C p (X, Z) Kevin Michael Drees. A Dissertation

Transcription

1 C p (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

2 ABSTRACT ii 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 C p (X, Z). We first deal with the basic properties of the ring C(X, Z) and the space C p (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 C p (X, Z) is itself a zero-dimensional space. Next, we consider some specific topological properties of the space C p (X, Z) that can be characterized by the topological properties of X. We show that if C p (X, Z) is topologically isomorphic to C p (Y, Z), then the spaces X and Y are homeomorphic to each other, this is much like a the theorem by Nagata from We show that if X is a zero-dimensional space, then there is a zero-dimensional space Y such that X is embedded in C p (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 C p (X, Z). We then consider when the space C p (X, Z) are G δ - and F σ -subsets of the collection of all functions from X to Z (a G δ -subset is a countable intersection of open subsets and a F σ -subset is a countable union of closed subsets). We make classifications for when C p (X, Z) is a discrete space, metrizable space, Fréchet- Urysohn space, sequential space, and k-space. We end with some results on cardinal invariants and the relationships between the tightness and Lindelöf numbers of related spaces.

3 iii God created the integers, all the rest is the work of Man. - Leopold Kronecker

4 ACKNOWLEDGMENTS iv 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. Christopher 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.

5 v Table of Contents CHAPTER 1: Preliminaries Introduction General Topology Groups, Rings, and Modules Cardinality CHAPTER 2: Properties of C(X, Y ) and C p (X, Y ) Topological Properties of C p (X, Y ) and C p (X) Algebraic Properties of C(X, A) Topological Properties of C p (X, Z) CHAPTER 3: New Results on C p (X, Z) Evaluation Maps and C p (C p (X, Z), Z) The Space L p (X, Z) Density and C p (X, A), Where A is a Ring G δ, F σ, and the Baire Category Theorem CHAPTER 4: Classification of C p (X, Z) Into Different Categories of Spaces Metrizablility of C p (X, Z) Fréchet-Urysohn Spaces and C p (X, Z) When is C p (X, Z) a Sequential Space?

6 4.4 Tightness and Lindelöf Properties for C p (X, Z) vi CHAPTER 5: Closing Remarks Conclusions Open Questions BIBLIOGRAPHY 117

7 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 Ĉech 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

8 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, ε, f) = {g C(X) : f(x) g(x) < ε, for all x F } where f C(X), ε > 0, and F is a finite subset of X. For notational convenience, we write C p (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 product topology on x X 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 C p (Y ). A. V. Arkhangel skiĭ wrote the book Topological Function Spaces [4] detailing the topological structure of the space C p (X). The structure of C p (X) is very well known at this point in time. It is know when C p (X) is metrizable, a Fréchet-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 C p (X) is Lindelöf,

9 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. In the early 1960 s Alling [1] and Pierce [20] wrote the first papers on C(X, Z). They were concerned with the ring structure. Their research was inspired by Gillman and Jerson s book [13]. However, when making the change of R to Z as the codomain for the functions it has a drastic change on the structure of the ring. The trace of a maximal ideal in C(X) is a minimal prime ideal in C(X, Z). It also inherits some properties from Z; C(X, Z) is always a Bézout ring, that is, every finitely generated ideal is principle. However, to have a plentiful amount of elements in C(X, Z) to make it interesting you need to assume that the space X is disconnected. In fact, it will be seen that the space X can be assumed to be a zero-dimensional space when dealing with C(X, Z). This conclusion is parallel to the assumption that the space X is a Tychonoff space when dealing with C(X). As in the case of C(X) assuming the space X to be Tychonoff, Pierce [20] demonstrated that for C(X, Z) one can assume the space is totally disconnected. Since C(X, Z) is a subset of C(X) we can endow it with the subspace topology of pointwise convergence inherited from C p (X). It is straightforward to check that this is the same as the subspace topology inherited from x X Z. When we want the properties of this topology we will write C p (X, Z). A basic open set of C p (X, Z) is a set of the form {f C(X, Z) : f(x i ) = n i, i = 1,..., k} where x i X and n i Z, and n N. The papers of Pierce [20], Alling [1], and Martinez [17] have shown that C(X, Z) behaves differently than C(X) with respect to its algebraic properties. We will show that C p (X, Z) behaves differently with respect to topological properties. For example C p (X, Z) is zero-dimensional for any space X while C p (X) is never zero-dimensional. Though they do have their similarities. A number of natural properties that the topological space may have will be studied in regard to C p (X, Z). There will be a characterization for when C p (X, Z) is a discrete space, is a metrizable, is a

10 4 Fréchet-Urysohn space, when it is a sequential and when it is a k-space. Along with these the connection between the Lindelöf number and tightness will be shown. We will consider other cardinal invariants such as the character and weight, and place a bound on the the density of certain spaces. Along the way we will find that there are a number of results that can be generalized to the object C p (X, A), where A is a zero-dimensional topological ring. 1.2 General Topology There are several standard sets that will be used, they are (i) N, the set of natural numbers, (ii) Z, the set of integers, (iii) Q, the set of rational numbers, and (iv) R, the set of real numbers. There will be more said about these four standard sets as we proceed. We can construct new sets out of given sets. Let A and B be sets; B A is the set of all functions from A to B. Note that B A may also be viewed as the product Π a A B, that is, we can view the element f B A as the element (f(a)) a A Π a A B. For a set S the power set, denoted P(S), is the collection of all subsets of S, i.e P(S) = {A : A S}. If A and B are sets and f B A is the constant function defined by f(a) = b, for all a A, we will denote f by b, that is, b B A means b(a) = b for all a A. The characteristic function for a subset S of a subset X, is the function χ S : X {0, 1} given by 1 if x S χ S (x) = 0 if x X\S.

11 5 Later we will define this in greater generality. If S and T are sets with a function f : S T and A is a subset of S, then the restriction of f to A, denoted f A, is the function f A : A T such that f A (x) = f(x) for all x A. Let S be a set, a partial order on the set S is a relation on S such that (i) a a, for all a S, (reflexive) (ii) if a, b S with a b and b a, then a = b, (anti-symmetric) and (iii) if a, b, c S with a b and b c, then a c (transitive). A partially ordered set (or poset for short) is a set S with a partial order relation. Example Let X be a set, and let F be a family of subsets of X. The inclusion relation on F makes (F, ) into a poset. If we have a particular kind of relation on a set we can construct a topology on the set called the ordered topology. First, we need to define a linear order; a linear order on a set the X is a relation < on X which has the following properties for x, y, x X: (i) If x < y and y < z, then x < z. (ii) If x < y, then the relation y < x does not hold. (iii) If x y, then x < y or y < x. The set X together with the linear order relation < is called a well-order set. We have that the power set of a set S is a collection of subsets of S, we can consider other collections of subsets S and what properties that they may have. Definition A topological space is a set X with a collection T of subsets of X which has the following properties: (i), X T,

12 (ii) if O α T for α I, then α I O α T, where I is any index set, and 6 (iii) if O i T for i = 1,..., n, then n i=1o i T. We denote a topological space as (X, T ), where T is called the topology on the set X. We define a point as an element in the set for the topological space. It then follows that for every set X there always exists a topology, namely the power set. The discrete topology on set X is the topology on X which is precisely P(X). The set X is said to have the indiscreete topology if the topology is the collection {, X}. Note that there are many other topologies for the set X, in general. Remark When we say X is a space we mean that X is a set equipped with a topology, i.e. we suppress the topology notation. A notational note: throughout X, Y, and Z will be topological spaces, unless otherwise stated. Given a topological space (X, T ) any set A T is called open and the complement of an open set is called closed. A set which is both open and closed is clopen. The closure of a subset S in a space X is cl X (S) = {A : S A and A is a closed subset of X}. Observe that the closure of a subset S is the smallest closed subset containing S. Lemma (Theorem 17.5, [19]). If (X, T ) is a space and S is a subset of X, then cl X (S) = {x X : for all x O T when O S } Remark If X is a space with A and B subsets of X such that A B, it is then clear from the definition of closure that cl X (A) cl X (B). Let X be a space, and let S be a subset of X. We say that x X is a limit point of S (also known as an accumulation point or cluster point) if every open subset of X containing x also contains a point in S other than x. A sequence is a map f from N into a set S,

13 7 if f(n) = x n we will simply write {x n } n N as the sequence. We say the sequence {x n } n N converges to the point x X provided that for all open subsets U of X containing x there is a N N such that x n U for all n N. If the sequence {x n } n N converges to X we use the notation x n x as n. We say that S is dense in X if cl X (S) = X. S is called co-dense in X if X\S is dense in X and S called nowhere dense in X if cl X (S) is co-dense in X. Proposition (Proposition 1.3.5, [11]). Let X be a space and S a subset of X, S is dense in X if and only if every non-empty open subset of X contains a point of S. Let X be a set. A base is a collection, B, of subset of X such that the collection of arbitrary unions of elements of B form some topology on X. Let (X, T ) be a space. A base for the topology T is a subcollection B of T such that (i) For all x X there is a U B such that x U. (ii) If x U 1 U 2, where U 1, U 2 B, there is a U 3 B such that U 3 U 1 U 2. If (X, T ) is a space and B is a base for the space we say that B generates the topology T. We then have the following useful lemma. Lemma (Lemma 13.2, [19]). Let X be a space. If C is a collection of open sets of X such that for each open set U of X and x U, there is an element C C such that x C U, then C is a base for X. A subbase is a collection of sets in the topology such that the set of finite intersections of members of the collection forms a base for the topology. Let x be a point in space X, let B x be a collection of open subsets of X containing x, if for every open set O containing x there is a U B x such that x U O we then say that B x is a base at point x, i.e. a base at a point. Definition A topological space is said to be zero-dimensional if it has a base of clopen sets.

14 A few standard topological spaces, with their bases, which will be used are: 8 (i) R, with the base {(a, b) : a < b, a, b R}, (ii) Q, with the base {(a, b) Q : a < b, a, b R}, (iii) Z, with the base {{k} : k Z}, and (iv) N, with the base {{n} : n N}. For the four spaces above the topology given will be called the usual topology for the respective spaces. Example A few examples of zero-dimensional spaces are N, Z, and Q all with the usual topology. However, R with the usual topology is not zero-dimensional. Also, it is clear that every discrete space is a zero-dimensional space. We can make new topological spaces based on old topological spaces. If X is a topological space with topology T and Y is a subset of X, then the set T Y = {U Y : U T } is a topology on Y called the subspace topology. If X is a space and Y is a subset of X with the subspace topology we call Y a subspace. If P is a property of subsets of topological space, then a P subspace should be understood to mean a subspace that also satisfies P. Let {X i } i I be a collection of topological spaces, with index set I, then the product topology is the topology on Π i I X i given by the base {Π i I U i : U i is an open subset of X i and U i = X i for all but finitely many i I}. There is a another way to create a new topology based on old ones, that is the sum of spaces. Let {X i } i I be a collection of spaces such that X i X j = when i j, the sum of the spaces is the set X = i I X i where U is an open set in X if U X i is open in X i for all i I. The sum of the spaces {X i } i I is denoted i I X i or X 1... X n if I = {1,..., n}.

15 9 If X is a set linearly ordered set we define a bounded interval in X as (x, y) = {z X : x < z < y} for x, y X. Thus, we define the order topology on the set X with the base {(x, y) : x, y X, x < y}. Take note that the usual topology for Z and N is the discrete topology and that the topology on Q, Z, and N all have the subspace topology of R. Moreover, the usual topologies on N, Z, Q, and R are all ordered topologies using the usual ordering of the real numbers. Lemma If X is zero-dimensional, then every subspace is zero-dimensional. Proof. Since X is zero-dimensional it has a clopen base B X. Since B X is a base for X it is clear that the collection B Y = {Y O : O B X } is a base for the subspace topology on Y. Now, we need just to verify that B Y is a collection of clopen sets of Y. We already have that Y O is open in Y for all O B X by the definition of a subspace. So we need to show that Y O is a closed subset of Y for all O B X. Since all O B X are closed subsets of X we have that Y O are closed subsets of Y, thus Y O is clopen in Y, for all O B X. Hence, it follows that B Y is a clopen base of Y. Therefore Y is zero-dimensional. Let (X, T ) and (Y, S) be topological spaces. A function f : X Y is continuous at the point x in X if for every V S with f(x) V, there is a U T such that x U and f(u) V. A continuous function is a map f : X Y such that f 1 (O) T for all O S. For example the constant functions are always continuous. A map is said to be bicontinuous if it is continuous with a continuous inverse map. An open map is a function f from the space X into the space Y such that f(o) is an open subset of Y whenever O is an open subset of X. Likewise, a closed map is a function f from the space X into the space Y such that f(v ) is a closed subset of Y whenever V is a closed subset of X. The notion C(X, Y ) denotes the set of all continuous functions from the space X into the space Y. Moreover, when dealing with real-valued continuous functions we write C(X) instead of C(X, R). If X is a space and f C(X), f is said to be a bounded function if there exists M R such that f(x) M for all x X. The set of bounded real-valued

16 continuous functions on X is denoted C (X). Likewise, C (X, Z) is the set of all bounded integer-valued continuous functions. 10 Lemma Let X, Y, and Z be spaces with continuous maps f : X Y and g : Y Z. The composition map g f is also a continuous map. Proof. Let O be an open subset of Z. We want to show (g f) 1 (O) is an open subset of X. By the continuity if g we have that g 1 (O) is an open subset of Y. Then, by the continuity of f, f 1 (g 1 (O)) is an open subset of X. However, (g f0 1 = f 1 (g 1 (O)) giving g f be a continuous function. Let X and Y be spaces, a function f from X onto Y is a quotient map when f 1 (U) is open in X if and only if U is open in Y (refer to page 137 in [19]). A special map which is also continuous is the identity map on a space X is the map id X : X X defined by id X (x) = x, for all x X. Remark Let X be a discrete space and Y any space. Any map f : X Y is a continuous map since f 1 (U) is a subset of X, hence open in X, for any U Y. Thus, it follows that C(X, Y ) = Y X. A function between two topological spaces is called a homeomorphism if it is bijective and bicontinuous. If there is a homeomorphism between spaces the X and Y, then X and Y are said to be homeomorphic. Consider the following characterization. Proposition (Proposition , [11]). Let X and Y be spaces. For a continuous bijective map f : X Y the following are equivalent: (i) The map f is a homeomorphism. (ii) The map f is an open map. (iii) The map f is a closed map.

17 Now for some facts about restrictions of continuous functions. 11 Proposition (Theorem 18.2, [19]). Let X and Y be spaces. (i) If f : X Y is continuous and A is a subspace of X, then f A : A Y is also continuous. (ii) The map f : X Y is continuous on X if X can be written as the union of open sets {U i } i I such that f Ui is continuous for all i I. Lemma (Theorem 18.3, [19]). (Pasting Lemma) Let X and Y be spaces. If V 1 and V 2 are closed subsets of X and f 1 : V 1 Y and f 2 : V 2 Y are continuous maps such that f 1 (x) = f 2 (x) when x V 1 V 2, then the map f : V 1 V 2 Y defined by f 1 (x), if x V 1 f(x) = f 2 (x), if x V 2 is a continuous map. We have the following characterization for clopen subsets. Proposition Let X be a space and U a subset of X. The set U is a clopen subset of X if and only if χ U is a continuous map. Proof. First, we consider {0, 1} equipped with the discrete topology. ( ) Suppose that U is a clopen subset of X. It follows that the functions f 1 : U {0, 1} given by f 1 (x) = 1 and f 2 : X\U {0, 1} given by f 2 (x) = 0 are both continuous since they are constant functions. Thus, it follows from the Pasting Lemma that χ U is continuous since f 1 (x), χ U (x) = f 2 (x), if x U if x X\U.

18 12 ( ) Suppose that χ U is a continuous map. We have that {0} and {1} are both open in {0, 1}, thus we have that χ 1 U ({1}) = U and χ 1 U ({0}) = X\U are both open subsets of X. Hence, we have that U is a clopen subset of X. A useful collection of function when dealing with the product topology are the projection functions. Let {X i } i I be a collection of spaces and Π i I X i the product of the spaces. The j th projection map is π j : Π i I X i X j such that π j ((x i ) i I ) = x j, for all j I. Notice that π j is surjective for each j I. Also notice that π j is continuous for all j I and that it is an open mapping. We then have the following. Proposition (Theorem 19.6, [19]). Let {X i } i I be a collection of spaces with index set I and Y a space with f j : Y X j for all j I, and f : Y Π i I X i by f(y) = (f i (y)) i I. f is continuous if and only if f j is continuous for all j I. A subset Y of a space X is connected if there are no disjoint open subsets U 1 and U 2 of X such that Y = U 1 U 2, Y is disconnected if such open subsets exist. Proposition (Theorem 23.5, [19]). The continuous image of a connected set is connected. Let X be a space, an open cover of S, a subset of X, is a collection {U i } i I of open subsets of X such that S i I U i. An open subcover is a subcollection of an open cover which is also an open cover. We say a space X is a compact space if for every open cover there is a finite open subcover. A space is countably compact if every countable open cover has a finite subcover. A space is called a Lindelöf space if for every open cover there is a countable open subcover. A space X is called locally compact if for every x X there exists an open subset U of X containing x such that cl X (U) is a compact subspace of X. There are different ways that spaces can be separated; we will list a few typical spaces that will be used and introduce additional spaces as need. A space X is a Hausdorff space if

19 13 for distinct points x 1 and x 2 in X there are open subsets O 1 and O 2 of X such that x 1 O 1 and x 2 O 2 with O 1 O 2 =. Example Let X = {a, b, c}, and let T = {, {a}, X}. The topological space (X, T ) is a non-hausdorff space. Remark A few remarks about Hausdorff spaces. (i) Each of the spaces R, Q, Z, and N is a Hausdorff space under the usual topology. (ii) A finite Hausdorff space is discrete. (iii) For each x X, the subset {x} is a closed subset of X. From this point on we assume that all spaces are Hausdorff. Another class of spaces which are used in the study of C(X) are the Tychonoff spaces. A Tychonoff space is a Hausdorff space X that has the property that for every closed set V of X and a point x X\V, there exists a f C(X) such that f(v ) = {0} and f(x) = 1. We will then notice the following propositions. The second of these tells us the relationship between zero-dimensional space and Tychonoff spaces. Proposition Every zero-dimensional space is a Tychonoff space. Proof. Let X be a zero-dimensional space. Take V to be a closed set of X and let x X\V. Since V is closed we have that X\V is open, thus there exists a clopen subset U of X such that x U X\V. So the map χ X\U has the properties that χ X\U (V ) = {1}, χ X\V (x) = 0, and this is a continuous real-valued function. A G δ -set is a set which is a countable intersection of open sets. A P -space is a space where every G δ -set is open. Lemma Let X be a space. X is a P -space if every countable union of closed sets is closed.

20 Proof. Let {O n } n N be a countable collection of open subsets of X. Set V n = x\o n for all n N. Thus, n N V n is a closed subset of X. However, we have that following: 14 O n = n ) = X\ n N n N(X\V V n. n N Therefore, n N O n is an open subset of X implying that X is a P -space. A Tychonoff space X is called pseudocompact if C(X) = C (X). A Tychonoff space X is called Z-pseudocompact if C(X, Z) = C (X, Z). A space X is called locally compact if for every x in X there exists an open subset O of X containing x such that cl X (O) is a compact subspace of X. Example Here are a few examples of spaces with different compactness properties. (i) The interval [0, 1] with the subspace topology from R is a compact space. (ii) R with the usual topology is a Lindelöf space, which is not compact. (iii) Let X be a countable set and let p X. Define a topology on X such that the open subsets of X are X and all the subsets not containing p. It follows that this is a countably compact space that is not compact. Refer to page 47 in [21]. Remark Every compact space is a countably compact space. Also, the continuous image of a compact set is compact. Now for a result dealing with Lindelöf spaces and P -spaces. Lemma (Proposition 4.4.9, [2]). If X is a Lindelöf P -space, then so is X n for all n N. Let X be a space, a partition of X is a collection {U i } i I of subsets of X such that X = i I U i and U i U j = when i, j I and i j. A clopen partition of the space X is a partition {U i } i I where U i is a clopen subset of X for all i I.

21 15 We can construct a new topological space based on a a given one when we have an equivalence relation. Let X be a space, and let be an equivalence relation on the set X. We say that [x] = [y] if and only if x y, that is they are in the same equivalence class. We will use the notation X/ to represent the set of all equivalence classes of X under the relation and define the function q : X X/ defined by q(x) = [x], the equivalence map of x. We define the quotient topology on X/ to be the family of all subset U of X/ such that q 1 (U) is open in X. It is clear by the definition of the quotient topology that q is a continuous map. Let X be a set. We call ρ a metric if ρ is a function from X X to [0, ) which satisfies the following properties for all x, y, z X: (i) ρ(x, y) = 0 if and only if x = y, (ii) ρ(x, y) = ρ(y, x), and (iii) ρ(x, z) ρ(x, y) + ρ(y, z). If X is a set and ρ is a metric on X we say that (X, ρ) is a metric space with the topology generated by basic open subsets of the form B r (x) = {y X : ρ(x, y) < r}, where x X and r > 0. A topological space is metrizable if there is a metric which generates the topology (refer to page 47 in [5]). The class of zero-dimensional spaces are going to be used extensively through out the following sections, thus there are several properties about zero-dimensional that will be needed for later use. Firstly, the product of zero-dimensional spaces is a zero-dimensional space. Proposition If X is zero-dimensional space and I is an index set, then X I is zero-dimensional space. Proof. Since X is zero-dimensional space, X has a base B of clopen sets. We want to show

22 that B I, where 16 B I = {Π i I U i : U i B for i F and U i = X for i I\F, where F is finite}, forms a clopen base for X I. Clearly all the members of B I are clopen in X I. Now let x = (x i ) i I X I and let U be an open subset of X I containing x. Since U is open in X I there is a finite subset F of I such that U = Π i I U i where U i is open in X, for i F, and U i = X for i I\F with x i U i, for all i I. Since each U i is open in X, for i F, there is a O i B such that x i O i U i, for all i F. Thus, x Π i I E i U, where E i = O i for i F and O i = X for i I\F. Therefore, X I has a clopen base, namely B I. Next we a result about a clopen subbase. Lemma If X is a space with a subbase of clopen subsets of X, then X is zerodimensional. Proof. Suppose that X is a space with a subbase S of clopen sets. Since S is a subbase we then have that B = n N{U 1... U n : U i S, i = 1,..., n} is a base for X. Let U 1,..., U n S for n N. Notice that U 1... U n is a clopen subset of X since U i is clopen for each i = 1,..., n. Thus, X is zero-dimensional as it has a clopen base. We also have the following result that will be used repeatedly without mention. Proposition If X is a zero-dimensional Hausdorff space with x 1,..., x n X are distinct, then there exists V 1,..., V n basic clopen subsets of X, such that x i V i, for i = 1,..., n, with V i V j = whenever i j. Proof. Let x 1,..., x n X be distinct. Since X is Hausdorff there exists disjoint open subsets O i,j and O i,j of X such that x i O i,j and x j O i,j, for i j. Set U i = j i O i,j, for all

23 17 i = 1,..., n, we see that U i is an open subset of X such that x i U i, for all i, and x j / U i, for j i. Let V i be a basic clopen subset of X such that x i V i U i, for all i. Therefore, we have that x j / V i, for i j, and n i=1v i =. 1.3 Groups, Rings, and Modules A binary operation on a set S is a function φ : S S S. A group is a set G with a binary operation which satisfies for all a, b, c G (i) (a b) c = a (b c) (associativity), (ii) there is an element e G G such that e G a = a e G = a (e G is called the identity of G), and (iii) there exists a a 1 G such that a a 1 = a 1 a = e G. A group G with binary operation is called Abelian if it also satisfies a b = b a, for all a, b G. The notation for a group is (G, ), if the group is an Abelian group we use the notation (G, +); we will use a to mean the inverse of a in an Abelian group and b a to mean b + ( a). A subset of a group which is also a group is called a subgroup under the given operation. A ring is an Abelian group (A, +) with a second binary operation which satisfies for all a, b, c A (i) a (b c) = (a b) c, (ii) a (b + c) = a b + a c, and (iii) (a + b) c = a c + b c. The notation for a ring is (A, +, ). The identity element of the Abelian group (A, +) will be denoted as 0 A, i.e. the additive identity. If there is an a A such that a b = b a = b for

24 18 all b A, such an element if called the multiplicative identity and will be denoted by 1 A. A ring A is commutative if for all a, b A, a b = b a. An element a of a ring A with identity is called a unit if there exist a b A so that a b = 1 A. A subring is a subset of a ring, with 1 A, which is itself a ring with the same binary operations. A field is a commutative ring where every element is a unit. We will assume all rings are commutative with the identity element 1 A and 0 A 1 A. Remark The set Z is a commutative ring with 1 with respect to the usual addition and multiplication. The sets Q and R are both fields with respect to the usual addition and multiplication. However, N is not a ring under usual addition and multiplication as it is not a group under addition. Suppose A is a ring, a non-zero element a A is called a zero divisor if there is a non-zero element b A such that a b = 0 A. A ring A is called an integral domain if it has no zero divisors. Let A be a ring. An ideal of A is a non-empty subset I of A which has the following properties: (i) r a I for all r A and a I, and (ii) a b I for all a, b I. A maximal ideal of A is an ideal M of A such that if N is another ideal of A with M N, then M = N or N = A. A minimal ideal is an ideal I of A such that if J is another ideal of A with J I, then I = J or J = {0 A }. A prime ideal is an ideal I such that if a b I, then a I or b I. An A-module over a ring A is an Abelian group (M, +) with a map : A M M which satisfies for all a, b A and m, n M (i) a (m + n) = a m + a n,

25 (ii) (a + b) m = a n + b m, 19 (iii) (ab) m = a (b m), and (iv) 1 A m = m. Let M be an A-module. If m 1,..., m k M and a 1,..., a k A, then the sum k i=1 a im i is an A-linear combination. A subset E of an A-module M is called linearly independent whenever 0 M = a 1 m a k m k, for a i A and m i E, then a 1 =... = a k = 0. A subset E of an A-module M is an A-basis if each element of M can be expressed as a linear combination of elements of E and E is linearly independent. A homomorphism is a function between groups (rings, fields, modules) which preserves the operation of the groups (rings, fields, modules), i.e. if f is a group homomorphism between (G,, 0 G ) and (H,, 0 H ), then f(a b) = f(a) f(b). If there is a homomorphism between two groups (rings, fields, modules), then the groups (rings, fields, modules) are said to be homomorphic. An isomorphism is a bijective homomorphism. If there is an isomorphism between two groups (rings, fields, modules) G and H, then G is isomorphic to H and denote thus by G = H. Let f be a group (ring, field) homomorphism from G to H. The kernel of f is the set ker(f) = {g G : f(g) = e G } and the image of f is the set im(f) = {f(g) : g G}. Take note that if f is a homomorphism on ring A, then ker(f) is an ideal of A. We can construct new rings from given ones; the first of these constructions is called the quotient ring. Let A be a ring and I an ideal of A, then we call A/I the quotient ring with elements of the form a + I for a A, where a + I = {a + i : i I}. The quotient ring has the operations (a + I) + (b + I) = (a + b) + I and (a + I) (b + I) = a b + I

26 20 for all a, b A. Another way to create a new ring is the direct product. Let {A i } i I be a collection of rings. The direct product of rings is the set Π i I A i together with componentwise addition and multiplication. The following theorem gives the relationship between the kernel and image of a ring homomorphism. Theorem (Theorem 7.3.7, [10]). (First Isomorphism Theorem for Rings) Suppose that A and B are rings and φ is a homomorphism from A to B, then A/ker(φ) = im(φ). We can classify certain ideals based on the characteristics of their quotient ring. The first classifiers a maximal ideal. Proposition (Proposition , [10]). Let A be a ring and M an ideal of A. M is a maximal ideal if and only if A/M is a field. We also have a ways of classifying a prime ideal. Proposition (Proposition , [10]). Let A be a ring and P an ideal of A. P is a prime ideal if and only if A/P is a integral domain. Since groups, rings, fields, and modules are based on a set, together with some operations, a topology may be given to the sets which form these algebraic structures. A topological group is a group (G, ) that is also a topological space such that : G G G and 1 : G G are continuous functions. Likewise, a topological ring is a ring (A, +, ) that is also a topological space such that + : A A A, 1 : g g, and : A A A are continuous maps. Note that a topological ring isomorphism is a bicontinuous ring isomorphism of topological rings. A topological module is an A-module M on a topological ring A and (M, +) is a topological group such that : A M M is a continuous map. A linear homeomorphism is a homomorphism between two topological modules such that it is a homeomorphism. Two topological modules are linearly homeomorphic if there is a linear homeomorphism between the topological modules.

27 Remark Clearly Z, Q, and R are all topological rings under the usual addition and multiplication. 21 If A is a topological ring and I is an ideal of A we will give the ideal I the subspace topology induced by the topology of A; it is clear that I is also a topological ring. Moreover, if A is a topological ring and I is an ideal of A, then the ring A/I will be given the quotient topology, with the map q : A A/I given by q(r) = r + I. Proposition (Problem 2.5(d), [19]). If A is a topological ring and I is an ideal of A, then the quotient ring A/I with the quotient topology is a topological ring. From this point onward we will assume that ideals of topological rings have the subspace topology and that quotient rings have the induced quotient topology. We also assume that our topological rings are Hausdorff. Since we now have an algebraic structure which has a topology on it we can talk about continuous functions into the algebraic structure. Let (A, +, ) be a topological ring; we can consider C(X, A) for a space X. Since A is a ring we see that C(X, A) is also a ring with the pointwise addition and multiplication of the ring, that is (i) (f + g)(x) = f(x) + g(x) and (ii) (f g)(x) = f(x) g(x) for all f, g C(X, A) and for all x X. Thus, with this in mind we can start to talk about the algebraic structure of C(X, A). Since we are dealing with rings we can consider certain subsets of the space X. For all topological rings A and f C(X, A), we define z(f) = {x X : f(x) = 0 A } to be the zero set of f. A cozero set, denoted coz(f), as that is coz(f) = X\z(f) for all f C(X, A). Notice that z(f) is a closed subset of X and coz(f) is an open subset of X for all f C(X, A) since singleton subsets are closed in a Hausdorff space.

28 1.4 Cardinality 22 We assume the Axiom of Choice. The cardinality of a set, loosely, indicates the size of the set. We will express the cardinality of a set S as S. If S and T are sets with a bijective function f : S T, we say that S = T. If there is a bijection between the set S and the set {1,..., n}, then S = n, where n N. The cardinality of N is denoted ℵ 0. If a set has cardinality ℵ 0 it is called countably infinite. The cardinality of the first infinite uncountable set is denoted ℵ 1. Remark The sets Z and Q both have cardinality ℵ 0, that is, they are both countably infinite. Let S be a set with cardinality τ. If κ τ, then we define [S] <κ = {T P(S) : T < κ}. Thus, it follows that [S] <ℵ 0 is the collection of all finite subsets of S. It is possible to add and multiply cardinalities. Suppose that τ and γ are infinite cardinals, without loss of generality, assume τ < γ. Define (i) γ + τ = τ + γ = γ and (ii) γτ = τγ = γ. Let S and T be two disjoint sets where S = τ and T = γ, the set S T has cardinality γ + τ. Also, if S and T where S = τ and T = γ, the cardinality of S T is γτ. Refer to [15] for more on the addition and multiplication of cardinals. Define a cardinal invariant as a function F which assigns to each space X a cardinal F (X) such that F (X) = F (Y ) whenever X and Y are homeomorphic spaces. We now define several cardinal invariants. Our base references for cardinal invariants are [4] and [11]. Definition The Lindelöf number of X, denoted l(x), is defined to be the smallest infinite cardinal τ such that any open cover of X contains a subcover with cardinality less than or equal to τ.

29 It is obvious from the definition that l(x) = ℵ 0 if X and only if X is a Lindelöf space. The following lemma will be useful to us later. 23 Lemma If X is a space and Y is a closed subspace of X, then l(y ) l(x). Proof. Suppose that l(x) τ. Let η Y be a open cover of Y. Since Y is closed we have that X\Y is open in X. It follows that η X = η {X\Y } is a cover for X. Since η X is an open cover for X there is a subcover η X of η X for X such that η X τ. Let η Y = {U U Y, where U η X }. Clearly ηy η Y and ηy τ. Thus, l(y ) l(x). Definition The tightness of X denoted t(x) is the smallest cardinality τ such that for any set A X and any point x cl X (A) there is a set B A for which B τ and x cl X (B). Proposition If X is a space and Y is a closed subspace of X, then t(y ) t(x). Proof. Assume t(x) = τ. Let A be a subset of Y and y cl Y (A). First, notice that A Y = A and cl X (A Y ) = cl Y (A) with y cl X (A Y ), since Y is a closed subspace of X. Now, since t(x) = τ there is a B A = A Y with B τ and y cl X (B). Further note cl X (B) = cl Y (B) is closed subset of X and Y, as Y is a closed subspace of X. Thus, we have that cl X (B) is closed in Y. Therefore, B A with B τ with y cl Y (B) and consequently t(y ) τ. Let us now define the character and weight of a space. For a space X and x in X, the character of a point x in X, denoted χ(x, X), is the least infinite cardinal number of a base of neighborhoods around x. The character of a space X, denoted χ(x), is the least infinite cardinal number τ such that χ(x, X) τ for all x X. The weight of a space X, denoted w(x), is the infinite cardinality of a minimal base for a space Remark If X is a space and Y a subspace of X, then w(y ) w(x).

30 24 A space X is said to satisfy the first axiom of countability, or is first countable, if χ(x, X) ℵ 0 for all x X. Also, a space X is said to satisfy the second axiom of countability, or is second countable, if w(x) ℵ 0. A space X is called separable if there is a subset A with A = ℵ 0 such that cl X (A) = X. There is a more general concept of separable, the density of a space X is minimal infinite cardinality of the subset A with cl X (A) = X, we denote the density of a space X by d(x). Clearly, if X is a separable space, then d(x) = ℵ 0. A network in a space X is a collection S of subsets of the set X such that any point x X and any open set U containing x there is a P S such that x P U. There is a second type of weight which is considered, this is the concept of network weight for a space X, denoted nw(x), this is the minimal cardinality of the networks of X. Lemma If X is a space, then d(x) nw(x). Proof. Suppose that nw(x) = τ. Let S be a network of X such that S = τ and let S be indexed by the set I. Thus, it follows that I = τ. Take x i P i for all i I where P i S and let A = {x i } i I. Suppose, by way of contradiction, cl X (A) X. Take x X\cl X (A), notice that X\cl X (A) is an open set in X. Since x X and X\cl X (A) is open there is a P j S such that x P j X\cl X (A). Thus, x j X\cl X (A) and x j A by definition, giving a contradiction. Hence, cl X (A) = X, so d(x) A = τ. Therefore, it follows that d(x) nw(x). We also have the concepts of τ-monolithic, a space X is said to be τ-monolithic if nw(cl X (A)) τ for every A X such that A τ. In particular, a space X is ℵ 0 - monolithic if the closure of every countable subset in X is a space with a countable network weight. A space is call monolithic if it is a τ-monolithic for every cardinal τ, that is for every Y X we have d(y ) = nw(y ).

31 25 CHAPTER 2 Properties of C(X, Y ) and C p (X, Y ) 2.1 Topological Properties of C p (X, Y ) and C p (X) In this section we are going to provide a number of general properties for the collection of continuous functions with the topology of pointwise convergence. We will also state some known results for C(X) which we will prove analogous results to in our study of C(X, Z) and C p (X, Z). Let X and Y be topological spaces. Define C p (X, Y ) as the set C(X, Y ) endowed with the topology of pointwise convergence, also called the pointwise topology. A basic open subset of C p (X, Y ) has the form W (x 1,..., x n, U 1,..., U n ) = {f C(X, Y ) : f(x i ) U i, i = 1,..., n}, where x 1,..., x n X and U 1,..., U n are open subsets of Y, and n N. We will state some general properties for C p (X, Y ), first of which deals with the base for C p (X, Y ). Proposition Let X and Y be spaces and let B be a base for the space Y. The collection {W (x 1,..., x n, U 1,..., U n ) : x i X, U i B, i = 1,..., n, n N}

32 is a base for the space C p (X, Y ). 26 Proof. Let f W (x 1,..., x n, U 1,..., U n ), where where x 1,..., x n X and U 1..., U n are open subsets of Y. For each i = 1,..., n there is a basic open set O i B such that f O i U i. Consequently, f W (x 1,..., x n, O 1,..., O n ). Notice W (x 1,..., x n, O 1,..., O n ) W (x 1,..., x 1, U 1,..., U N ). Therefore, the collection {W (x 1,..., x n, U 1,..., U n ) : x i X, U i B, i = 1,..., n, n N} is a base for the space C p (X, Y ). The next property relates subspaces of Y to certain subspaces of C p (X, Y ). Proposition If X and Y are spaces and Z is a subspace of Y, then C p (X, Z) is a subspace of C p (X, Y ). Furthermore, if Z is a closed subspace of Y, then C p (X, Z) is a closed of C p (X, Y ). Proof. Let f C(X, Z); it is clear that f C(X, Y ). Observe that for x 1..., x n X and U 1,..., U n open subsets of Z we have, with a straightforward argument, that W (x 1,..., x n, U 1,..., U n ) = C p (X, Z) W (x 1,..., x n, V 1,..., V n ) where V i is an open subset of Y with V i Z = U i. Now, assume that Z is a closed subset of Y. Let f C p (X, Y )\C p (X, Z). There is a x X such that f(x) Y \Z. Then f W (x, Y \Z). Notice there for g W (x, Y \Z) we have g / C p (X, Z). Hence, C p (X, Y )\C p (X, Z) is an open subset of C p (X, Y ), so C p (X, Z) is a closed subspace.

33 27 Let X, Y, and Z be spaces and f : X Y be a continuous map, denote the composition map by f : C(Y, Z) C(X, Z) and given by f (ϕ) = ϕ f, for all ϕ C(Y, Z). We will now state a useful fact about the composition map f from C p (Y, Z) into C p (X, Z), for a fixed f C(X, Y ); more properties of the composition map will be stated later. Proposition If X, Y, and Z are spaces and f : X Y is a continuous map, then f is a continuous map from C p (Y, Z) to C p (X, Z). Proof. Let ϕ C p (Y, Z) and f (ϕ) = ψ. Consider a basic open set W (x 1,..., x n, U 1,..., U n ) of C p (X, Z) containing ψ. Then ϕ(f(x i )) = (ϕ f)(x i ) = f (ϕ)(x i ) = ψ(x i ). It is clear that we have (ϕ f)(x i ) = ψ(x i ) U i for all i = 1,..., n. With some rewriting we see that ϕ(f(x i )) U i. Thus ϕ W (f(x 1 ),..., f(x n ), U 1,..., U n ), where W (f(x 1 ),..., f(x n ), U 1,..., U n ) is a basic open subset of C p (Y, Z). Let g W (f(x 1 ),..., f(x n ), U 1,..., U n ), it follows f (g)(x i ) = (g f)(x i ) U i for all i = 1,..., n. Providing f (g) W (x 1,..., x n, U 1,..., U n ). Hence f (W (f(x 1 ),..., f(x n ), U 1,..., U n )) W (x 1,..., x n, U 1,..., U n ). Consequently, f is continuous at the point φ. Since φ is arbitrary, the inverse image of an open subset is open. Hence, f is a continuous map.

34 28 We now recall two theorems involving the pointwise topology, the product space, and the topological sum. Theorem (Proposition , [11]). If X and Y are spaces and I is an index set, ( then C ) p X, Y I is homeomorphic to C p (X, Y ) I. Theorem (Proposition 2.6.9, [11]). If X and Y are spaces and X is a topological sum, say X = i I X i, then C p (X, Y ) is homeomorphic to Π i I C p (X i, Y ). We will now consider the case when the codomain of the continuous functions, on a space X, is a topological ring. Let A be a topological ring, we already know that C(X, A) is a ring, a natural consideration is whether or not C p (X, A) is a topological ring when given the pointwise topology. Theorem (Remark , [6]). Let X be a space, and let A be a topological ring. The direct production x X A of A is a topological ring when endowed with the product topology. Corollary Let X be a space and A a topological ring. The space C p (X, A) is a topological ring. Proof. By Theorem we have that x X A is a topological ring. Recall that C (X, A) is a subspace of x X A with the same operations. Thus, it follows that C p(x, A) is a topological ring since continuous functions restricted to subspaces are continuous. Now we will consider some facts about C p (X). Notice that C p (X) is a specific case of C p (X, Y ). Since we know that R has a base for the usual topology, we define a basic open subsets of C p (X) by a basic open subsets of R, this follows from Proposition Let f C(X), let x 1,..., x n X, and let 0 < ε. Define a basic open subset of C p (X) as W (f, x 1,..., x n, ε) = {g C(X) : f(x i ) g(x i ) < ε, i = 1,..., n} and {W (f, x 1,..., x n, ε) f C(X), x 1,..., x n X, ε (0, )} n N