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Topology for Analysis
By Albert Wilansky Dover Publications, Inc.
Copyright © 1998 Albert Wilansky
All rights reserved.
ISBN: 978-0-486-15075-8
Contents
DOVER BOOKS ON MATHEMATICS,
Title Page,
Copyright Page,
Dedication,
Preface,
Acknowledgments,
Topology for Analysis,
1 - Introduction,
2 - Topological Space,
3 - Convergence,
4 - Separation Axioms,
5 - Topological Concepts,
6 - Sup, Weak, Product, and Quotient Topologies,
7 - Compactness,
8 - Compactification,
9 - Complete Semimetric Space,
10 - Metrization,
11 - Uniformity,
12 - Topological Groups,
13 - Function Spaces,
14 - Miscellaneous Topics,
Appendix,
Tables of Theorems and Counterexamples,
Bibliography,
Index,
CHAPTER 1
Introduction
1.1 Explanatory Notes
There are certain standard and all-pervasive notations and terminologies used by mathematicians. In addition, we use a few special notations with less currency.
The following notations will be used:
R real numbers
Q rational numbers
J irrational numbers
ω positive integers
Z integers
[empty set] empty set
à the complement of A
{x: ...} the set of all x such that
x [member of] Ax is a member of A
x [not member of] Ax is not a member of A
A [union] B {x: x [member of] A or x [member of] B}
union of A and BA [union] B
A [intersection] B {x: x [member of] A and x [mmeber of] B}
intersection of A and BA [intersection] B
U {S: S [member of] [summation]} {x: x [member of] S for some S [member of] [summation]} ([summation] is a collection of sets)
U {Sα: α [member of] A} {x: x [member of] Sα for some α [member of] A} (A is some indexing set)
[intersection] {S: S [member of] [summation]} {x: x [member of] S for all S [member of] [summation]}
[intersection] {Sα: α [member of] A} {x: x [member of] Sz for all α [member of] A}
A [subset] B, B [contains] A every member of A is a member of B
A is included in B,
B includes A,
A is a subset of B,
A is a set in B,
B is a superset of A A [subset] B, B [contains] A
A is a proper subset of B A [subset] B, [empty set] ≠ A ≠ B
A \ B A [intersection] [??]
A [??] B A [intersection] B = [empty set]
A does not meet BA [??] B
A meets B A [intersection] B ≠ [empty set]
A meets B in x x [member of] A [intersection] B
singleton set with one member
{x} set whose only member is x
disjoint family a family of sets, each pair of which has empty intersection
[a, b] {x: a ≤ x ≤ b}
(a, b) {x: a< x< b}
[a, b) [a, b] \ {b}
(a, b] (a, b) [union] {b}
(-∞, a) {x: x< a}
[a, ∞) {x: x ≥ a}
characteristic function of S f, where f(x) = 1 if x [member of] S,
f(x) = 0 if x [not member of] S
f]S] {f(x): x [member of] S}
f-1[S] {x: f(x) [member of] S}
(f< a) {x: f(x) < a}
(f = a) {x: f(x) = a}
f[perpendicular to] (f = 0)
f is one-to-one x1 ≠ x2 implies f(x1) ≠ f(x2)
f: X -> Y is ontof]X] = Y
one-to-one correspondence function which is one-to-one and onto
We draw the reader's attention to the following:
A [??] B, read "A does not meet B" meaning A [intersection] B = [empty set].
When the notation à is used, it is assumed that a set X has been designated and à = X \ A. When the presence of X is not clear from the context, the notation X \ A will be used.
The words "space," "set," "family," and "collection" are synonymous.
When a space X has been designated the members of X will be called points.
The words "map," "mapping," and "function are synonymous. If f: X ->Y we call X the domain, and Y the range of f; "f" will sometimes be written as "x ->f (x)."
ITALICS. A word in italics is being used for the first time and is defined by the sentence in which it appears.
PROOF BRACKETS. Part of a discussion enclosed in square brackets means the statement immediately preceding the brackets is being proved. As an example, suppose the text reads, "Since x is not zero if x = 0, it follows that cos x = 1 contradicting the hypothesis], we may cancel x from both sides." The reader should first absorb "Since x is not zero, we may cancel x from both sides." He may then proceed with the text, or, if desired, return to the proof in brackets.
STARRED PROBLEMS AND EXAMPLES. Problems marked * must be done as they form part of the development of the text and, in extreme cases, are used later without citation. These are simple or are supplied with hints. Examples marked * are part of the development; those unmarked may be omitted, and those marked [??] are special and of limited interest.
100 PROBLEMS. In each section Problems 101, 102, ... are devoted to extending the text, and exposing interesting results beyond the scope of the book.
200 PROBLEMS. Problems 201, 202, ... may be extremely challenging.
END OF PROOF. The end of a proof is indicated by [??].
BIBLIOGRAPHY. References to the bibliography at the back of the book are indicated by [ ].
THE EMPTY SET. We state our conventions concerning the empty set [empty set]. These cannot be proved since we do not set up our set theory formally. There is only one empty sent [empty set]; [empty set] [subset] A for all sets A. (This makes the statement "A [subset] B implies [??] subset] Ã, for subsets of a space X," true if B = X since [??] = [empty set]) Wherever some property of sets is tested for a set A by examining an arbitrary point of A, then this property is true of the empty set. For example, "all positive integers a, b, c satisfying an + bn = cn for some integer n > 2 are larger than 3" is a true statement (and very easily proved!) even though there may be no such integers. (Here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] {(a, b, c): an + bn = cn, a, b, c [member of] ω}.) The empty set is finite.
FOUNDATIONS. We do not state our set theoretical foundations. These are based on a naive belief that we know what sets are like. In any particular argument the context will always indicate some fixed space of which all sets mentioned are subsets or members. The only explicit statements are those on the empty set, just given, and on maximal chains in Section 7.3. Our use of cardinal and ordinal numbers is restricted to examples marked [??].
Problems
* 1. If f: X ->Y, g: Y ->Z, define g [??] f: X ->Z by (g [??] f) (x) = g]f(x)]. If f, g are one-to-one and onto prove that (g [??] f)-1 = f-1 [??] g-1. (f-1 means the map from Y to X satisfying f-1f = ix, f [??] f-1 = iy where ix(x) = x for all x [member of] X.)
* 2. Let {Sα: α [member of] A} be a family of subsets of a set X, and let f: X ->Y. Prove that
a. f [[union] {Sα: α [member of] A}] = [union] {f.]Sα]: α [member of] A};
b. f [[intersection] {Sα: α [member of] A}] [subset] [intersection] {f]Sα]: α [member of] A};
c. if f is one-to-one, inclusion may be replaced by equality in (b), but not in general, even if A has only two members;
d. if f is one-to-one, f][??]] c {f]S]}~;
e. if f is onto, {f]S]}~ [subset] f][??]];
f. "one-to-one" cannot be omitted in (d), and "onto" cannot be omitted in (e).
* 3. Let {Sα: α [member of] A} be a family of subsets of Y, and let f: X ->Y. Then (in contrast with Problem 2), f-1[[union]Sα] = [union] f-1[Sα], f-1[[intersection] Sα] = [intersection]f-1[Sα], f-1[[??]] = {f-1[S]}~.
* 4. Let A, B be two collections of subsets of a set X with A B. Prove that [intersection] {S: S [member of] A} [contains] [intersection] {S: S [member of] B}. What is the corresponding result for union?
* 5. Let f: X ->Y and let S be a subset of X or Y. Prove that f]f-1 [S]] [subset] S, f-1[f [S]] [contains] S, f]{f-1[S]}~] [empty set] S.
* 6. What assumption about f would produce equality in the first two parts of Problem 5?
* 7. Let X, Y be sets. Let X x Y be the set of all ordered pairs (x, y) with x [member of] X, y [member of] Y. Show that R2 = R x R.
* 8. Show that à [subset] [??] if and only if A [contains] B.
* 9. Prove the following formulas (given by the 19th-century mathematician, A. de Morgan).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
* 10. Let f: X ->X. Let A, B be disjoint subsets of X, and set G = A [intersection] f-1[B]. Show that f [G] [??] G.
* 11. Let f: X ->Y, g: Y ->X. We say that g is a left inverse of f, and f is a right inverse of g if g [??] f is the identity on X. Show that f has a left inverse if and only if f is one-to-one, and a right inverse if and only if f is onto.
1.2 n-Space
The space R of real numbers will not be defined in this book. It will be assumed to allow the usual operations of arithmetic, to have its usual ordering (in short, it is a totally ordered field), and to have the property that every bounded set S has a least upper bound, written sup S. Such facts as x2 ≥ 0 for all x [member of] R will be used without scruple.
A set S is called countably infinite if it can be put in one-to-one correspondence with ω; that is, there exists f: S -> ω which is one-to-one and onto. A set is called countable if it is finite or countably infinite. A set which is not countable is called uncountable. We shall assume the existence of an uncountable set. (Some are shown in Problems 201, 202, etc., and R is also proved uncountable in Sec. 9.3, Problem 114.)
We denote by Rn the set of all ordered n-tuples of real numbers, where n is a positive integer; R1 is the same as R. For x, y [member of] Rn, say, x = (x1, x2, ..., xn), y = (y1, y2, ..., yn), we define
x
, pronounced norm x, by the formula
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and x • y, pronounced x dot y, by the formula
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
so that, in particular, x • x =
x
2.
Note that
x + y
2 = (x + y) • (x + y) = x • x + 2x • y + y • y. Thus, for all x, y,
2x • y =
x + y
2 -
x
2 -
y
2,
and, similarly
-2x • y =
x - y
2 -
x
2 -
y
2.
Since
x ± y
2 ≥ 0 we get ±2x • y ≤
x
2 +
y
2,hence,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.1)
Now let x, y be different from 0. (By 0 is meant the n-tuple (0, 0, ..., 0).) Let x' = x/
x
, y' = y/
y
. Then
x'
=
y'
= 1 and so, by (1.2.1), we have |x' • y'| ≤ 1, and so
|x • y| ≤
x
•
y
. (1.2.2)
Formula (1.2.2), called Cauchy's inequality (named for the famous 19th-century mathematician, A. Cauchy), was proved for x,y different from 0, but obviously holds in this case also.
We now have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Taking positive square roots we obtain
x + y
≤
x
+
y
. (1.2.3)
If now we define d(x, y) =
x - y
, the familiar distance (familiar at least for n = 1, 2, 3), we have, for any x, y, z,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
This function d is called the Euclidean distance forRn, also the Euclidean metric forRn
Problems on n-Space
* 1. Show that
x - y
≥ |
x - y
|.
* 2. Fix m, n with m >n and define f:Rn ->Rm by (x1, x2, ..., xn) -> (x1, x2, ..., xn, 0, 0, ..., 0). Is f one-to-one? Is it onto? Show that
f(x)
=
x
for all x [member of] Rn.
(Continues...)
Excerpted from Topology for Analysis by Albert Wilansky. Copyright © 1998 Albert Wilansky. Excerpted by permission of Dover Publications, Inc..
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