A Uniformities and Uniform Completion A uniform space is a set X together with a structure called a uniformity defined on it. This structure can be defined in three different but equivalent ways, each with its own particular advantages.1 The first two of the following originate in the seminal work of Weil [109]; the third was defined by Tukey [108].2 1. In terms of a family of pseudometrics on X. This definition is used by Gillman and Jerison in their monograph on Rings of Continuous Functions [43], “as they [pseudometrics] provide us with a large supply of continuous functions”. 2. As a filter on X × X that satisfies certain conditions, one of them being that each member of the filter contain the diagonal Δ = {(x, x)|x ∈ X}. This definition was given in its present form by Bourbaki [12]. 3. In terms of certain covers of X called uniform covers,inwhichX is covered by “sets of the same size”. This definition was given by Tukey [108], and is used extensively by Isbell in his monograph [53] on what, according to its author, “might be labelled fairly accurately [as the] intrinsic geometry of uniform spaces”. All three definitions are given below. It should be pointed out that there are slight differences in terminology in the literature, the most important 1 Willard [130] calls the second and third of these diagonal and covering unifor- mities, respectively, and we shall follow his terminology. In the same spirit, we shall call the first pseudometric uniformities. It should be borne in mind that the different names refer, not to different mathematical objects, but to different definitions of the same object. 2 Uniformities are treated in many standard textbooks on topology. All three defi- nitions may be found, either in the text or in the exercises, in the books by Kelley [57] and Willard [130]. The specialized monograph by Gillman and Jerison uses only pseudometric uniformities. The elementary textbook by James [55] discusses only diagonal uniformities. H.-J. Borchers and R.N. Sen: Mathematical Implications of Einstein–Weyl Causality, Lect. Notes Phys. 709, 157–167 (2006) DOI 10.1007/3-540-37681-X A c Springer-Verlag Berlin Heidelberg 2006 158 A Uniformities and Uniform Completion being that a few authors (cf. [53]) assume the Hausdorff property to be an integral part of the definition of a uniformity. A.1 Equivalent Definitions of Uniformities A.1.1 Pseudometric Uniformities We begin by recalling the notion of a pseudometric. Definition A.1.1 A pseudometric on a set X is a function d : X × X → R that satisfies the following conditions for all x, y, z ∈ X: 1. d(x, y) ≥ 0; 2. d(x, x)=0; 3. d(x, y)=d(y, x); and 4. d(x, z) ≤ d(x, y)+d(y, z). A pseudometric differs from a metric only in that d(x, y) = 0 need not imply x = y. Example A.1.2 Let f : X → R be a real-valued function on X. The function d(x, y)=|f(x) − f(y)| is a pseudometric on X. Note that f does not have to be continuous. Notations A.1.3 If A is a nonempty subset of X,thed-diameter of A is defined to be d{A} =supd(x, y) . x,y∈A Definition A.1.4 A pseudometric uniformity on X is defined to be a non- empty family G of pseudometrics on X satisfying the following conditions: a) If d1,d2 ∈G then d1 ∨ d2 ∈G,where d1 ∨ d2 =sup(d1,d2). b) If d is a pseudometric and δ>0(δ is otherwise arbitrary), and there exists a d ∈Gand δ > 0 such that d(x, y) <δ ⇒ d(x, y) <δ,then d ∈G. Condition b) may be expressed as follows: d{A} <δ ⇒ d{A} <δ. A pseudometric uniformity G is called Hausdorff if for x = y there exists a d ∈Gfor which d(x, y) =0. A.1 Equivalent Definitions of Uniformities 159 A.1.2 Diagonal Uniformities In the following, the notations E−1 and F ◦F are to be understood as relations on X. Recall that if E = {(x, y)} is a relation on X, i.e., a subset of X × X, then the inverse relation E−1 is defined to be the subset {(y, x)} of X × X. If U and V are relations on X, then their composition U ◦ V is defined to be the set of all pairs (x, z) such that, for some y,(x, y) ∈ V and (y, z) ∈ U. Definition A.1.5 The diagonal of a set X ×X is the subset Δ = {(x, x)|x ∈ X}.Adiagonal uniformity on a set X is a filter E on X × X consisting of subsets of X × X called entourages or surroundings such that: a) If E ∈E then Δ ⊂ E. b) If E ∈E then there exists an entourage F ∈E such that F ⊂ E−1. c) If E ∈E then there exists an entourage F ∈E such that F ◦ F ⊂ E. Note that E ∈E implies that E−1 ∈E. A diagonal uniformity E on X is called Hausdorff (or separated, or separating) iff E = Δ. E∈E Condition 3) of Definition A.1.5 has been described by Kelley [57] as “a vestigial form of the triangle inequality”. It imposes more structure than a topology on X.GivenanycoverofX, one can always define a topology by taking finite intersections and arbitrary unions. Take now a family of subsets of X × X each containing the diagonal. Add all finite intersections to the family, and define a filter by taking supersets. This filter may fail to satisfy condition 3) of Definition A.1.5. See [55] for an example. Examples A.1.6 1. On any set X, the family consisting of all supersets of the diagonal Δ = X × X defines a uniformity called the discrete unifor- mity. 2. On any set X, the family consisting of the single set X × X defines a uniformity called the trivial uniformity. A.1.3 Covering Uniformities We begin with a few preliminary definitions: Definition A.1.7 Let U and U be covers of X. U is said to refine U if every U in U is contained in some U in U, U ⊂ U for some U ∈U. 160 A Uniformities and Uniform Completion Definition A.1.8 If U is a cover of X and A ⊂ X, then the star of A with respect to U, written St (A, U), is the union of all members of U that intersect A: St (A, U)= {U ∈U|A ∩ U = ∅} . Definition A.1.9 Let U and V be covers of X. One says that U star-refines V (is a star-refinement of V ),iff,foreachU ∈U,thereissomeV ∈V such that St (U, U) ⊂ V . Definition A.1.10 A covering uniformity on a set X is a family μ of covers of X such that a) If V, W∈μ then there exists U∈μ that refines both V and W. b) If U refines V and U∈μ,thenV∈μ. c) Every element of μ has a star-refinement in μ. Definition A.1.11 A base for a covering uniformity μ on X is any subcol- lection μ of μ such that μ = {U|U covers X and U refines U for some U ∈ μ} . A covering uniformity μ on X is called Hausdorff if for any two distinct points x, y ∈ X,thereisacoverU∈μ such that no element of U contains both x and y. A.2 Equivalence Theorems The results given below establish that covering uniformities are equivalent to i) pseudometric uniformities, and ii) to diagonal uniformities. The equivalence of pseudometric and diagonal uniformities follows from these. Lemma A.2.1 Let μ be a covering uniformity for X. Then there exists a pseudometric ρ on X such that U = {Uρ(x, ) | x ∈ X} is a uniform cover (i.e., U ∈ μ) for each >0. Here Uρ(x, )={y | y ∈ X, ρ(x, y) >}. The family of pseudometrics {ρα | α ∈ A} that generates a covering uni- formity μ is called the gage G of the uniformity μ. The gage G of a uniformity μ has the properties a) and b) of Definition A.1.4. Conversely: A.3 The Uniform Topology 161 Theorem A.2.2 Any collection G of pseudometrics on X satisfying the con- ditions a) and b) of Definition A.1.4 is a gage for some covering uniformity μ on X. The correspondence between gages and uniform covers is one-to-one. To state the result that establishes the correspondence between diagonal and covering uniformities, we need a definition: Definition A.2.3 AbaseB for a diagonal uniformity E on X is a filter base for X × X that satisfies conditions a) – c) of Definition A.1.5. Theorem A.2.4 Let μ be a family of covers of X that satisfies the conditions a)–c) of Definition A.1.10.ForU∈μ,define DU = {U × U | U ∈U} and B = {DU |U ∈μ} . The collection B is a base for a diagonal uniformity E on X. The uniform covers of X are precisely the elements of μ. A.3 The Uniform Topology The three distinct but equivalent definitions of uniformities lead to three dis- tinct but equivalent ways of defining new concepts, and stating and proving new results. We shall not give proofs, and shall confine ourself to only one de- finition each of the concepts that will be introduced below.