Cp(X; Z) Kevin Michael Drees A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2009 Committee: Warren Wm. McGovern, Advisor Christopher Rump, Graduate Faculty Representative Rieuwert J. Blok Kit C. Chan ii ABSTRACT Warren Wm. McGovern, Advisor We examine the ring of continuous integer-valued continuous functions on a topological space X, denoted C(X; Z), endowed with the topology of pointwise convergence, denoted Cp(X; Z). We first deal with the basic properties of the ring C(X; Z) and the space Cp(X; Z). We find that the concept of a zero-dimensional space plays an important role in our studies. In fact, we find that one need only assume that the domain space is zero-dimensional; this is similar to assume the space to be Tychonoff when studying C(X), where C(X) is the ring of real-valued continuous functions. We also find the space Cp(X; Z) is itself a zero-dimensional space. Next, we consider some specific topological properties of the space Cp(X; Z) that can be characterized by the topological properties of X. We show that if Cp(X; Z) is topologically isomorphic to Cp(Y; Z), then the spaces X and Y are homeomorphic to each other, this is much like a the theorem by Nagata from 1949. We show that if X is a zero-dimensional space, then there is a zero-dimensional space Y such that X is embedded in Cp(Y; Z). Thus every zero-dimensional space can be viewed as a collection of integer-valued continuous functions. We consider and prove the collection of all linear combinations of characteristic functions on clopen (open and closed) subsets is a dense subspace of Cp(X; Z). We then consider when the space Cp(X; Z) are Gδ- and Fσ-subsets of the collection of all functions from X to Z (a Gδ-subset is a countable intersection of open subsets and a Fσ-subset is a countable union of closed subsets). We make classifications for when Cp(X; Z) is a discrete space, metrizable space, Fr´echet- Urysohn space, sequential space, and k-space. We end with some results on cardinal invari- ants and the relationships between the tightness and Lindel¨ofnumbers of related spaces. iii God created the integers, all the rest is the work of Man. - Leopold Kronecker iv ACKNOWLEDGMENTS There are a number of people that I would like to acknowledge at the completion of this manuscript, and my work on my dissertation. I am going to list them numerically, however the order in which names appear does not reflect the importance of their contribution to this work. 1. I would like to thank my parents, Richard and Karen, for all of their support during the endless years of college. My parents are the ones who made it possible for me to make it this far. I would also like to thank my siblings Scott, Jerry, and Sarah for putting up with me as a little brother. 2. I thank my advisor and friend Dr. Warren Wm. McGovern for trying to answer my endless stream of questions during my dissertation research. When seeking out a dissertation adviser you never know ahead of time what your relationship a person is going to be like; I am pleased that everything turned out positively. I am thankful that Dr. McGovern took me as a student; it would have been hard (impossible?) to find an adviser as talented as he. 3. I thank my Dissertation Committee, Dr. Rieuwert Blok, Dr. Kit Chan, and Dr. Christo- pher Rump, for putting up with my many typos and bad grammar. 4. I thank my best friend and classmate Papiya Bhattacharjee for all the help she has given me on my research and for all the time that she has listened to me talk about it. 5. I would like to thank Dr. K.T. Arasu, Dr. Qingbo Huang, and Dr. Steen Pedersen of Wright State University, and Dr. Kit Chan, Dr. Corneliu Hoffman, Dr. Warren Wm. McGovern, and Dr. Sergey Shpectorov of Bowling Green State University for teaching me how to be a mathematician. 6. I would also like to thank the Department of Mathematics and Statistics at Bowling Green State University for funding me during my time as a student. v Table of Contents CHAPTER 1: Preliminaries 1 1.1 Introduction . 1 1.2 General Topology . 4 1.3 Groups, Rings, and Modules . 17 1.4 Cardinality . 22 CHAPTER 2: Properties of C(X; Y ) and Cp(X; Y ) 25 2.1 Topological Properties of Cp(X; Y ) and Cp(X) . 25 2.2 Algebraic Properties of C(X; A)......................... 30 2.3 Topological Properties of Cp(X; Z) ....................... 40 CHAPTER 3: New Results on Cp(X; Z) 53 3.1 Evaluation Maps and Cp(Cp(X; Z); Z)...................... 53 3.2 The Space Lp(X; Z)................................ 65 3.3 Density and Cp(X; A), Where A is a Ring . 75 3.4 Gδ, Fσ, and the Baire Category Theorem . 80 CHAPTER 4: Classification of Cp(X; Z) Into Different Categories of Spaces 88 4.1 Metrizablility of Cp(X; Z) ............................ 88 4.2 Fr´echet-Urysohn Spaces and Cp(X; Z)...................... 94 4.3 When is Cp(X; Z) a Sequential Space? . 99 vi 4.4 Tightness and Lindel¨ofProperties for Cp(X; Z) . 107 CHAPTER 5: Closing Remarks 115 5.1 Conclusions . 115 5.2 Open Questions . 116 BIBLIOGRAPHY 117 1 CHAPTER 1 Preliminaries 1.1 Introduction During the late 1950's the object C(X), the set of all real valued continuous functions on the topological space X, became of interest because of its algebraic and topological properties. The original concept of the ring of continuous functions was brought to the attention of the mathematics community when Weirstauss considered C(X) to be an algebra. According to Henriksen there was a renewed interest in this object brought about by Cech^ and Stone in 1937, who independently wrote the papers On Bicompact Spaces and Applications of the Theory of Boolean Rings to General Topology, respectively (refer to [8] and [22]). Since that time C(X) has been used as a source for examples and counter-examples of statements about rings with desired algebraic aspects. Furthermore, C(X) was investigated from an analytical point of view. First, we will discuss the algebraic aspects of C(X). The book Rings of Continuous Functions by Gillman and Jerison [13] is a milestone in the study of C(X) as a ring. Since C(X) is a collection of real-valued functions, addition and multiplication can be defined on the set by pointwise addition and multiplication of functions. Hence C(X) forms a ring. The ring theoretic structure of C(X) has a direct relationship with the topological properties 2 that the space X possesses. When studying C(X) it is usually assumed that the space X is a Tychonoff space, since given any space X there exists a Tychonoff space Y such that C(X) is ring isomorphic to C(Y ). Recall that a space X is Tychonoff if is Hausdorff and completely regular. Refer to [11] for the definitions of the above terms. Gillman and Jerison [13] gives a detailed look at the ring structure of C(X) but does not consider a topology on the set C(X). In the late 1970's a number of people began studying the topology of pointwise convergence on C(X). The topology of pointwise convergence has as basic open subsets, sets of the form W (F; "; f) = fg 2 C(X): jf(x) − g(x)j < "; for all x 2 F g where f 2 C(X), " > 0, and F is a finite subset of X. For notational convenience, we write Cp(X) when the set C(X) is endowed with the topology of pointwise convergence. Note that the topology of pointwise convergence is the subspace topology on C(X) inherited from the Q product topology on x2X R. The topology of pointwise convergence is considered because it is the weakest of the many natural topologies that can be placed on C(X). For example, if X is a compact space, then C(X) is a Banach space under the uniform norm and one of the natural topologies to consider would be the weak topology. However, the major reason why one would use the topology of pointwise convergence is that every Tychonoff space X can be viewed as a collection of real-valued continuous functions on a space Y with the topology of pointwise convergence (refer to page 9 of [3]). That is, for every Tychonoff space X there exists a Tychonoff space Y such that X is homeomorphic to a subspace of Cp(Y ). A. V. Arkhangel0ski˘ıwrote the book Topological Function Spaces [4] detailing the topo- logical structure of the space Cp(X). The structure of Cp(X) is very well known at this point in time. It is know when Cp(X) is metrizable, a Fr´echet-Urysohn space, a sequential space, a k-space, and countably tight; all of these are based on the choice of X. There are other properties that have been characterized, for example, when the space Cp(X) is Lindel¨of, 3 σ-compact, σ-countably compact, and normal. Thus, the topological side of C(X), with the pointwise topology, is a well studied object also. Our interest in this dissertation is to given a detailed investigation of the lesser studied C(X; Z), the set of integer-valued continuous functions on X.