Rings of Continuous Functions

by Leonard Gillman, Meyer Jerison

Overview

Designed as a text as well as a treatise for active mathematicians, this volume begins with an unusual notice: "The book is addressed to those who know the meaning of each word in the title." As such, it constituted the first systematic account of the theory of rings of continuous functions, and it has retained its secure position as the basic graduate-level book in this area.
The authors focus on characterizing the maximal ideals and classifying their residue class fields. Problems concerning extending continuous functions from a subspace to the entire space play a fundamental role in the study, and these problems are discussed in extensive detail. A thorough treatment of the Stone-Čech compactification is supplemented with a number of related topics: the Ulam measure problem, the theory of uniform spaces, and a small but significant portion of dimension theory. Hundreds of problems of varying difficulty appear throughout the text, providing additional details, describing counterexamples, and outlining new topics.

Product Details

ISBN-13:	9780486816883
Publisher:	Dover Publications
Publication date:	01/16/2018
Series:	Dover Books on Mathematics
Pages:	320
Product dimensions:	6.00(w) x 9.00(h) x (d)

About the Author

American mathematician Meyer Jerison (1922–1995) was on the faculty of Purdue University from 1951 until his retirement in 1991.
Leonard E. Gillman (1917–2009) taught at the University of Rochester and the University of Texas and was President of the Mathematical Association of America in 1987–1988.

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CHAPTER 1

FUNCTIONS ON A TOPOLOGICAL SPACE

1.1. The set C(X) of all continuous, real-valued functions on a topological space X will be provided with an algebraic structure and an order structure.

Since their definitions do not involve continuity, we begin by imposing these structures on the collection Rx of all functions from X into the set R of real numbers. Addition and multiplication are defined by the formulas

(f + g)(x) = f(x) + g(x), and (fg)(x) = f(x)g(x).

It is obvious that both of the operations thus defined are associative and commutative, and that the distributive law holds: these conclusions are immediate consequences of the corresponding statements about the field R.

In fact, it is clear that Rx is a commutative ring with unity element (provided that X is not empty). The zero element is the constant function 0, and the unity element is the constant function 1. The additive inverse -f of f is characterized by the formula

(-f)(x) = -f(x).

The multiplicative inverse f-1 — in case it exists — is characterized by the formula

f-1(x) = 1/f(x).

1.2. The partial ordering on Rx is defined by:

f ≥ g if and only if f(x) ≥ g(x) for all x [member of] X.

That this is a partial ordering relation follows from the fact that R is ordered. It is clear that for every h, f + h ≥ g + h if and only if f ≥ g. Hence the ordering relation is invariant under translation. In addition, f ≥ 0 and g ≥ 0 implies fg ≥ 0. Therefore Rx is a partially ordered ring (0.19).

Next, for any f and g, the function k defined by the formula

k(x) = f(x) [disjunction] g(x)

satisfies: k ≥ f and k ≥ g; furthermore, for all h such that h ≥ f and h ≥ g, we have h ≥ k. Therefore f [disjunction] g exists: it is k. Dually, (f [conjunction] g)(x) = f(x) [conjunction] g(x). Thus, Rx is a lattice-ordered ring (0.19). The function [absolute value of f], defined as f [disjunction] -f, satisfies

[absolute value of f](x) = [absolute value of (f(x))].

course, R is also totally ordered (so that r [disjunction] s is simply max {r, s}), but Rx is not if X contains at least two points.

The ambiguous use of the various symbols, which refer sometimes to R and sometimes to Rx, should cause no difficulty.

1.3. The set of all continuous functions from the topological space X into the topological space R is denoted by C(X) — or, for short, by C.

The sum of two continuous functions is, of course, continuous; so is the product. And if f belongs to C, then so does f. Therefore C(X) is a commutative ring, a subring of Rx. The constant function 1 belongs to C and is its unity element.

It is easy to see that if f is continuous, then the function [absolute value of f] is also continuous. Since

f [disjunction] g = 2-1(f + g + [absolute value of (f - g)]),

f, g [member of] C implies f [disjunction] g [member of] C. Therefore C is a sublattice of Rx (0.5 and 0.19).

The symbol fn (n [member of] N) is used as in any ring. (Recall that N denotes the set of positive integers.) If f ≥ 0, then, more generally, f has a unique, nonnegative rth power (r [member of] R, r > 0), denoted by fr and defined by

fr(x) = f(x)r (x [member of] X);

and if f is continuous, then fr, as a composition of two continuous functions, is also continuous. In like manner, if n is odd (n [member of] N), then f1/n may be defined as a function in C, for any f [member of] C.

If the space X is discrete, then every function on X is continuous, so that Rx is the same as C(X). Conversely, if Rx = C(X), then the characteristic function of every set in X is continuous, which shows that the space is discrete.

1.4. The subset C* = C*(X) of C(X), consisting of all bounded functions in C(X), is also closed under the algebraic and order operations discussed in 1.3. Therefore C* is a subring and sublattice of C.

It can happen that the subring C*(X) is all of C(X) — i.e., every function in C(X) is bounded. When this is the case, X is said to be pseudocompact. Every compact space is pseudocompact, as is well known.

More generally, as we shall now prove, every countably compact space is pseudocompact. By definition, X is countably compact provided that every family of closed sets with the finite intersection property has the countable intersection property — i.e., every countable open cover has a finite subcover. Suppose, now, that X is countably compact, and consider any function f in C(X). The sets

{x: [absolute value of (f(x))] < n},

for n [member of] N, constitute a countable open cover of X. Hence a finite subfamily covers X, i.e., f is bounded.

A pseudocompact space need not be countably compact; see 51.

PROSPECTUS

1.5. A major objective of this book is to study relations between topological properties of a space X and algebraic properties of C(X) and C*(X). It is obvious that each of these function rings is completely determined by the space X. One of the main problems will be to specify conditions under which, conversely, X is determined as a topological space by the algebraic structure of C(X) or of C*(X). In other words, what restrictions on X and Y, if any are needed at all, will allow us to conclude that X is homeomorphic with Y, when we are given that C(Y) is isomorphic with C(X) or, perhaps, that C*(Y) is isomorphic with C*(X)?

Another type of problem is that of determining the class of topological spaces whose function rings satisfy some natural algebraic conditions, or, conversely, of determining the effects on the function ring of imposing some natural topological condition on the space. An example that might fit into either classification is given in 1B: X is connected if and only if C(X) is not a direct sum of proper subrings. Other classes of problems are to discover algebraic properties common to all function rings and to find relations between C(X) and C*(X) for a given X.

INVARIANTS OF HOMOMORPHISMS

1.6. Even before embarking upon a detailed study of function rings, we can observe quickly that several important properties of the family of functions that may not seem to be determined by the ring structure are, in fact, so determined (see Notes). The most significant of these properties is the order structure. To describe the order, it is enough to specify the nonnegative functions; but the condition f ≥ 0 is simply the algebraic requirement that f be a square, i.e., f = k2 for some k. It follows, moreover, that [absolute value of f] is determined algebraically: it is the unique nonnegative square root of f2.

We have just proved that every isomorphism from C(Y) into C(X) preserves order. Moreover, if f is bounded and f = k2, then k is bounded; hence an isomorphism from C*(Y) into C(X) also preserves order. Here is a more conclusive result:

Theorem. Every {ring) homomorphism t from C(Y) or C*(Y) into C(X) is a lattice homomorphism.

PROOF. Since g = l2 implies tg = (tl)2, t sends nonnegative functions into nonnegative functions, i.e., t is order-preserving. Next,

(t [absolute value of g])2 = t([absolute value of g]2) = t(g2) = (tg)2.

and since t[absolute value of g] ≥ 0, we have t[absolute value of g] = [absolute value of tg]. Combining this with the formula

(g [disjunction] h) + (g [disjunction] h) = g + h + [absolute value of (g - h)],

we get

t(g [disjunction] h) + t(g [disjunction] h) = tg + ih + [absolute value of (tg - th)] = (tg [disjunction] th) + (tg [disjunction] th).

But t(g [disjunction] h) and tg [disjunction] th are real-valued functions (defined on X), and therefore t(t [disjunction] h) = tg [disjunction] th.

1.7. Boundedness of functions is another property determined by the algebraic structure of C. More generally, we have the following result.

Theorem. Every (ring) homomorphism t from C{Y) or C*(Y) into C(X) takes bounded functions to bounded functions.

PROOF. As with any homomorphism, t1 = t(1 x 1) = (t1)(t1), so that the function t1 in C(X) is an idempotent. Therefore it can assume no values on X other than 0 or 1. Hence for each n [member of] N, the function

tn = t1 + ... + t1

assumes no values other than 0 or n. Consider, now, any function g in C*(Y). Since [absolute value of g] ≤ n, for suitable n [member of] N, we have [absolute value of tg] ≤ tn ≤ n.

1.8. Corollary. If X is not pseudocompact, then C(X) is not a homomorphic image of C*( Y), for any Y.

In particular, C(X) and C*(X) are isomorphic only if they are identical.

1.9. Another consequence of Theorem 1.7 is that an isomorphism from C(Y) onto C(X) carries C*(Y) onto C*(X). This is also a corollary of the next theorem.

Theorem. Let t be a homomorphism from C(Y) into C(X) whose image contains C*(X). Then t carries C*(Y) onto C*(X).

PROOF. We prove first that t1 = 1. Let k [member of] C(Y) satisfy tk = 1; then t1 = (tk)(t1) = t(k x 1) = tk = 1. It follows that tn = n for each n [member of] N.

Now, given f [member of] C*(X), we are to find g [member of] C*(Y) such that tg = f. Choose h [member of] C(Y) for which th = f, and choose n [member of] N satisfying [absolute value of f] ≤ n. Now define g = (-n [disjunction] h) [conjunction] n. Then g [member of] C*(Y), and, by Theorem 1.6, tg = (-n [disjunction] f) [conjunction] n = f.

ZERO-SETS

1.10. In studying relations between topological properties of a space X and algebraic properties of C(X), it is natural to look at the subsets of X of the form

(a) f(r) = {x [member of] X: f(x) = r} {f [member of] C, r [member of] R).

Clearly, these sets are closed.

We notice that if s is any real number, then

{x: f(x) = r) = {x: (f - s)(x) = r - s}.

Consequently, the family of sets of the form (a) obtained by allowing f to run through all of C, and r through all of R, can also be obtained by holding r fixed. The algebraic aspect of the situation points to the choice of the number 0 as the fixed value of r to be considered.

The set f(0) will be called the COP[zero-set]COP of f. We shall find it convenient to denote this set by Z(f), or, for clarity, by ZX(f):

Z(f) = ZX(f) = {x [member of] X: f(x) = 0} (f [member of] C(X)).

Any set that is a zero-set of some function in C(X) is called a zero-set in X. Thus, Z is a mapping from the ring C onto the set of all zero-sets in X.

Evidently, Z(f) = Z([absolute value of f]) = Z(fn) (for all n [member of] N), Z(0) = X, and Z(1) = 0. Furthermore,

Z(fg) = Z(f) [union] Z(g),

and

Z(f2 + g2) = Z([absolute value of f] + [absolute value of g]) = Z(f) [intersection] Z(g).

If f [member of] C and g = [absolute value of f] [conjunction] 1, then g [member of] C* and Z(g) = Z(f). Hence C and C* yield the same zero-sets.

The formula

[MATHEMATICAL EXPRESSION OMITTED]

shows that every zero-set is a Gδ, i.e., a countable intersection of open sets. Conversely, in a normal space, every closed Gδ is a zero-set (3D.3). This need not be so if the space is not normal, however (3K.6). On the other hand, in a metric space, every closed set is a zero-set, as it consists precisely of all points whose distance from it is zero.

(Continues…)

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Table of Contents

Foreword
1. Functions on a Topological Space
2. Ideals and Z-Filters
3. Completely Regular Spaces
4. Fixed Ideals. Compact Spaces
5. Ordered Residue Class Rings
6. The Stone-Cech Compactification
7. Characterization of Maximal Ideals
8. Reralcompact Spaces
9. Cardinals of Closed cSets in BX
10. Homomorphisims and Continuos Mappings
11. Embedding in Products of Real Lines
12. Discrete Spaces. Nonmeasurable Cardinals
13. Hyper-Real Residue Class Fields
14. Prime Ideals
15. Uniform Spaces
16. Dimension
Notes
Bibliography
List of Symbols
Index