The Lattice of Compactifications of a Locally Compact Space By Luke A. Eichelberger Submitted to the Department of Mathematics and the Graduate Faculty of the University of Kansas in partial fulfillment of the requirements for the degree of Master of Arts Jack Porter, Chairperson Committee members Estela Gavosto Daniel Katz Date defended: 5-13-2016 The Master’s Thesis Committee for Luke A. Eichelberger certifies that this is the approved version of the following master’s thesis : The Lattice of Compactifications of a Locally Compact Space Jack Porter, Chairperson Date approved: 5-13-2016 ii Abstract This is an expanded version of [5] by Magill. The results of [5] are proven with greater detail and any result stated in [5] but not proven is proven here. Let K (X) and K (Y) be used to indicate the lattice of Hausdorff compactifications of locally compact, non-compact spaces X and Y with X and Y Tychonoff. This paper primarily concerns how a lattice isomorphism between K (X) and K (Y) exists if and only if a homeomorphism between particular extensions of X and Y exists with specified properties. On the way to proving the main results, we prove several lemmas about b − f amilies of compact extensions of Tychonoff spaces. Some of the Lemmas slightly generalize corresponding lemmas in [5]. Efforts are made to make this paper self- contained. iii Acknowledgements I would like to thank my advisor Professor Jack Porter for showing me how interesting topology can be, and for eagerly agreeing to work with me when I wanted to deepen my knowledge of the subject. His abundance of help, guidance, and support made working on this project feasible and rewarding. I would also like to thank the members of my committee: Professor Estela Gavosto, who reminds me that mathematics is about ideas, both big and small, and who has been there for me when I needed extra support; and Professor Daniel Katz, who goes out of his way to help students who seek additional knowledge, and whose high standards have encouraged me in grasping concepts that initially seemed beyond reach. I wouldn’t be where I am without the support of family and friends. The encourage- ment of Avary Kolasinski, Nick Ma, and John Reynolds helped me to push forward when my doubts were too large to ignore. My family is a source of endless support, and reminds me there is more to life than mathematics. Finally, I would like to thank Lori Springs, who cares deeply for graduate students, and makes sure to take care of us. iv Contents 1 Introduction 1 2 Preliminaries 3 2.1 Basic Concepts . .3 2.2 Compactifications and Extensions . .4 3 Lemmas 9 3.1 b − f amilies .....................................9 3.2 The Lattice of Compactifications . 10 3.3 Comparing Compactifications . 12 3.4 Constructing Compactifications . 15 3.5 Dual Points . 22 4 Main Results 27 4.1 Lattice Isomorphism and Homeomorphism . 27 4.2 Lattice Structure and The Automorphism Group . 39 5 Appendix 43 5.1 Function Results . 43 5.2 Lattice Theory . 44 v Chapter 1 Introduction Hausdorff Compatifications have been studied since the 1920s, if not earlier, when the founda- tions were established through papers of Alexandroff, Cartan, Cech,ˇ Lubben, Stone, Tychonoff, Urysohn, and Wallman between 1929 and 1941 [1]. It was through Cechˇ [1937] that the Stone- Cechˇ compactification used in the present paper was established. While competing notions of compactness were considered in the early 1900s, including what we now call countable compact- ness and sequential compactness, the modern definition seems to have been accepted by the 1940s. Some impetus for this was Tychonoff’s Theorem (published in 1937) that the arbitrary product of compact spaces is compact. This property is not shared by countably compact or sequentially compact spaces. That the modern definition of compactness was “correct" is underscored by a va- riety of theorems that hold for compact spaces but not spaces in general, such as the Extreme Value Theorem (that every continuous function on a compact space is bounded and attains a maximum value). In order to prove the results in this paper, we must assume a space is Hausdorff and Completely Regular. Let K (X) indicate the collection of all Hausdorff compactifications of a Tychonoff space X. We will say that Hausdorff compactifications aX and gX are equivalent (and identified as the same) if there exists a homeomorphism h : aX ! gX such that h(x) = x for all x 2 X. It is known (but we will show) that K (X) is not only a collection but a set. We can then define a relation ≤ 1 on K (X) by a1X ≤ a2X iff there exists a continuous onto function from a2X to a1X that fixes X. This relation is a partial order. In addition, (K (X);≤) is a complete lattice if and only if X is locally compact. Now consider a lattice isomorphism G between K (X) and K (Y) where X and Y are locally compact but not compact. This paper investigates the relationship between G and a homeomor- phism between bXnX and bYnY where bX indicates the Stone - Cechˇ compactification of X. In addition, two corollaries are proven concerning the automorphism group A (K (X)) of the lattice K (X). The precise statements of these results are given in 4.1 through 4.6. We now discuss the material from general topology used within the paper. 2 Chapter 2 Preliminaries 2.1 Basic Concepts We begin by defining the separation axioms. Most separation axioms are, as the name suggests, axioms that allow us to enclose points or sets by disjoint sets that are typically open or closed. That is, we can separate points or sets by other sets. Note: The word space will be used to mean topological space in all that follows. Definition 2.1.1. (a) A space X is called T1 if for distinct x;y 2 X, x 2= clfyg and y 2= clfxg (b) A space X is called Hausdorff if for any two points x;y 2 X, there exist disjoint open sets U;V 2 t(X) such that x 2 U and y 2 V. (c) A space X is called completely regular if for each closed set A and p 2 XnA, there is a continuous function f : X ! R such that f (p) = 1 and f [A] ⊆ f0g. A completely regular T1 space is called Tychonoff. (d) A space X is called normal if for any pair of disjoint closed sets A and B, there exist disjoint open sets U ⊇ A and V ⊇ B. We now define other terms from topology that are used freely in this paper. In addition, we prove some basic results involving these terms. The results will be cited when used. 3 Definition 2.1.2. Let X be a space and A ⊆ X with x 2 X. (a) The point x is called a closure point of A if U \ A 6= /0for each open set U such that x 2 U. (b) The set of all closure points of A is denoted by clA or clX A to emphasize the underlying space. Definition 2.1.3. (a) Let X be a space and x 2 X. A subset N of X is a neighborhood (nhb) of x if there is an open set U such that x 2 U ⊆ N. (b) A family Vx is a neighborhood base of x if Vx ⊆ Nx and for each N 2 Nx, there is V 2 Vx such that V ⊆ N. Definition 2.1.4. Let X be a space, Y a set, and g : X ! Y a function. We say that Y has the quotient topology induced by g if U is open in Y if and only if g [U] is open in X. It is easily seen with complements that Y has the quotient topology induced by g if and only if A is closed in Y if and only if g [A] is closed in X. Definition 2.1.5. Let X be a space. A collection A of subsets of X is a cover of X if X = SfA : A 2 A g. An open cover is a cover consisting of open sets. Definition 2.1.6. A space X is said to be compact if every open cover has a finite subcover. That is, Sn if A is an open cover of X, there exists a subcollection fU1;U2;:::;Ung ⊆ A such that X ⊆ i=1 Ui Definition 2.1.7. A space X is said to be locally compact if every point of X has a neighborhood base of compact closed neighborhoods. Definition 2.1.8. A space X is said to be connected if X is not the union of two nonempty disjoint open sets. A subspace A ⊆ X is said to be connected if A is connected as a subspace of X. 2.2 Compactifications and Extensions Definition 2.2.1. A space Y is an extension of a space X if X is a dense subspace of Y (Recall that X is dense in Y if clY (X) = Y). Any extension which is Hausdorff is called a Hausdorff extension. Definition 2.2.2. Let a1X and a2X be two extensions of a space X. We say that a1X is isomorphic to a2X if there is a homeomorphism f : a1X ! a2X such that f (x) = x for all x 2 X. 4 Definition 2.2.3. A compact extension Y of a space X is called a compactification of X.