SOME PROPERTIES OF QUASI-UNIFORM SPACES By R. STOLTENBERG [Received 17 September 1965] 1. Introduction In a recent publication Kelly (3) studied some of the properties of quasi- metric spaces. 1.1. DEFINITION. A quasi-pseudo-metric for a set X is a non-negative real-valued function d defined on X x X such that, for x, y, z in X, (a) d(x,x) = 0, (b) d(x,y) ^d{x,z) + d(z,y). d is a quasi-metric if, also, (c) d(x, y) = 0 if and only if x = y. Let d be a quasi-(pseudo-)metric on a set, and define d': X x X -> R by the equation d'(x,y) = d{y,x). d' is a quasi-(pseudo-)metric on X. d' and d are called conjugate quasi-(pseudo-)metrics. In this paper the pair (X, d) will be called a quasi-(pseudo-)metric space, and the triple (X, d, d') will be called a bi-quasi-(pseudo-)metric space. The quasi-(pseudo-)metric topology is the family of all sets T such that for each x in T there exists e > 0 with 8e{x) = {y: d(x, y) < e} c T. This topology will be denoted Kelly ((3) 75) defined a sequence {xn: n e N} to be d-Cauchy if and only if for every e > 0 there exists n0 such that d(xm, xn) < e for m ^ n ^ n0. The question as to whether or not (X, d) had a ^-completion was left open. It is the purpose of this paper to answer this question. More generally, the question of whether or not a quasi-uniform space has a ^-completion is answered. 1.2. DEFINITION. A quasi-uniformity for a set X is a collection tfl of subsets of X x X which satisfies the following axioms: (a) A c U for each U in <%\ (b) if U, V G <% then there exists U' in ^ such that U' £ U n V; (c) if U c V and U e <% then V eW; (d) if U e <% then there exists F in % such that Fo F c tf. Quasi-uniform spaces have been studied by Csaszar (1) and Pervin (4). If ^ is a quasi-uniformity for a set X then W = {U~x: U e %}, where XJ-1 = {(x, y): (y, x) e U}, is a quasi-uniformity for X. °tt and W are called conjugate quasi-uniformities. The pair (X,1^/) will be called a Proe. London Math. Soe. (3) 17 (1967) 226-40 SOME PROPERTIES OF QUASI-UNIFORM SPACES 227 quasi-uniform space, and the triple (X, <%,<%') will be called a bi-quasi- uniform space. The quasi-uniform topology generated from % is the family of all subsets T of X such that for each x in T there exists U in °ll with U[x] = {y: (x, y) e V) c: T. The quasi-uniform topology will be denoted by r^-. If (X,d) is a quasi-(pseudo-)metric space then 38 = {Udr: r > 0}, where Udr = {(x, y): d(x, y) < r}, is a base for a quasi-uniformity tfl for X such that T« = Td- ^ is a base for a quasi-uniformity °U if and only if for each UinW there exists B in 3d such that B <=,V. We shall denote the quasi-uniformity generated in this way from d by %d. 1.3. DEFINITIONS. If / is a function from a quasi-uniform space (X,tft) into a quasi-uniform space (7,y) then / is quasi-uniformly continuous relative to % and Y if and only if for each F in V the set If /: (X, °U) -» (F, y) is a one-to-one quasi-uniformly continuous function onto Y such that /^x is quasi-uniformly continuous, then / is called a quasi-uniform isomorphism, and (X, %) is said to be quasi- uniformly isomorphic to (Y,ir). A quasi-uniform space (X, °U) can be embedded in a quasi-uniform space {Y,y) if and only if there exists a quasi-uniform isomorphism from (X, W) onto a subspace of (7, Y). For/: (X,#) ~> (7,^), define /:XxX->7x7 by f(x,y) = (/( / is called the map induced by /. The following are equivalent: (a) /: (X,%) -* (7, y) is quasi-uniformly continuous; (b) if /is the induced map then for each V in y there exists U in ^ such that/[C/] c V; (c)ifSf is a sub-base for % then/-1[F] £ ^ for each V in T^". Each quasi-uniformly continuous function is continuous relative to the quasi-uniform topology. If f:X-±(Y,y), where (7, y) is a quasi-uniform space, then {/"^[F]: V e y} is a base for a quasi-uniformity for X. The induced quasi-uniformity is the smallest one that makes / quasi-uniformly con- tinuous. In general, if F is a family of functions such that each/ in F is a mapping from X into a quasi-uniform space (Yf, yf), then the family of all sets of the form /^[F], for finF and F in yf, is a sub-base for a quasi- uniformity °U for X. % is the smallest quasi-uniformity for X such that each map / in F is quasi-uniformly continuous. 1.4. DEFINITION. If (Xa, <%a) is a quasi-uniform space for each member a of an index set A, then the product quasi-uniformity for II (Xa: a e A} is 228 R. STOLTENBERG the smallest quasi-uniformity such that each projection is quasi-uniformly continuous. It follows from what has been said above that sets of the form {(x, y): (xa, ya) e Ua}, for a in A and J7a in ffla, is a sub-base for the product quasi-uniformity. It is easy to see that the topology generated from the product quasi-uniformity is the product topology. Also, a function / from a quasi-uniform space to a product of quasi-uniform spaces is quasi- uniformly continuous if and only if the composition paof is quasi- uniformly continuous for each a in A. The following theorem is taken from Kelley ((2) 185). 1.5. THEOREM. Let {Un: n = 0,1,...} be sequence of subsets of XxX such that Uo = X x X, Un+1 o Un+1 o Un+1 £ Un for each n, and each Un contains the diagonal. Then there is a quasi-pseudo-metric on X such that n Un g {(x, y): d{x, y) < 2~ } c C^,_1 for each positive integer n. As a corollary of Theorem 1.5 and a theorem of Pervin ((4) 317), we have the following quasi-metrization theorem. 1.6. THEOREM. A topological space (X, r) is quasi-pseudo-metrizable if and only if there exists a quasi-uniformity % for X which generates r and has a countable base. A topological space (X, T) is quasi-metrizable if and only if (X, T) is Tx and there exists a quasi-uniformity % for X which generates r and has a countable base. A quasi-uniformity for a set X may be derived from a family P of quasi-pseudo-metrics. For p in P, let Upr = {(x, y): p(x, y) < r). The family of all sets of the form Upr, for p in P and r > 0, is a sub-base for a quasi-uniformity fy for X. The quasi-uniformity <%t is called the quasi- uniformity generated by P. Each quasi-uniformity for a set X is generated by a family of quasi-pseudo-metrics. 1.7. THEOREM. Each quasi-uniform space (X, <%) can be embedded in a product of quasi-pseudo-metric spaces. The proof of Theorem 1.7 follows from Theorem 1.5 and what has been said above. If (X,^) can be embedded in a quasi-uniform space (Y,ir), then there exists a quasi-uniform isomorphism / mapping {X,^l) into (Y,ir). It is convenient in the rest of this paper to let X denote f[X] in (Y, 'f). 2. Main definitions Extending the definition of a Cauchy net to quasi-uniform spaces has been difficult, and no completely satisfactory definition has been found. SOME PROPERTIES OF QUASI-UNIFORM SPACES 229 There are at least two ways in which ^-Cauchy nets can be defined. We give the definition which seems most natural in the sense in which we shall use it. 2.1. DEFINITION. A net {Sn: n e D] in a quasi-uniform space {X,°U) is fy-CaucKy if and only if for each U in % there exists n0 in D such that (Sn, 8m) E U whenever m ^ n0, n ^ n0, m ^ n, and m, n e D. 2.2. DEFINITION. A quasi-uniform space (X, °tt) is ^-complete if and only if each ^-Cauchy net in X converges to a point in X relative to r^. 2.3. DEFINITION. A quasi-uniform space (X, °U) has a W-completion if and only if (X, <%) can be embedded in a ^-complete quasi-uniform space (X*,4t*) in such a way that X is dense in Note if (X} d) is a quasi-pseudo-metric space the following terminology will be used. If a net is 4^-Cauchy we shall say it is d-Cauchy. If {X, d) is ^-complete we shall say it is d-complete. A ^-completion for (X,d) will be called a d-completion. This is in accord with the definitions given by Kelly ((3) 75). We say that a quasi-pseudo-metric space (X, d) is d-sequentially complete if and only if every d-Cauchy sequence converges to a point in X.