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## Über dieses Buch

The purpose of this book is to provide an integrated development of modern analysis and topology through the integrating vehicle of uniform spaces. It is intended that the material be accessible to a reader of modest background. An advanced calculus course and an introductory topology course should be adequate. But it is also intended that this book be able to take the reader from that state to the frontiers of modern analysis and topology in-so-far as they can be done within the framework of uniform spaces. Modern analysis is usually developed in the setting of metric spaces although a great deal of harmonic analysis is done on topological groups and much offimctional analysis is done on various topological algebraic structures. All of these spaces are special cases of uniform spaces. Modern topology often involves spaces that are more general than uniform spaces, but the uniform spaces provide a setting general enough to investigate many of the most important ideas in modern topology, including the theories of Stone-Cech compactification, Hewitt Real-compactification and Tamano-Morita Para­ compactification, together with the theory of rings of continuous functions, while at the same time retaining a structure rich enough to support modern analysis.

## Inhaltsverzeichnis

### Chapter 1. Metric Spaces

Abstract
The study of metric spaces preceded the study of topological spaces. The emerging awareness of the significance of the so-called open sets in metric spaces led to the concept of a topological space. Although many properties of topological spaces that have been studied extensively are motivated by our understanding of metric spaces, there are many topological spaces that are quite different from metric spaces. When we relax the conditions on a space so that we no longer have a metric we may get some surprising (and unpleasant) properties.
Norman R. Howes

### Chapter 2. Uniformities

Abstract
The concept of a metric space leads naturally to the concept of a uniform space, especially if one approaches the topic from the point of view of covering uniformities. The first development of uniform spaces by A. Weil titled Sur les espaces à structure uniforme et sur la topologie générale published in Actualities Sci. Ind. 551, Paris, 1937 took a different approach. It involved a family of pseudo-metrics that generate the topology of the space as opposed to a family of coverings. Weil’s original approach was rather unwieldy and was soon replaced by two others.
Norman R. Howes

### Chapter 3. Transfinite Sequences

Abstract
In the theory of metric spaces, sequences play a fundamental role. Recall that a function from one metric space to another is continuous if it preserves convergent sequences (Proposition 1.12) and that a metric space is compact if each sequence has a convergent subsequence (Theorem 1.10). Furthermore, it is possible to characterize the topology in metric spaces by means of convergent sequences (e.g., Proposition 1.10 and Corollary 1.6).
Norman R. Howes

### Chapter 4. Completeness, Cofinal Completeness And Uniform Paracompactness

Abstract
In 1915, A paper by E. H. Moore appeared in the Proceedings of the National Academy of Science U.S.A. titled Definition of limit in general integral analysis. This study of unordered summability of sequences led to a theory of convergence by Moore and H. L. Smith titled A general theory of limits which appeared in the American Journal of Mathematics in 1922. In 1937, G. Birkhoff applied the Moore-Smith theory to general topology in an article titled Moore-Smith convergence in general topology, which appeared in the Annals of Mathematics, No. 38, pp. 39-56. In 1940, J. W. Tukey made extensive use of the theory in his monograph titled Convergence and uniformity in topology published in the Annals of Mathematics Studies series. Tukey worked with objects that were generalizations of sequences that he referred to as phalanxes. They were a special case of the objects that are usually called nets today.
Norman R. Howes

### Chapter 5. Fundamental Constructions

Abstract
In this chapter we consider some important constructions of uniform spaces from other uniform spaces. Our first concern will be to consider the so called classical constructions that are studied for most spaces and algebraic structures that arise in the study of mathematics, namely subspaces, sums, products and quotients. Our approach will be to derive these constructions from a few fundamental concepts. These fundamental concepts take the form of limits of collections of uniformities. We will make these concepts precise in the next section.
Norman R. Howes

### Chapter 6. Paracompactifications

Abstract
In the late 1950s and during the 1960s, K. Morita and H. Tamano worked (independently) on a number of problems that involved the completions of uniform spaces. We state some of these problems here even though we have not yet defined some of the terms used in the statements of these problems.
Norman R. Howes

### Chapter 7. Realcompactifications

Abstract
From Stone’s characterization of βX (Theorem 6.2), it can be seen that if f is a real valued bounded continuous function on X, then f can be uniquely extended over βX. Consequently, any continuous function f:X → [0,1] has a unique continuous extension over βX. In fact, βX can be characterized by this property. To see this we need only establish that if each continuous function f:X → [0,1] has a unique continuous extension over βX, then any continuous function f:X → Y where Y is a compact Hausdorff space has a continuous extension over βX.
Norman R. Howes

### Chapter 8. Measure and Integration

Abstract
In this chapter and the next, the theory of integration in uniform spaces will be developed. This chapter will only be concerned with those aspects of integration theory that do not depend on the uniform structure of the space. In elementary analysis one encounters the concept of the Riemann integral. Intuitively, the process of Riemann integration in one, two and three dimensional Euclidean space corresponds to calculating lengths, areas and volumes respectively. The formalization of the Riemann integral occurred during the nineteenth century. Briefly, the main idea for one dimensional Euclidean space is that the Riemann integral of a function f over an interval [a,b] can be approximated by sums of the form
$$\sum\nolimits_i^n = 1f(x_i )\Delta (I_i )$$
where I1I n are disjoint intervals whose union is [a, b], Δ(I i ) denotes the length of I i and x i I i for each i = 1…n.
Norman R. Howes

### Chapter 9. Haar Measure in Uniform Spaces

Abstract
In 1933, in a paper titled Die Massbegriff der Theorie der Kontinuierlichen Gruppen published in the Annals of Mathematics (Volume 34, Number 2), A. Haar established the existence of a translation invariant measure in compact, separable, topological groups. Translation invariance of a measure μ in a topological group G means that if E is a measurable set then μ(E + x) = μ(E) for each xG. Here, E + x = {yG | y = a + x for some aG}. E + x is called the x-translate of E. The transformation T x defined on G by T x (y) = y + x is called the x-translation or simply a translation. Topological groups will be defined later in the chapter and these concepts will be developed formally.
Norman R. Howes

### Chapter 10. Uniform Measures

Abstract
In 1945, L. Loomis introduced an interesting generalization of the Haar measure to locally compact metric spaces that satisfy a property he called the congruence axiom in a paper titled Abstract congruence and the uniqueness of Haar measure (Annals of Mathematics, Volume 46). Loomis’ concept of invariance in this paper did not involve a translation (transformation), but rather the property that for a measure μ, μ(S1) = μ(S2) whenever S1 and S2 are compact spheres with the same radius. Because of this distinction, we will refer to these measures as uniform measures rather than Haar measures. As we shall see, Loomis’ concept, when applied to uniform spaces, generalizes the concept of Haar measure as presented in the last chapter. In this paper, Loomis showed there exists a unique, regular, invariant (uniform) measure on the Lebesgue ring (Borel sets that can be covered by countably many compact sets) on locally compact, metric spaces that satisfy the congruence axiom.
Norman R. Howes

### Chapter 11. Spaces of Functions

Abstract
In this chapter, we will be investigating several types of spaces constructed from functions on a given space X to some other space Y. The spaces X and Y will usually be uniform spaces, but the theory of function spaces is more general than that. Much of this chapter, is independent of uniform spaces. The spaces we will study first will be needed in the next chapter to develop the concept of uniform differentiation. In fact, much of the content of this chapter and the next is needed simply to develop the machinery from classical analysis so that we can discuss uniform differentiation in isogeneous uniform spaces.
Norman R. Howes

### Chapter 12. Uniform Differentiation

Abstract
At the time of writing this chapter, very little is known about uniform differentiation. It is an area where all the good theorems remain to be proven. It is also an area where we know the approximate form we would like for some of these yet unproven theorems to take. This is because we expect the uniform derivative to be equivalent to the Radon-Nikodym derivative when it exists. In this chapter we develop both the concepts of the Radon-Nikodym derivative and the uniform derivative. However, the assumptions we make about the spaces on which the uniform derivatives are defined, and perhaps about the uniform derivative itself, are probably unnecessarily restrictive. This is because we do not know how to prove the equivalence of the uniform derivative and the Radon-Nikodym derivative at the present time without them.
Norman R. Howes

### Backmatter

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