.The bottom half of Page 23 is really dealing with a one-to-one correspondenceand its inverse.

In the paragraph above the definition there is a process that takes an equiv-alence relation on A and derives from it a partition ofA. Thus ifE is the set ofequivalence relations on A and P is the set of partitions on A, we have created afunction f : E P. In the paragraphs after the definition a function is defined(that we may as well call g) that goes in the other direction.

The paragraph at the bottom of the page shows something about this con-struction. What is it showing? Is it about g? Or is it about f? And whatexactly is being shown? Once this is done, you should realize that all this couldhave been made a lot simpler by showing that f and g are inverses of each otherand using Problem 5 of the previous section. In fact, you should take as anexercise that f and g are inverses of each other.

The next point is that functions give partitions and thus equivalence rela-tions. If f : A B is a function, then

{f1(b) | b B}

is a collection of subsets of A which (if you ignore the empty sets that showup if f is not onto) is a partition of A. If f is onto, then the empty sets dontshow up. The corresponding equivalence relation on A is defined by x y ifff(x) = f(y).

The book does not bring in the map from a set with an equivalence relationto the set of equivalence classes. This relates to the partition that comes from afunction in the previous paragraph, but this can wait until it arises. It is relatedto the observations in Problem 4 in this section.

The subsection on order relations is loaded with definitions. Unfortunatelyfor you, they are all in common use and need to be memorized. The dictionaryorder relation will need some thought to get used to and you should devote someenergy to that. It comes with several alternate names. My favorite is lexicalorder since it uses few letters. Lazier people call it lex. People that like lots

of syllables call it lexicographical order.To show the care needed in learning the definitions, note that not all sets

that are bounded above have a largest element. Find an example.The main point of the least upper bound property is that the real numbers

have this property. That will just be assumed in this course. In another course,it can be proven if a given construct of the reals is assumed.


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The problems are starting to get a little harder. Do as many as you can.Problems in this section break into two categories. One category is of general

sounding problems. These are 2, 4, 7, 9, 11, 13, 14. The other category (theones not mentioned) are less general and are about specific examples. Bothtypes are valuable and you should not stick only to one type if you dont end updoing all the problems. Note that some problems (13 is a good example) cannot

just be done by relying on the formalities to get you through. Some thoughtneeds to be involved. This requires complete familiarity with the definitions,and often an understanding of the correspondence between the definition andthe idea behind the concept.

1.3.1 Uniqueness

If something s is claimed to be unique with respect to a property, that means

that any candidate c with that property must have c = s. This looks like piecesof the definition of one-to-one, and it should. The definition of one-to-one saysthat an element that is mapped to a particular value is unique.

One shows uniqueness by assuming two items with a given property andproving that they must be the same.

Show that a largest element of a set is unique.

This section does not cover partial orders. They show up in the last sectionof the chapter in an exercise.

1.4 The integers and the real numbers

This section is a rather mixed assemblage of facts.

1.4.1 The reals

The assumptions on the reals (1)(8) from the bottom of Page 30 to the topthird of Page 31 seem to leave some things out. I dont know how to prove thatevery x R has a x R so that x + (x) = 0. I dont know how to provethat every x = 0 in R has an x1 so that x x1 = 1.

After the positive integers are defined it is possible to give one more propertyof the reals that says:

(9) For every real number x there is a positive integer n so that x < n.

This turns out to be provable from (1)(8) and this is done on Page 33, soit doesnt really need its own number.

But, I think you can prove (8) from (1)(7). You might need to assume1 > 0 which is part of Exercise 2. Try to prove (8). I can give a hint if you needone.

The problem with the listed properties is that no one really refers back tothem every time they use the real numbers. However, I claim that every factabout the real numbers follows from properties (1)(8), even though it would


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1.4. THE INTEGERS AND THE REAL NUMBERS 13

be a pain to prove all these facts from these basic properties. The exercises giveenough samples of things that follow from these assumptions.

To see the importance of (9), try to prove that there is no positive realnumber that is a lower bound for the set

1

n| n Z+

first by assuming (9) and then by not assuming (9). Of course, (9) is provablefrom (1)(8), but by not assuming (9) I mean by not proving (9) first.

1.4.2 Induction

The other major topic in the section is induction. You should be familiar withinduction, but are probably not familiar with it in the form given in the book.

Item (2) in the middle of Page 32 is the key. If you have a statement P(n)about elements ofn Z+ and you wish to prove that

n Z+, P(n)

then the usual technique is to prove P(1) and then prove that

n Z+,

P(n) P(n + 1)

.

The step above is called the inductive step.The connection of this usual form of induction with (2) on Page 32 of the

book is to let T be defined by

T = {n Z+ | P(n)}.

In words T is the set of all n Z+ that make P(n) true.The usual induction proof is essentially a proof that T is inductive. That

is, 1 T and for every x T, we have x = 1 T. Now by (2) on Page 32 ofthe book, we get that T is all of Z+ and we have that P(n) is true for everyn Z+.

Note that this connection is used in the proof of Theorem 4.1. Read theproof of 4.1 carefully to find the use of this connection.

If we call the procedure of proof by induction that you are familiar withordinary induction, then Theorem 4.2 (after translation) gives strong induc-tion. The translation from 4.2 to a method of induction is the same as thetranslation from (2) on Page 32 to ordinary induction.

To make the translation smoother, let us describe ordinary induction astrue for 1, together with true for n implies true for n + 1 implies true for alln Z+.

The translation of 4.2 gives true for everything less than n implies true forn implies true for all n Z+. Of course this is too brief. Expanding it saysthe following.


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If for each n Z+, you can prove P(n) by assuming P(j) for each j Z+with j < n, then you have proven P(n) for every n Z+. The justification

from 4.2 is the same as the justification of ordinary induction from (2) on Page32. Let T be the set of all n Z+ that make P(n) true and show that truefor everything less than n implies true for n means that T satisfies all thehypotheses stated about A in Theorem 4.2, and so T = Z+.

The main point is that strong induction is justified by Theorem 4.1 which isproven using ordinary induction. So strong induction is as justified a techniqueof proof as ordinary induction.

So why is strong induction called strong. It is because it is easier to use. Toprove P(n), you are allowed to assume all of P(j) for j Z+ with j < n. Inordinary induction you are only allowed to assume P(n 1).

To appreciate the difference try to prove the fundamental theorem of arith-metic with strong induction and without it. The fundamental theorem of arith-metic says that every n Z+ is either 1, a prime or a product of a finite numberof primes.

One last comment on strong induction is how to prove P(n) under the as-sumption that P(j) is true for all j Z+ with j < n when n = 1. You will notethat there are no j Z+ with j < 1. So you are supposed to prove P(1) underno assumptions at all. That is, you are just supposed to prove P(1), period.This is the same as in ordinary induction except that in ordinary induction ithas to be stated separately. In strong induction, the fact that you have to proveP(1) directly is there, but it is hidden in the main part of the statement ofstrong induction.

1.4.3 The exercises

The theme of the exercises is that you really can prove everything you knowabout the reals from (1)(8). Once that point is made, it is not all that necessaryto go on.

In Problem 1, you should go on until bored. Much of what is in Problem 1belongs in an algebra course.

Problem 2 deals with order. It might be good to go at least as far as (g).Problem 3 justifies everything on Page 32 and is important. The theme of

(a) will be repeated in many settings in many courses and is crucial. The themewill be identified later. A different part of the theme continues into (b) and is

just as important.Problem 4(a) is a bore, but 4(b) is important.With a few exceptions 5 through 11 continue the theme of everything follows

from (1)(8) and get less important as you plow through them. They dont

teach you anything new about the reals, and the techniques used have little todo with a topology course. They are more appropriate for an analysis coursefor some and for an algebra course for others.

Note that 8(a) is a repeat of 13 from the previous section.8(b) was already mentioned in the discussion in these notes about the first

part of this section. 8(c) is more of an analysis problem.


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1.5. CARTESIAN PRODUCTS 15

Problem 9 might be of more interest, but once again, there are no surprisingstatements. The surprise might be that the parts might require a lot of thought.

Lastly, unless I missed something, I do not find an exercise that says: foreach n Z+, there is no j Z+ with n < j < n + 1. (Although, Exercise 5(c)comes close.) Try proving the statement that I gave for just n = 1. If you getn = 1, then the other values of n follow quickly.

1.4.4 A pedagogical point

The exercises build on each other. If you try to prove every exercise by alwaysgoing back to (1)(8), you will do way too much work. One hard part of learninga mathematical subject is that the things that are proven true one day may becrucial a day later. This means that as time goes on, you have a larger andlarger set of facts that you need to remember so that you can function as you

go deeper into the subject. This is often the aspect of mathematics that givesstudents the greatest problem. To avoid this problem, work to remember thefacts as they accumulate.

1.5 Cartesian products

This is a major section.

First feature: a second way to refer to lots of sets. In Section 1.1 on Pages 11and 12, the concept of a set of sets was introduced for the purpose of discussingarbitrary unions and arbitrary intersections. Now we have the concept of anindexed family of sets. This is just a function whose domain is a set and whoserange is a set of sets. The notation is a bit more like the notation for a sequence.

This requires a digression which is not really a digression since it is really aspecial case.

A sequence is a function whose domain is the positive integers Z+ (if thesequence is an infinite sequence) or a set like {1, l2, . . . , n} if the sequence is afinite sequence. See the discussion on Page 37. Ifs is the sequence, we write siinstead ofs(i).

In an indexed family of sets, the domain is the indexed set J and ifA is thefamily, then we write A for a given J instead of A().

Indexed families of sets are not needed in a discussion of unions or intersec-tions since intersecting the sets X and X and Y is the same as intersecting thesets X and Y. That is, it doesnt matter that X was mentioned twice the firsttime and not the second. The result is the same.

However, X X Y is not the same as X Y. It makes a difference thatX is mentioned twice the first time and not the second.The idea that a product of sets is built out of tuples is not hard to get

used to. The idea that tuples can be be of arbitrary large size and are reallyjust functions on an indexing set takes work to get used to. You should allocateenough time to to this.


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This book starts gently by first making the jump from finite tuples totuples indexed over Z+. Later, tuples indexed over arbitrary indexing sets

will be introduced.In the exercises 1 and 2 are boring but worth doing. In 3, (a) and (b) are

good; (c) is not really a good introduction to a large topic, but you should lookanyway; (d) needs careful thought. Problem 4 needs the notation defeined inthe middle of Page 38 and should be atempted. Problem 5 is reasonable. Thenotation is that of Example 3 on Page 38.

1.6 Finite sets

There is little in this section worth looking at. Even the definition on Page 39is of minor importance. The purpose of the chapter is to demonstrate that alot of intuitive statements can acutally be proven rigorously. Those that find

this interesting should read the section. The rest shoulc read the statements ofCorollaries 6.3 through 6.8 carefully, but not the proofs. Do not do the exercises.

1.7 Countable and uncountable sets

The important thing to develop is a way to see reasonably quickly which infinitesets are countable and which infinite sets are uncountable. This is not to implythat such questions are always easy, but often they are.

First point: appreciate the definitions on Pages 44 and 45.Next appreciate the techniques in Examples 1 and 2 on the same pages.

Example 1 is quite easy, but important. Example 2 is a bit more invovled, butits arguments are incredibly standard.

The book leaves out the important notion of having the same cardinality.Two sets have the same cardinality if there is a bijection between them. Ofcourse that means that this is not exactly a new notion. It is simply giving anew word to the relation has a bijection between. However, the importantpoint is that the relation is an equivalence relation (check that it is) and inparticular is transitive. Thus to show that two sets have the same cardinality itsuffices to show that they have the same cardinality of some third set. This turnsone problem into two problems, but sometimes each of the two new problems ismuch easier than the original.

Theorems 7.1 and 7.2 are important in statement, and the proofs are almostas important. The proofs are not as important as material that comes laterthat can be used to replace some of the proofs given. The easy argument for

(1) (2) does not get replaced later and should be read carefully as it is.The material on Pages 47 and the top of Page 48 is rather overblown andshould be read only once and not that critically.

The statements 7.3 through 7.7 are critical. They need to be committed tomemory and used when needed. THe proofs are nice too and you should learnthem.


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1.8. THE PRINCIPLE OF RECURSIVE DEFINITION 17

The key to 7.7 is the parenthetical remark at the top of Page 50. You haveto decide how that justifies the sentence that follows the remark.

The technique in the proof of 7.7 is almost always referred to as Cantorsdiagonal argument. This is the same technique used in the proof of the equallyimportant 7.8. The important part of the proof of 7.8 is that g in the secondparagraph cannot be surjective. The formula for B in the middle of the proofwould have been more clearly stated as

B = {a A | a / g(a)}.

Every book likes to give the argument that B is not in the image of g thatis given in the last paragraph of the proof. I dont quite like it. I prefer a moredirect argument that shows

a A, B = g(a).

You should write out what this argument would consist of. It does not needproof by contradiction.

You should also decide why the proof of 7.8 is also called the Cantor diagonalargument.

In spite of the fact that the last paragraph in the section downplays the useof the Cantor diagonal argument to prove that R is uncountable, it is goodpractice to build such a proof. Prove that the set of all real numbers in (0, 1)whose decimal expansion uses only the digits 5 and 6 is uncountable using aCantor diagonal argument.

1.7.1 Exercises

Do Problem 1 in steps. If you try to do it in one step, you will be missing someof the points of the section.

Problem 2 is not that rewarding, but it pays to check that you can do it.

Problem 3 is worth thinking about.

The work in 4 is in (a). Again, prove it in steps, and be sure to use resultsfrom the section.

Problem 5 is good practice.

Problem 6(b) is famous (it is named) and the statement should be memorizedeven if you dont do the problem.

The rest of the problems are optional.

1.8 The Principle of Recursive Definition

This should be skipped.


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1.9 Infinite sets and the axiom of choice

1.10 Well-ordered sets

1.11 The maximum principle

These sections put too much work into the material. I will talk about what isneeded in class.


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Chapter 2

Topological spaces andcontinuous functions

The title of this chapter is the subject of topology. Unfotunately, the bookdefers the definition of continuous functions until the seventh section of thischapter. This causes problems in motivation and understanding. Rather thantry to change the order of the book, I will just point out that the practicalityand motivation of several of the definitions will not become apparent until thesection on continuous functions.

The outline of the first six sections is partly laid out in the second sentenceof the chapter. To add more detail, topological spaces are first defined, then asection tells how they are often specified, then a section gives one construction oftopological spaces, then two sections tell how new topological spaces can be built

from existing topological spaces, and then a section discusses some fundamentalconcepts associated to topological spaces.

The other half of the subject of topology comes after the six sections justmentioned. Continuous functions will be discussed together with whatever elseis introduced from that point on in the book.

2.12 Topological spaces

Since topological spaces form half of the subject and since the other half ofthe subject, continuous functions, have topological spaces as their domains andranges, it is clear that the definition of a topological space must be learnedthoroughly and learned immediately. Unfortunately, the definition is hard to

motivate. We will not do so here, but will give some comments.1. There are no interesting topologies on finite sets. Thus putting energy

into building funny topologies on a set with, say, seven elements is a waste oftime. If the first sentence of this comment is too strong and sweeping for some,I will modify it and say that there are no interesting topologies on finite sets inthis course.

19


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20 CHAPTER 2. TOPOLOGICAL SPACES

2. There are two useful topologies on finite sets. If you find them interesting,then you are allowed to disagree with the point I make above. The two useful

topologies are introduced in Example 2 on Page 77. One of them is extremelyimportant. Even though it is important, I still say that it is not interesting. Thetwo topologies will be referred to often. Example 3 is also somewhat useful, butmostly as an example.

3. Any definition by properties, such as the definition of a topology, mustbe immediately followed by examples. Otherwise the definition remains tooabstract. This means that you should learn all the examples as thoroughly aspossible. However, given my first point above, you are allowed to forget thematerial in Example 1 as soon as you read it.

4. The definition of a topology allows for many strange cooked up exam-ples. The power of the definition is that it covers such a wide sweep of naturalexamples that are useful to mathematical investigation. The fact that the defi-nition is flexible enough to build funny structures that no one would look at isnot part of its power, nor any evidence of any weakness.

5. The definitions of coarser and finer are given a bit too early. They carryno intuition until continuous functions are defined. In spite of the fact that thedefinitions carry no intuition, they can be useful. Since they talk about one set(the topology) being contained in another, the concepts can be used to showthat two topologies are equal. This is done in the next section.

6. If you have been introduced to open subsets of the real line in a previouscourse but not in a course on topology, then the word open hear can lead toconfusion. In a topology, the open sets are those that are declared to be openwhen the topology is specified. In the section on bases, there will be a topologyon the real line where the open sets are not the open sets that you are used to.This example is important for several reasons, and its strange open sets is one

of the reasons.7. The book does not adopt the usual convention that a topological spacealways refers to a pair (X, T) where X is a set and T is the topology on X.There is good reason for not insisting on the pair point of view. Very few peopleadhere to the pair view in the way that they write and the way that they talk.However, the pair point of view lies behind the way most people refer to atopological space. The book gives one sentence to this view immediately afterthe definition of a topological space. However, it is much more important thanthis. The wording of Problem 5 on Page 92 suffers tremendously because thepair point of view is so thoroughly suppressed in the book.

You will note that there are no exercises in this section.

2.13 Basis for a topologyTopologies should be specified by saying what is in the topology. This is oftenhard. So what is often done is to specify enough of what is in the topology sothat the rest of the topology will follow naturally from the part that is specified.This is similar to giving a basis for a vector space or a generating set for a group


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2.13. BASIS FOR A TOPOLOGY 21

or subgroup. One technique is referred to as giving a basis for a topology. Theuse of the same word as is used in linear algebra is deliberate but there are

no parallels in the details. There is no notion of linear combination, linearindependence, and so on, and you should not look for them. The use of bases isnot the only technique to specify a topology and a more fundamental techniqueis labeled by the word subbasis. In spite of the fact that it is more fundamental,it is used less often. However, it is used, and must be learned.

There are two key set theory points in the section.One is that if A and B are collections of sets, then

AA

A

BB

B

=

(A,B)AB

(A B).

This goes back to the older way of denoting a collection of sets. You

should prove the statement above. It is quite easy.The book abandons the non-indexed collection notation in spite of thefact that it uses it before the indexed collection notation. Thus you shouldalso state (and prove) the parallel statement in which A for I and B for J are given and we wish to consider

I

A

J

B

.

You might not notice that this is essentially proven on Page 79.The reason for introducing this is that the section does not emphasize unions

of sets. It gets around this by the use of (2) in the definition of a basis. Theconnection between (2) and unions of sets is the following.

Lemma 2.13.1 Let A be a collection of sets. Then S equals a union of setsfrom A if and only if for every x S there is an element A of A so thatx A S.

You should do several things with this lemma. You should prove it. However,first you must understand it. The statement that S equals a union of sets fromA does not mean that it is the union of all the sets in A. To give notation tothe notion, it means that there is a subset A of A and

S =

AA

A.

You should then prove that the definition of basis in which (2) is replacedby (2a): Every intersection of two basis elements is a union of basis elementsis equvalent to the definition of basis given in the book.

You should then use this equivalent definition and the facts given aboveabout intersections of unions to give an argument different than the one givenon Page 79 that the collection T defined there is a topology.


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You should then use the lemma to shorten part of the proof of Lemma 13.2.Lemma 13.3 together with Example 4 illustrates how the notions of coarser

and finer can be used.The topologies introduced on the bottom of Page 81 and the top of Page 82

are important. One, the standard topology, is important because it is standard.The others are important because they are not standard.

The notion of subbasis is given near the bottom of Page 82. The notionshould be learned. Do not worry if the motivating question given before thedefinition never occurred to you. I doubt it occurred to very many people justafter they learned the definition of basis.

You should have the impression at this point that the subject is dominatedby manipulation of sets. Any weakness that you have with the manipulation ofsets will hold you back from this point on.

2.13.1 ExercisesExercise 1 is a test to see if you have absorbed the lemma that is given above.If you have, then the exercise can be done in one line.

I see no use for Exercise 2.Exercise 3 is worth doing.Exercise 4(a) and (b) are useful. The smallest topology part of (b) is related

to the notion of basis and subbasis. See the next problem. If you find (c) useful,then you should do it.

The relevance of the part of 4(b) that uses with the word smallest is shownin Exercise 5. It explains the use of the phrase generated by right after thedefinition of basis.

You should do as much of the various parts of Exercises 6, 7, and 8 as needed

to check your understanding. Parts involve tools about the real line from theprevious chapter.

2.14 The order topology

This section and the two that follow it give ways of building topologies. Thatthere are no exercises given until after the three sections indicates the commontheme that runs through the three sections. These three sections do not giveevery technique for building topologies. There are more given later in the chap-ter, more given later in the book, and more that are not even covered in thebook.

The order topology is an important technique, but not a dominant one. It

shows up with some regularity, but not with great frequency. However, it is agreat source of examples. Questions like Why dont properties P, Q and Rimply property X? can often be answered with a topology built from someorder.

The material should be learned. Pay particular attention near the end ofthe section to the use of basis and subbasis.


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2.15. THE PRODUCT TOPOLOGY ONX Y 23

2.15 The product topology on X Y

The product topology on X Y seems natural. It is. But it is not naturalfor the obvious reason that the definition is natural. The obvious reason isthat its basis is the usual (wrong) first guess as to what would be a reasonabletopology to put on X Y if X and Y have topologies. This leads to somepoints.

The usual first guess is given as the basis in the definition of the producttopology on Page 86. My apologies if this would not have been your first guess.You should understand the books example as to why it is wrong to use thebasis as the full topology.

The product of two topological spaces is generalized later to an arbitraryproduct of topological spaces. The sophistication gained after the study of theproduct of two spaces leads to a usual first guess for a topology on an arbitrary

product. This usual first guess is not wrong in that it does not form a topology.The usual first guess is wrong because it is not the one given by the usualdefinition. The reason for this is based on an omission in the current section.The current section completely omits motivation. It cannot give any motivationbecause there has been no mention of continuous functions. So you will have tolearn the material in this section without motivation. If you want motivation,you will have to peek ahead to Theorem 18.4 on Page 110 and Theorem 19.6 onPage 117. The motivation from these theorems is not complete enough for me.When we get there, I will give more motivation.

Theorem 15.2 gives a better idea of what to expect from the topology that isput on an arbitrary product, but it will not be clear why it should give a betteridea.

All the material in the section should be learned.

2.16 The subspace topology

Subspace topologies are everywhere. The previous two sections can build quite afew topologies, but with the subspace topology, the supply of examples explodes.More topologies are introduced by the subspace topology than by any othertechnique. I have no way to justify this statement, but I dont particularly care.Perhaps there can be doubts about this statement, but no one can claim thatit is obviously false.

Subspace topologies introduce the word in that cannot be omitted fromthis point on. Since any subset of a topological space can be a topological space,

and since the notion of open changes as you move from one space to another,it is now forbidden to use the word open without putting the word in behindit.

If R is given the order topology then [0, 1) is not open in R, it is not openin (1, 1), but it is open in [0, 1), it is open in [0, 2), it is open in [0, ) if all of(1, 1), [0, 1), [0, 2) and [0, ) are given the subspace topology.


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All of the material in the section should be learned. This includes the ex-amples.

Lemma 16.2 can be shortened to say open in open is open which is cuteand worth remembering, but it does voilate the rule of always following theword open by the word in.

2.16.1 exercises

Exercise 1 is not hard, but it is a study in being careful with definitions. Thesame applies to Exercise 2.

Exercise 3 is a study in the proper use of the word in.Exercise 4 is sadly misplaced without the introduction of continous functions,

but it is important anyway and should be done. It is the first exercise involvinga function in this chapter and it would be good if you can figure out whichproperties of functions and their interactions with sets are used in the arguments.

The wording of Exercise 5 is confusing. The book says let X and X denotea single set in . . . , but it really means that X and X denote the same set in. . . . Or, if that is too wordy, then it simply means X = X. Then the booksays in the topologies . . . which is hard to interpret. Are they sets in a largertopological space? I dont see any reason why they should be. The problemmakes perfect sense if instead of the wording given in the problem it said thatT is a topology on X and T is a topology on X with X = X. Similartreatment would apply to Y and Y. Of course, if the pair point of view hadbeen emphasized, then (X, T) and (X, T) could have been given as topologicalspaces and the commonality of the two sets would have been clear.

Exercises 6 through 10 test understanding of the definitions. The wordcompare in 9 and 10 is vague. I suppose that it might mean that you should

see if the words finer or coarser apply to the pair of topologies mentioned.Other forms of comparison get into the realm of comparison of properties, andproperties of topological spaces and their subsets have not yet been covered.

2.17 Closed sets and limit points

This is a large section with a lot of material. It is all important.

2.17.1 Closed sets

Some comments.1. Closed sets are complements of open sets, but closed is not the opposite

of open. The book makes this point emphatically and you should internalizeall the discussion that goes around it.2. Just as open must be followed by in, so must closed be followed by

in for the same reasons.3. Closed has a connection to limit and convergence that is easier to absorb

than the connection of these notions to open. If something that is capable


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2.17. CLOSED SETS AND LIMIT POINTS 25

to converging (sequence, for a familiar example, or net or filter for unfamiliarexamples) is contained in a closed set and it manages to converge to a limit,

then the limit will be in the closed set. Thus elements outside a closed set arefar away from the closed set.

4. Statements about closed sets come with proofs. At this point, it is just asimportant to know the proofs as it is to know the statements. All the basic factsare triumphs of basic facts from set theory. For example, Theorem 17.1 is builton the DeMorgan laws. Theorem 17.2 is built on the way that set differencecopperates with intersection.

5. Theorem 17.5 is important since it motivates the definition of limit pointon the next page. It also motivates the other definition of closed which isgiven as the second half of the statement of Corollary 17.7.

The phrase U is a neighborhood of x is discussed on Page 96. On Page97, the book points out that different authors mean different things by theword neighborhood. The book adopts one convention, but I would like to notcontradict the book but at least not adopt the convention. Thus I would liketo always use the phrase u is an open neighborhood of x to add emphasis(and perhaps clarity in the presence of authors of other books) to the opennessassumption of the book.

More on that phrase. Many people (including me) say U is open about xwith the word about implying that x is in U. Of course this means that U isan (open) neighborhood ofx. This sort of wording often gets used in proofs withwords like Let U be open about x to mean that U is an open neighborhoodof x.

In the definition of limit point is the provision other than x itself. This isannoying. Get used to it. Study the example. The point (pun unavoidable) isthat a set A can contain an element that really does not deserve to be a limit

point ofA.The key to the alternate definition of closed given by Corollary 17.7 isTheorem 17.6.

By the end of Corollary 17.7, you will have a train of ideas that leads tothe corollary starting with the defintion of a closed set. You should familiarizeyourself with the entire train of ideas. That is, know the proofs.

2.17.2 Hausdorff spaces

We encounter our first restriction. The property Hausdorff is a property thata topological space can either have or not have. The book defines Hausdorffspace without defining Hausdorff. The word is used without being followedby space so often (see Problem 9 on Page 101) that it should be introduced

that way. So we say that a topological space is Hausdorff if it is a Hausdorffspace.

The word Hausdorff becomes a restriction if one only looks at Hausdorffspaces. Or it is a restriction if one has a theorm that needs Hausdorff as ahypothesis. Restrictions are common. The definition of topological space is soflexible that it is hard to find theorems that apply to all topological spaces. The


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interesting (but restrictive) hypotheses are the ones that show up as hypothesesof the largest number of theorems. The hypothesis is Hausdorff is one of the

most interesting from this point of view.As often happens (and as mentioned in the book) it is possible to modify a

hypothesis or defintion to get definitions with similar but not identical effect.The book points out that T1 is close to Hausdorff. Problem 15 on Page 101shows that there is a definition of T1 equivalent to the one given on Page 99that makes T1 look even more like Hausdorff.

One of the motivations the book uses for the notion of Hausdorff is thebehavior of the limit of a sequence. The definition of limit of a sequence is acombination of the definition of converges to on Page 98 and the definitionat the top of Page 100. You could try to compare this to the defintion of limitof sequence of real numbers that you are used to, but that can wait until abit later. Theorem 17.10 about limits of sequences is a good illustration of thepower of the assumption that a space is Hausdorff.

Theorem 17.11 is good evidence that Hausdorff topologies are not all thatrare.

The role of the open sets in the definition of convergence of a sequence willbe discussed in class.

2.17.3 Exercises

Exercise 1 is a converse to Theorem 17.1. Exercises 2 and 3 have correspondingfacts about open sets to compare to. Exercises 4 and 5 are basic.

Exercises 6 and 7 are important in that it is easy to get parts wrong andit is important to know why. Part (b) of 6 and Problem 7 need be consideredcarefully.

Exercises 8 and 9 are a good reviews of the definitions.Exercises 10 through 12 review the basic definitions. Exercise 13 is a good

test of how thoroughly you have learned the material and how flexible you arein approach. There is more than one way to approach this problem and someare easier than others. This makes the problem sound hard. It is not, but itcan appear hard if you are not yet used to the terms.

Exercises 14, 16, 17 and 18 are tests of how well you can work with theconcepts of the section.

Exercise 15 was mentioned above as giving another way to define T1.

This book sometimes introduces important notions in exercises. Exercises19 and 20 do that with the notion of boundary. They are important exercises.

Exercise 21 has a star. It is somewhat harder than the others and less

fundamental. It is more of a puzzle than an exercise. However, the no morethan 14 aspect can be handled neatly with a few key observations. In a certainsense, one must develop an understanding of the algebraic behavior of thetwo operations closure and complement. To do this, give each a symbol(obviously it will not do to use c for both) and see what happens when theyare combined in various ways.


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2.18. CONTINUOUS FUNCTIONS 27

2.18 Continuous functions

2.18.1 Immediate concernsCertain exercises are imperative after reading the definition of a continuousfunction.

1. In the exercises, Exercise 1 is an implication. Both directions are true andshould be written out.

2. Another is the content of the first paragraph on Page 103.

3. Another is the content of the second paragraph of Page 103.

4. Another is the equivalence of (1) and (3) of Theorem 18.1.

5. Another is the equivalence of (1) and (4) of Theorem 18.1.

Of course most of the above have been done for you already. You shouldlearn them well and perhaps try to prove them without looking at the book.

Item (2) of Theorem 18.1 is nice, but not used often. However, it is a goodexercise in certain techniques.

Other aspects of the section are as follows.

2.18.2 Restrictions and inclusions

Theorem 18.2 is full of trivialities, but they are important in that they reviewthe definitions of several concepts, most importantly the concept of the subspacetopology. Item (b) in that theorem lies behind a couple of the other items.

2.18.3 Continuity in piecesItem (f) of Theorem 18.2, Theorem 18.3 and Exercise 9 go together. Theycombine into one result something that can be split into two steps.

Step 1. There is a notion of generating a topology that is not the sameas generating a subgroup or the notion summarized by the words basis orsubbasis. Let X be a set with topology T. Let A be a family of subsets ofX. Each A A gets the subspace topology from T. We say that A generatesthe topology T if for every B X, we have that B is in T if and only if forevery A A, it is true that B A is open in A under the subspace topologyon A. The only if direction will always be true and is useless. It is the ifdirection that is important. It basically says that testing for openness can bedone on small sets (the sets in A).

1a. You can show that if A is a collection of open sets in X whose union isall of X, then A generates the topology T. This is the content of Item (f) ofTheorem 18.2 and the proof can be derived from the proof of this item.

1b. You can show that if A is a finite collection of closed sets in X whoseunion is all of X, then A generates the topology T. This is the content ofExercise 9(a) and if you can do one, you can do the other.


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1c. You can show that if A is a locally finite collection of closed sets in Xwhose union is all of X, then A generates the topology T. This is the content

of Exercise 9(c) and if you can do one, you can do the other.Step 2. You can show that in the setting of Step 1, ifA generates the topology

T, then a function f : X Y is continuous if and only if its restriction to everyA A is continuous using the subspace topology on A. Of course only the ifdirection is interesting. This is the easier of the two steps. It basically says thattesting for continuity can be done on sets in a collection if testing for opennesscan also be done on the sets in the collection.

The summary is that Item (f) of Theorem 18.2, Theorem 18.3 and Exercise9 are really statements about collections that generate a topology.

Theorem 18.3 and its mild generalization Ecercise 9(a) are used all the time.The claim that

f(x) = 2x, x 1,

x + 1, x 1is continuous is usually argued in calculus by saying that the two parts of thedefinition agree at x = 1. This check only argues that f is a function. Whatis really going on is the pasting lemma (18.3). It is clear from the formulas(both are differentiable) that the restrictions off to the closed sets (, 1] and[1, ) are both continuous, and the union of these two sets is all of R. Now thepasting lemma guarantees the continuity of f on all of R.

2.18.4 Homeomorphisms

Homeomorphisms are extremely important. They are to topological spaces whatisomorphisms are to algebraic structures (groups, vector spaces, etc.). A home-

omorphism f from X with topology T to Y with topology S carries open setsin T as a one-to-one correspondence to the open sets in S. (This sentence iseven worth proving as a simple exercise.) Thus homeomorphisms carry thetopological structure of one space over to the topological structure of the other.

By and large, anything defined in a topology course using only open sets(we will see definitions using other means later that behave badly under home-omorphisms) that apply to one space will apply to a space homeomorphic to itif properly carried over by a homeomorphism. So for example, if f, X and Yare as in the last paragraph, and A X and x X are given, then x is a limitpoint of A if and only if f(x) is a limit point of f(A). Find an example wheresome aspect of this last sentence fails if f is only known to be a continuousfunction.

The book defines topological property rather obliquely. It does so because

this is the right definition. The definition is in terms of an effect rather than acause. If you cannot change something by a homeomorphism, then whatever itis that is not changing must be a topological property. Thus the definition oftopological property is defined in terms of the behavior of the property andnot on the manner in which the property was defined. This takes some gettingused to.


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2.19. THE PRODUCT TOPOLOGY 29

Definitions that discuss only open sets are obviously topological and the factthat they are topological surprises no one. We will meet quite a few obvious

ones. There are various other properties that can be defined on a topologicalspace where the definition does not look all that topological. We will not meetthe less obvious ones in this course, but you will in algebraic topology. Forexample, it was not known for a while that the homology groups (wait untilalgebraic topology) as defined before (say) 1940 were topological properties.They are and the proof took some doing.

2.18.5 Products

Theorem 18.4 partly justifies the definition of the product topology on X Y.It does not completely justify it. It leaves open the question of whether thereis another topology that could be defined on X Y that would also have theproperty guaranteed by Theorem 18.4. We will discuss this question in duetime.

2.18.6 Exercises

Exercise 1 has been mentioned and the content of Exercise 2 has been mentionedwhen f is a homeomorphism. Exercise 2 only assumes that f is continuous.Some properties are preserved by continuous functions. It is not necessary tohave homeomorphisms all the time.

Exercise 3 combines larger and smaller topologies with continuous functions.It is easy and should be done.

Exercise 4 is used later and should be done and also remembered.Exercise 5 is easy if you think about the definitions carefully.

Exercise 6 is easy, tricky if you have not seen things like it before and notcrucial.Exercise 8 is a good review of facts, definitions and previous results.Exercise 9 has been discussed.Exercise 10 is useful.Exercises 11 and 12 go together, are nice but not crucial.Exercise 13 is a nice argument that uses a lot of techniques that are useful.

It will be gone over. You should attempt it.

2.19 The product topology

When the number of factors of a product is infinite, then the obvious topology

is no longer the topology with the desired properties. This is explored to someextent in the book. The right definition is given minor motivation before thedefinition, and given much stronger motivation after the defintion. But it isnever explained why the right definition is the only right definition.

The discussion through the top half of Page 113 is fine as far as it goes.The definition of the Cartesian product of an indexed family of sets has been


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mentioned in class before and is necessary for the discussion. The break of thedefinition into two parts (the two definitions on Page 113) is good.

The comments in the second paragraph of Page 114 need to be emphasizedmore. The comments say that there are two ways of looking at the set offunctions from J to X. One way is just what the words say: it is the set offunctions from J to X. The other way is to think of it as a product of copiesof X indexed over J. The reason that this is important is that this section willput a topology on that product if X is a topological space. That means the setof functions from J to X will be given a topology. This lets us talk about aspace of fuctions rather than just a set of functions. This will come up againin important ways.

The defintions on Page 114 are needed. The box topology must be un-derstood only as far as knowing that it is the topology with the undesirableproperties.

The basis for the product topology discussed at the top of Page 115 is moreimportant than the subbasis that is used in the definition at the bottom of Page114. The subbasis does have some uses in arguments, but the basis is muchmore likely to be used when trying to prove something.

Theorem 19.1 is important only in that its statement efficiently discusses thedifferences between the box and product topologies.

Theorem 19.2 is a typical theorem of the type it suffices to consider a basisinstead of all open sets in a definition.

The next three theorems are straightforward.Theorem 19.6 and the example that follows makes clear the important dif-

ferences between the box and product topologies. The importance of Theorem19.6 is hidden in the fact that it has an if and only if statement in it, andnot just an if statement. Exercise 10 goes a little way towards justifying the

product topology as the right topology to put on a Cartesian product. To com-plete the justification, we need some preliminary exercises. We dont need themall, but once we start, we may as well give them all.

Let f : A B be a function. If A and B have topologies, then we candiscuss the continuity of f. We can discuss what makes it easier for f to becontinuous and what makes it harder.

Which makes it easier for f to be continuous, a large topology on B or asmall topology? If a topolgy is given on A what is the smallest topology on Bthat makes f continuous? What is the largest? Is there a topology on B thatguarantees the continuity of f no matter what topology A has?

Which makes it easier for f to be continous, a large topology on A or asmall topology? If a topolgy is given on B what is the smallest topology on Athat makes f continuous? What is the largest? Is there a topology on A that

guarantees the continuity of f no matter what topology B has?The last two paragraphs asked for examples of topologies that guarantee

continuity no matter what the other topology is. Give an example of a func-tion that is guaranteed to be continuous no matter what topologies are used onboth domain and range? Show that there is such a function between any twosets?


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2.19. THE PRODUCT TOPOLOGY 31

In the setting of Theorem 19.6 a test is made which determines the continuityof a function f. The result of the test has to do with the topology on the

product, but the details of the test (checking the continuity of the f) do not.So if Theorem 19.6 is to be true, then this forces the topology on the productto make a certain set of functions continuous. This is from the if part ofthe conclusion. Does this force the product topology to be large or small? (Orperhaps it should be asked whether it forces the product topology to be nottoo large or not too small?)

Note that if Theorem 19.6 is to be true, then any function that does notpass the test must not be continous. This is from the only if part of theconclusion. Does this force the topology to be large or small? (Or perhaps itshould be asked whether it forces the product topology to be not too large ornot too small?)

We now try to make large and small more specific. To one of thequestions just asked, the answer is the product topology cannot be too small.This can be made more specific by stating what the topology must contain. Tothe other question, the answer is the product topology cannot be too large.This can be made more specific by stating what the topology must be containedin.

It can be shown that the same collection of sets can be used in both answers.In an attempt to make things less confusing, let P be the collection of sets in theproduct topology. Of course we know that P is what the book calls the producttopology, but let us pretend that we dont. It is a perfectly good topology andwe can use it no matter what we call it. Then we have the following.

Consider the identity function from a product to itself. Since the productis both domain and range, we have to use the words domain and range todistinguish the two roles that the product plays. Put the topology P on the

domain. This is to play the role ofA in the statement of Theorem 19.6. Assumethat there is a topology on the range that satisfies the conclusions of Theorem19.6. That is, any function into the range is continuous if (we just use the ifpart here) all the f are continuous. The important observation at this pointis that all the f are in fact continuous because we are using P on the domain.Thus if Theorem 19.6 is to hold, the identity must be continous. Argue thatthis means that the product topology (i.e., the right topology on the productthat cooperates with 19.6) must be contained in P. This makes specific thenot too large aspect of the product topology.

Now we deal with the not too small aspect. Let Q be a topology on theproduct that is contained in P but that is strictly smaller than P. Argue thatfor some the topology Q does not have one of the subbasic sets defined atthe bottom of Page 114. Then argue that for this , the projection is not

continuous.Now we return to the identity function from the product to the product. Put

the topology Q on both domain and range and pretend that we think that Q isthe right topololgy to be called the product topology on the range. Argue thatthis would violate Theorem 19.6 because it would make a fuction continuous(the identity) that does not pass the test. Conclude that the right topology


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to put on the product cannot be strictly smaller than P.The the only topology on the product that cooperates with all aspects of

Theorem 19.6 must both contain P and be contained in P. This justifies callingP the product topology.

2.19.1 Exercises

Exercises 1, 2 and 3 are routine and are good checks of understanding. Exercise2 requires care in interpreting definitions.

Exercise 4 is a check that definitions are understood. What is the homeo-morphism?

Exercise 5 is another check of definitions, and such checks should be gettingtiresome by now.

Exercises 6, 7 and 8 are more advanced checks of understanding the topolo-gies. They are also more fun.

Exercise 9 is too trivial to spend any time on. If you dont get it skip it.Exercise 10 is not worth it at this point. If you went through the justification

of the product topology, then the concepts in the exercise wont add much. Someof the ideas get repeated later.

An extra exercise is worth it.Let {X}J be an indexed collection of topological spaces and let

Y =J

X

be given the product topology. Let c be an element of Y and let be in J.We define a function f : X Y. Since f(x) for x X will be an elementof a product, we must say what its coordinates are. That is, we must define

f(x)() for every J. We set

f(x)() =

x, = ,

c(), = .

Sometimes the image of f is called the slice parallel to X through c. Provethat f is a homeomorphism onto its image (which inherits the subspace topologyfrom Y).

2.20 The metric topology

The metric topology is important enough to get two sections in the book. Ev-

erything on Pages 119 and 120 are important.The top two paragraphs of Page 121 are opinions and are somewhat ques-

tionable.The two definitions on Page 121 are important as is the discussion of the

example of a property that is not topological. Another more compelling exampleof a property that is not topological will come up later.


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2.20. THE METRIC TOPOLOGY 33

The note at the top of Page 122 is important. It says that small sets deter-mine the topology of a metric space.

The definitions on Page 122 are important. The comment about the Eu-clidean metric is valid. (The book consistently uses names without capitalizingthem. Most books follow the tradition that all names are capitalized with oneexception. The word abelian, named for Niels Henrik Abel, is never capital-ized.) The square root in the Euclidean metric makes it a pain to work with.The square metric has other names and is much easier to work with.

The argument that the two metrics considered on Page 122 give the sametopology is important. Lemma 20.2 is a key lemma and it should be appreciatedfor the fact that several containments are involved (one metric contained inanother, one ball contained in another) and that the containments might not goin a direction that you might guess without thinking it through. It is importantthat you dont remember which way the containments go. Instead you should

get familiar enough with the pictures to have in mind so that you can tell whichway the containments go in just a few seconds.

Theorem 20.3 does two things. It shows that two metrics give the sametopology, and it shows that they give the same topology as the product topology.The techniques of the second part are much more important than the techniquesof the first part.

The calculations behind the first part really come from the pictures of theballs in the two metrics. The pictures are not given in the book and you shouldwork out what they are.

The second part is really an exercise in working with the word max. Sincethe word max is never used in the argument, you should see where it fits in.

The discussion on Page 124 is important and justfies the need for Theorem20.1.

Recall that XJ can be thought of in two ways. One is that it is a set offunctions and the other is that it is a product of copies of X indexed over J.The rest of the section discusses topologies to put on this set of fuctions (givingdifferent spaces of functions) when X = R. We already have one topology, theproduct topology. The uniform metric gives another.

The uniform metric is more important than Theorem 20.4 would have youbelieve. We got used to the idea that the product topology is the righttopology to put on a product and that the box topology is wrong. However,that does not mean that they are both useless. The box topology has its uses,rare though they may be. The uniform topology is used quite often in spite ofthe fact that it differs from the product topology. The uniform metric is definedhere for functions into R. It can be defined for functions into any metric spacewith no extra work, but R will do for now.

The proof of Theorem 20.5 is long. However, it is important in that it givesmore examples of how to work with sup and max.


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2.20.1 Exercises

Exercise 1(b) introduces what is often known as the taxi metric. It is not crucial.Part (b) is important for analysis. It is not crucial here either.Exercise 2 is easy, but not crucial.Exercise 3 is more important.Exercise 4 is good if you want to understand the topologies. It is more

important to understand the examples with respect to the product and uniformtopologies, but if you are going to do that, you may as well include the boxtopology.

Exercise 5 is a good test of understanding. The notation R is terrible.Exercise 6 is a one idea exercise. What is the key idea?I have no idea how important Exercise 7 is.Exericse 8, is very important in analysis. Exercise 9 shows how much work

is needed to deal with the Euclidean metric. Exercise 10 is also more important

for analysis. None of these is crucial to this course, but they might be for others.Exercise 11 can be skipped unless you think it is cute.

2.21 The metric topology (continued)

The first section on the metric topology concentrated on the metric and a biton continuous functions. In a sense, the exercises on sequences were exerciseson continuity. (Give the set {0, 1, 2, . . . , } with larger than any n N theorder topology. Then an infinite sequence x is a function on {0, 1, 2, . . .}. Aninfinite sequence x converges to L if and only if extending the function x to take to L is continous.)

This section adds other topological notions to the discussion.The rather quick statements in Page 129 are important. Exericse 1 forces

you to do one statement. The claim about the Hausdorff property should bedone out in extreme detail just to see how it works. The bottom of the pageshows that metric spaces can handle continuity in much the same way thatcontinuity is handled in calculus.

Lemma 21.2 and Thoerem 21.3 have a common theme: in metric spacessequences determine properties. That this is false in non-metric spaces is crucialto know. That this is true in the more general setting of first countable spaces isalso important to know. The contruction ofBn near the top of Page 131 showsthat in first countable spaces, one can insist that the countable basis at eachpoint be a sequence of nested sets.

Lemmas 21.4 and 21.5 verify the obvious and should be done, but can be

skipped if you are feeling overwhelmed.The topic of uniform convergence that starts near the bottom of Page 131

and continues through Theorem 21.6 is very important. Exercise 7 ties it totopology. This point is made just below the middle of Page 132. (Recall thatthe book only defines the uniform metric for functions into R. Exercise 7 holdsin much more generality with no more work.) The argument in the proof of


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2.21. THE METRIC TOPOLOGY (CONTINUED) 35

Theorem 21.6 is often called the epsilon over three argument. The result ofTheorem 21.6 together with Exercise 7 and Lemma 21.2 says that in the uniform

metric on the set of functions from a topological space to a metric space, thecontinuous functions form a closed set.

The book misses a point. It also is not careful with the way that it writesout quantified sentences. The right way to say that the sequence of functions(fn) converges uniformly to f is to say

> 0, N N, x X, n > N, (d(fn(x), f(x)) < ).

This allows us to compare this definition to the defintion that the sequenceof functions (fn) converges pointwise to f. This reads

> 0, x X, N N, n > N, (d(fn(x), f(x)) < ).

The difference is the reversal of a in uniform convergence to in point-wise convergence. So in uniform convergence, one n works for every x, but inpointwise convergence each x can have its own private N that works for it.

The relevance to the current topics is that there is a parallel to Exercise 7for pointwise convergence.

Exercise 7 12 . Let X be a set, and let fn : X R be a sequence of functions.Let RX have the product topology. Show that the sequence (fn) convergespointwise to the function f : X R if and only if the sequence (fn) convergesto f as elements of the topological space RX with the product topology.

This begins to justify the existence of the uniform metric. The uniformmetric leads to a topology that is larger than the product topology. Thatmakes it harder to be continuous into the uniform metric than into the producttopology. For sequences, it makes it harder for sequences to converge. Thus in

the uniform metric it is not possible for a sequence of continous functions toconverge to a non-continous function. But in the product topology (the topologyof pointwise convergence) it is. There is an example in the exercises.

2.21.1 Exercises

Exercise 1 justifies a quick remark at the beginning of the section.Exercise 2 is trivial but it introduces an important concept.Im not sure about Exercise 3. I suppose it is different from Theorem 20.5,

but I cant see it as an important difference. The main change is from R toarbitrary metric spaces. Perhaps it is just a good review of Theorem 20.5.

Exercise 4 is nice but not crucial.Exercise 5 is hard to get excited about.

Exercise 6 is relevant if you have done Exercise 712 . What is the limit of the

sequence (fn) in the product topology (topology of pointwise convergence)? Isit continuous?

Exericse 7 has already been mentioned.It is hard to get excited about the rest of the exercises. Do any that look

interesting.


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2.22 The quotient topology

We will cover this section in spite of the fact that the book labels it optional.It is used too often to skip.

The quotient topology is easy to define and a little harder to visualize. It isalso often set up in a form that seems different from its definition.

If f : X Y is a surjection from a topological space to a set, then there isa largest topology on Y that makes f continous. This topology is the quotienttopology.

You should show that the definition in the book has this property.

The word saturated is useful and is defined on Page 137. The notionsof open map and closed map are moderately useful and their definitions areeasy to remember. Open maps are seen discussed more often than closed maps.The projection from a product to a factor is an open map. This was an earlier

exercise. However, neither notion is crucial.The verification that the quotient topology is a topology comes down to thenice behavior of inverses of functions and is discussed on Page 138.

Page 139 is where the quotient topology is set up in a form that seemsdifferent from the definition. We will argue that it is really not different.

Let f : X Y be a surjection. Let X be partitioned by the sets f1(t)for t Y. As a partition, it might be suspected that these sets are equivalenceclasses under some equivalence relation. They are. The equivalence relation onX is the relation x is related to y iff(x) = f(y). The properties of = makeit trivial that this is an equivalence relation.

Now it is also trivial that there is a one-to-one correspondence betweenthe set of equivalence classes and Y. Further the one-to-one correspondenceis derived from f in that if [x] is the equivalence class containing x (it is the

class of all elements of X that map to the same place as x under f), then [x]corrsponds to f(x) in Y.

So the definition at the top of Page 139 is not all that special. Every sur-jection comes from a function that is like taking each element to an equivalenceclass that contains it. On the other hand, every partition (set of equivalenceclasses from an equivalence relation) leads to a surjection to the set of equiva-lence classes by sending each element to the eqivalence class that contains it.

We continue the discussion without topologies for a minute.

In the situation above, Y is universal for the partition given. If g : X Zis a function that preserves classes (if x and y are in the same class in thatthey map to the same element of Y under f), then there is a unique functionh : Y Z so that hf = g. Just let h send f(x) to g(x). This is well defined

by the hypotheses, it is the only choice that works, and it defines h on all of Ysince f is onto.

If X is a topological space, then the quotient topology on Y lets us do thediscussion above all over again with topologies. We get that f is continous, andifg is continous, then the unique h discussed above is continous by the definitionof the quotient topology.


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2.22. THE QUOTIENT TOPOLOGY 37

If every surjection comes from a partition (eqivalence relation) and everyequivalence relation leads to a partition, why have both approaches? After all,

they lead to the same thing. The answer is convenience. In many situationsthe description of the equivalence classes (partition elements) is quite easy. If Iwant to glue a square to a disjoint triangle along an edge, then I can say thefollowing.

S A

TB

Let i be a one-to-one correspondence from side A of square S to side B oftriangle T. The obvious one will do. Now we put a partition on S T. Every

point not in A B gets its own private equivalence class consisting of just thatpoint. All other equivalence classes have two points each. These come frompoints in A. For each x A we form the class {x, i(x)}. Now we take thequotient topology. We say that we have glued T to S along A by the fuctioni.

This can be varied tremendously. For example, we can use functions thatare not one-to-one correspondences.

The book goes to great lengths to show how to visualize a quotient topology.You must keep in mind that the cases considered are quite special. There is nogeneral way to visualize all quotient topologies. They can be quite messy.

(I just discovered that the book has been using x y since Page 85 for pairsof elements in R R to avoid confusion with open intervals. I suppose this isacceptable, but it will be found in no other book.)

Example 4 is less important than Example 5. This is especially true becausethe book confesses that in Example 4 one can show that the quotient space isthe 2-sphere. Later when we are ready to show that this is true, we will comeback to the example. The picture of Example 5 is easier to deal with since theequivalence classes are smaller. In Example 5, the largest equivalence class hasfour points.

The points made at the bottom of Page 141 are worth knowing, but notdwelling over. The main point is that very little is preserved by quotient maps.The exact examples are not important for now.

Theorem 22.1 is probably not needed for a first look at quotient spaces.

Theorem 22.2 was discussed above and gives a crucial property of quotienttopologies. Corollary 22.3 is an easy conequence.

The examples on Page 143 and Page 144 can be skipped for now. They showhow bad quotient maps can be.

A point made earlier about topological spaces is relevant. There are manystrange topologies. The ability of the definition of a topology to include suchstrange spaces should not detract from the fact that the definition includesmany very useful topologies. Similarly, the ability of quotients to create strange


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