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Scholars' Mine Doctoral Dissertations Student Research & Creative Works 1970 Completeness and related topics in a quasi-uniform space John Warnock Carlson Follow this and additional works at: http://scholarsmine.mst.edu/doctoral_dissertations Part of the Mathematics Commons Department: Mathematics and Statistics Recommended Citation Carlson, John Warnock, "Completeness and related topics in a quasi-uniform space" (1970). Doctoral Dissertations. 2144. http://scholarsmine.mst.edu/doctoral_dissertations/2144 This Dissertation - Open Access is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. COMPLETENESS AND RELATED TOPICS IN A QUASI-UNIFORM SPACE by JOHN WARNOCK CARLSON, 1940 - A DISSERTATION Presented to the Faculty of the Graduate School of the UNIVERSITY OF MISSOURI - ROLLA In Partial Fulfillment of the Requirements for t h e Degree DOCTOR OF PHILOSOPHY in MATHEMATICS 1970 ii ABSTRACT Completions and a strong completion of a quasiuniform space are constructed and examined. It is shown that the trivial completion of a To space is T 0 • Ex- amples are given to show that a T 1 space need not have a T 1 strong completion and a T 2 space need not have a T 2 completion. The nontrivial completion constructed is shown to be T 1 if the space is T 1 structure is the Pervin structure. and the quasi-uniform It is shown that a space can be uniformizable and admit a strongly complete quasi-uniform structure and not admit a complete uniform structure. Several counter-examples are provided concerning properties which hold in a uniform space but do not hold in a quasi-uniform space. It is shown that if each mem- ber of a quasi-uniform structure is a neighborhood of the diagonal then the topology is uniformizable. iii ACKNOWLEDGEMENTS The author wishes to express his deep appreciation to Dr. Troy L. Hicks for his inspiration and guidance during these years of graduate study. The value of his many penetrating suggestions and consistent patience in the preparation of this thesis can not easily be measured. The author also wishes to express his appreciation to his wife and family for their understanding and encouragement. iv TABLE OF CONTENTS Page ABSTRACT ••••••••••• o • • • • ll o ••• ••••••••••••••••••• o • • lll INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 •o••····· 3 COMPLETIONS OF A QUASI-UNIFORM SPACE . . . . . . . . . . . . . . 5 A. Trivial Completion and Some Examples.......... 5 B. Constructions of a Completion................. 9 c. A Completion for a Pre-compact Structure and Further Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 16 D. Cauchy F i l t e r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 E. A Cour..'cer-example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 F. General Properties for a Completion . . . . . . . . . . . 25 SOME THEOREMS AND EXAMPLES REGARDING QUASI- UN IF 0 RM SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A. Neighborhoods of 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 B. U/\U- 1 . • • • • . • • . • • • • • . . . . . • . . . . • . • . . . . • . . • • • • . 34 c. Saturated Quasi-uniform Spaces . . . . . . . . . . . . . . . . 38 D. 0-Complete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 E. q-Ti Separation Axioms . . . . . . . . . . . . . . . . . . . . . . . . 45 F. A Counter-example in (F,W) . . . . . . . . . . . . . . . . . . . . 51 SUMMARY, CONCLUSIONS AND FURTHER PROBLEMS . . . . . . . . . 53 BIBLIOGRAPHY . . . . . . . . . . . . . ·. · . · · · · .. · · ·. · · · · · · · · · · · 55 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 o • o • o • o • o ACKNOWLEDGEMENT .. I. II. III. IV. V. o • o • o • o • • • • • • • • • • • • REVIEW OF THE LITERATURE . . . . . . . . . . . . . . . . . • • • • • l I. Let (X,U) INTRODUCTION be a quasi-uniform space. The primary problem in Chapter III is to construct a completion for (X,U). It is shown that it is impossible in general to construct a completion which preserves the Hausdorff separation property. A trivial completion for any (X,U) is given which preserves the T 0 separation property, but the trivial completion is not T 1 • Several nontrivial completions are given for special classes of quasi-uniform spaces. One of these constructions preserves the T 1 separation property when U is the Pervin quasi-uniform structure. A parallel discussion of strong completions is also considered, and an example is given to show that not every T 1 space need have a T 1 strong completion. A sufficient condition for the concepts of complete and strongly complete to coincide is given. An example of a uniformizable space which does not admit a complete uniform structure is shown to admit a strongly complete quasiuniform structure. In Chapter IV we consider some well-known properties of uniform space which fail to carry over for quasiuniform spaces. It is of interest to have conditions on U that will guarantee that U is compatible with a uniform structure. It is shown that if each U borhood of the diagonal in X with a uniform structure. ~ x E U is a neigh- X then U is compatible necessary and sufficient con- 2 dition for U sented. ~ _l U to be a quasi-uniform structure is pre- A class of separation properties which is depend- ent on the particular quasi-uniform structure under consideration is defined and some of their properties are studied. Finally, an example is given to show that a sequence of quasi-uniformly continuous, (continuous) functions may converge quasi-uniformly to a function g and g not be quasi-uniformly continuous, (continuous). The topological terminology found in this thesis is consistent with the definitions found in Gaal [11]. The definitions of concepts related to quasi-uniform spaces can be found in Murdeshwar and Naimpally [16], with the following exception. exists z EX such that By U o V we mean { (x,z) E U and (x,y) (z,y) is the definition of U o V found in Gaal E V}. [11]. there This 3 II. REVIEW OF THE LITERATURE In 1955, Krishnan [14] essentially showed that every topological space admitted a compatible quasi-uniform structure. In 1960, Csaszar [3] showed this more explicit- ly and in 1962, Pervin [19] gave a useful and simple construction of a compatible quasi-uniform structure for a given topology. Fletcher [7] gave a construction for a family of compatible quasi-uniform structures. In 1960, Csaszar [3] extended the notions of a Cauchy filter and completeness to a quasi-uniform space. [13] Isbell noted that the convergent filters were not necessarily Cauchy, so Sieber and Pervin [20] in 1965 proposed the def- inition of Cauchy filter which is now in use. They defined a space to be complete if every Cauchy filter converged. We will call a space strongly complete if it has this property. In this paper, they showed that every ultrafilter is Cauchy in a pre-compact space. They obtained the following generalization of the Niemytzki-Tychonoff Theorem [18]. A topological space is compact if and only if it is strongly complete with respect to every compatible quasi-uniform structure. In 1966, Murdeshwar and Naimpally [16] continued to use the definition of Cauchy filter proposed by Sieber and Pervin [20] but defined a quasi-uniform space to be complete if every Cauchy filter had a nonvoid adherence. 4 By making use of the fact that every ultrafilter in a pre-compact space is Cauchy, the corresponding generalization of the Niemytzki-Tychonoff Theorem carried over for this new definition of completeness. Sieber and Pervin [20] proposed the question, does every quasi-uniform space have a completion? [22] Stoltenberg in 1967 showed that every quasi-uniform space had a strong completion. However his construction left open the question of whether every Hausdorff quasi-uniform space had a Hausdorff completion. The techniques used by Liu [15] motivated the completions developed in this thesis. In 1965, Naimpally [17] showed that if (X,U) and (Y,V) were quasi-uniform spaces and Y is T 3 and V the Pervin structure then U and C are closed in (F,W), where F = YX and W is the quasi-uniform structure of quasi-uniform convergence and u tinuous (C) is the set of all quasi-uniformly con- (continuous) mappings from (X,U) to (Y,V). 5 III. A. COMPLETIONS OF A QUASI-UNIFORM SPACE TRIVIAL COMPLETION AND SOME EXAMPLES DEFINITION 1. Let X be a nonempty set. A quasi-uni- form structure for X is a family U of subsets of X x x such that: (x, x) { ( 1) /:, ( 2) if u ( 3) if U, ( 4) for each u v = 0 u E v : X E and uc u E u for each u X} C v, then v f then un E u v E E E u. ' u; u. ' there exists V E.: u such that v cu. DEFINITION 2. If U is a quasi-uniform structure for a set X, let tu = { A c X : if a s A there exists U that U [a] C A}. Then tu lS E U such the quasi-uniform topology on X generated by U. DEFINITION 3. Let (X,t) be a topological space and let U be a quasi-uniform structure for X. patible if t If (X,t) Then U is com- = tu. is a topological space and 0 s t, let S(O) = 0 X 0 u (X-0) In [19], Pervin showed that { S(O) X X. : 0 E t} lS for a compatible quasi-uniform structure for X. refer to this as the Pervin structure. a subbase We will The following def- inition is due to Sieber-Pervin [20] and is equivalent to the usual definition of a Cauchy filter in a uniform space. DEFINITION 4. Let (X,U) be a quasi-uniform space 6 and F a filter on X. F is U-Cauchy if for every u there exists x such that U[x] = x(U) DEFINITION 5. E E U F. A quasi-uniform space (X,U) is strongly complete if every U-cauchy filter converges. (X,U) is said to be complete if every U-Cauchy filter has an adherent point. DEFINITION 6. Let (X,U) and (Y,V) be quasi-uniform spaces and f a mapping from X toY. The mapping f is said to be quasi-uniformly continuous if for each V exists U E E U such that (a,b) E E V there U implies that (f(a) ,f(b)) V. DEFINITION 7. (Y,V) Two quasi-uniform spaces (X,U) and are said to be quasi-uniformly isomorphic relative to U and V if there exists a one-to-one mapping f of X onto Y such that f and £DEFINITION 8. (X,U) 1 are quasi-uniformly continuous. A completion of a quasi-uniform space is a complete quasi-uniform space (Y,V) such that X is quasi-uniformly isomorphic (relative to U and V) to a dense subset of Y. In [22], Stoltenberg proved that every quasi-uniform space has a strong completion. The proof is long and in- volved and it is not clear if any separation properties the space may possess carry over to the completion. The following construction shows that every quasi-uniform space has a rather simple completion. 7 Let (X,U) be a quasi-uniform space CONSTRUCTION 1. s and put X* = X U { S} where = S(U) Then B = { S(U) U U : U t: structure U* for X*. Us U, and S(U) [x] = F on X* converges to Also, u = { U* n X l X : { ~ X. (S,x) X 1 1 '-J U, let X*}. Note that S(U) [13] =X* for each U[x] s. if x t: X. Clearly, every filter Hence (X*,U*) is strongly complete. X : U* (X,U) -+ is a completion of and t u. * = t u. E t: U} forms a base for a quasi-uniform U*} and E (X*,U*) is a quasi-uniform isomorphism. (X*,U*) For U Since X is dense in X*, (X,U) In fact, X* is compact {X* } . Fletcher [7], working independently, also discovered the above construction. If X is T 0 , then X* is T 0 • X* is never T 1 since the only open set containing S is X*. Thus the following question naturally arises. quasi-uniform space have a T 2 (T 1 ) completion or strong completion? EXAMPLE 1. We give an example of a Hausdorff quasi- uniform space that does not have a Hausdorff completion and consequently does not have a Hausdorff strong completion. Let X denote the real numbers and A = { 1/n : n 1,2, ••• }. = Set t = { E-B : B c A and E is open in the usual topology}. (X,t) is Hausdorff but not T 3 • Let U denote the Pervin quasi-uniform structure generated by t. Suppose (X*,U*) 8 is a completion for n x and U = { U* (X,U). X : U* x We may assume that X = X* U*}. E Let F be the filter on X generated by { A,{1/2,1/3, ... }, {1/3,1/4, ... }, ... }. Then there exists an ultrafilter M Since X such that M ~ F. ~n 'V U is pre-compact, M is a Cauchy filter. trafilter on X* generated by M. U E E U so that there exists x Clearly U* [x] "'M. E Hence u* If Let M be the ul- U*, E u* n x X such that u [x] M is Cauchy. E x = x /d. Since (X*,U*) lS 'V complete, there exists x* s X* such that M converges to x*. Thus every open neighborhood of x* must meet every set of the form { 1/n, 1/ (n+1), ... }, (n vve show that N). E x* can not be separated by disjoint open sets. u = (X-A Then there exists U* n U*[o] that s > 0 o 0 1 and X* E u (A 02 • Let X) . X n U* such that u = U* E X X X. Now o ~ x*. Therefore, A= [0. E X-A) X and o n Then 0 1 such that ((-E,E)- A) C 0 X t E ~ow ()X. 1 so there exists there exists positive integers N and k such that 1/N Since 1/k ~ 02 E and 1/k £ n X (1/k- 8,1/k + 8) [ (- E Therefore, I E) - t A] () X n 1 /N ' 1I (N+ 1 ) ( . . . 1 . there exists 8 -A C 0 2 01 n 02 EXAJ'vlPLE 2 . E n 02 E n [ ( l /k ·- ~~and X, 0' o > such that Clearly 1 /k (X*,U*) + 8) - A] ~ 0. is not Hausdorff. We give an example of a quasi-uniform structure U for the set N of natural numbers such that: (1) tu is the discrete topology, and 9 (N,U) does not have a T 1 strong completion. ( 2) u n = { (x,y) : x = y or x ~ n}, B = { U n n Let N}, and let E U denote the quasi-uniformity generated by the base B. Suppose (N*,U*) is a strong completion assume that N N* and u= U* n N X ~or N. (N,U). If F = We may F is a {N}, "v Cauchy filter on (N,U). on (N*,U*) Let F generated by F. complete, there exists n* n*. Clearly, n* E ~enote Since E N* - N. (N*,U*) N* such that N = N. to n E F and 0* N E F. For each n E N, we have n E 0* for every open set in N* containing n*. B. P converges Then 0* containing n*. n is strongly Let 0* be any open set in N* 'V 0* the Cauchy filter Thus N* is not T 1 Thus • CONSTRUCTIONS OF A COMPLETION LEMMA 1. Let (X,U) be a quasi-uniform space and S Then a filter F converges to x if and a subbase for U. only if for each S E S we have S[x] ~ F. Throughout this section we will let (X,U) denote a quasi-uniform space with a base B such that for any V we have that V o V = V. E B If U has a subbase with this property then the base generated by the subbase also has this property. It is clear that the Pervin quasi-uniform structure has such a base as well as the class of quasiuniform structures introduced by Fletcher in [7]. will say that two Cauchy ultrafilters M1 and 1\l are equivalent provided U[x] E M1 2 We on X if and only if U[x] E M2 • 10 Certainly this is an equivalence relation on the set of all Cauchy ultrafilters on X. We will denote the equivalence class containing ,\! by 1'·.!. = { Set !\ M = on X} and X* M is a nonconvergent Cauchy ultrafilter D(V) XU!\. will denote the set of all mappings 6 from!\ to X such that V[6(M)J f Since AI is Cauchy, D (V) ~ M where V c for each V c B. c F~ For V B. B and 6 c D(V), we set = S(V,6) V U 6 U { = { S* LEMMA 2. (/d,y) S(V,6) ,\j : V t: and y II t: B, c') c c V [ 6 ( ,\!) ] } • D(V)} forms a subbase for a quasi-uniform structure U* on X*. It is clear that S* f PROOF. ~. Let S (V, ()) Then S (V, cS) ::;) 6 and we show that S (V, cS) Suppose (x*,y*) If x* = x (x,y) c X, c V, S(V,6) c then y* (y,z) c = and (y*,z*) S (V, cS) C S(V,6). c = z c c X and z* y o S*. 1: S (V, 6) • Case X. (a). Thus V and since V o V = V, we :1a- 2 (x*,z*) 7 h (X , Z ) have z* = S (V 1 6 ) . E z c c V and z* s = = (x*,z*) z (V, cS) X, y ' c V[6(i\,j)] s (V, cS) PROOF. M E c V [ 6 (,\1) ] , consequently 0 (b ) . (y*,z*) THEOREM 1. where CaSe and s = X* \J (V, 6) • ( y, z) and ( 6 (i\1), z) (x*,z*) If y* f, • c If y* = " .L V. t: (:\f,z) E ) V S(V,cS). ,lj we X, then ( 6 (t.l) , y) Iience c V since V = E = = V. Then Therefore, c s (V, 6) . I I (X*,U*) is complete. Let F be a Cauchy filter on X*. is an ultrafilter. If Jl! converges, adherent point and U* is complete. Then F C F has an Suppose tl is not .\l = 11 We show that X s M. convergent. Let S(U,6) U*. s s (U, 6) [x*] Since M is Cauchy, there exists x* s X* such that E M. "' = M1 If x* A, s then A M1 implies there exists S(V,6) M does not converge to s (V, 6) [ M1 ] such that ~ S* s Since M is Cauchy, there M. A exists z* f. M1 such that S (V ,y) [z*] s Now if z* s X ,\{. A there is nothing to show; so we suppose that z* A = M2 s A. A Then M1 f. M2 and we have A u (V[y(M2)] A (U[6(Ml)] {.\!2}) (l u {,\{1}) EM. M since Thus X s n X~ V[y(M2)] M0 = { M E U[6(Ml)] M : M C X} is an ultrafilter on X. We show that M0 is U-Cauchy. with V CU. Let 6 s D(V). exists x* s X* such that have V[x*] If U s U, there exist V s Trien S(V,6) s (V, 6) [x*] M and thus U[x*] s EM. s M0 s s AI. If x* • U* so there If x* s X we _M s then A, A V[6(;\,{)] U {M} s and since ,\1 Consequently, U [ o UA) ] s M0 s M, X we have V[6(M)] Thus M0 is Cauchy on X. • Either M0 is convergent on X or it is not. converges to x. V [x] s A! 0 • Let S (V, 6) Thus S (V, 6) [x] S*. s converges to M0 • Case (2). If S(V,6) 0 ] M0 = ,\! 0 Then M0 s A and we show that M E S*. A S(V,6) [M (1) and we have that M con- ,\-f s Case Then S(V,6) [x] verges to x which is a contradiction. is nonconvergent on X. sM. = V[o (M 0 )] U {M 0 }. B 12 Now V[o(M 0 )] s and consequently S(V,o) [J\,1 0 ] M0 sM. Therefore, M converges to M0 and this is a contradiction.// THEOREH 2. By theorem l, PROOF. Ms A and U* S(V ,o ) n n (X*,U*) is a completion for U*. E n n l = {M} l Since V.[o.(M)] n n l -:~ v. [o. (M)J l U V. [ 6 . ( ,1{) ] l n A c u*. for i = 1 Let ), ••• , We have 1, 2 , •.• n . = 1,2, ... ,n, we have Now ¢. l u* [ MJ l M for eacn i s l cv.l , ol. ) s l A S ( V . , o . ) [ M] is complete. Now there exists S(V 1 ,o S* such that c (X*,U*) (X,U). x => n en s cv. , o . ) ) [ MJ n A l l n n = S(V. ,o,) [M] l n x X l n = Thus M E l X and X= X*. l Let U' uniform structure of U* on X. V s B with V CU. v = s cv, o) n x x -:1¢. nv.[o.(M)J denote the induced quasiIf U s U there exists Let o s D(V). x. Thus v U' E Then S(V,o) and hence u s E U* and u· . Suppose U s U'. Then there exists U* c U* such that n Since U* c U* there exists S (V 1 , o 1 ) u = u* x x. x S(V ,o ) s S* such that U* n n ~ n n (n s (v , o ) ) n x l n (1 'l'herefore, that U = U' . mapping i: l V i S(V. ,o.). 1 x c u* n x x Consequently, 1 x x u. l U and hence U E Lf. Thus we have shown From this i t follows that the identity (X,U) isomorphism.// C ·~ , ••• , n (X,U* tl X x X) is a quasi-uniform 13 One can let A = { F filter on (X,U) }. F is a nonconvergent Cauchy Then using the same construction we 0 get a strong completion of 0 It is clear that (X,U). Denote it by 0 (X,U). 0 (X,U) is not in general the trivial strong completion given in construction l. A space EXA.l'\iPLE 3 . not T 1 (X,U) may be T 1 and Consider the space • (X*,U*) (N,U) described in example 2. It is clear that U o U = U for each U in the base B. n n n n There Thus we may construct the completion (N*,U*). M containing the filter base exists an ultrafilter Since U is pre-compact, \·Je have { {n,n+l, ... } : n s N}. ;ll E Since M is nonconvergent, M is a Cauchy filter. that N* . Let S(U ,6) Clearly, S*. s n 6(\l) > n. There- fore, s (u , n = u [ c5 c5 ) [ MJ n = Hence (N*,U*) 1s not T THEORE1'1 3. Let 1 N UA ) J U Ul} U { ;\1}. • (X,t) be a T 1 topological space and Then (X*,U*) U the Pervin quasi-uniform structure. a T 1 completion for that X E 0, y have O,W s S*. (X,U). ~ If x,y s X, x PROOF. t*. E: W, Let lS ~ W, X 11A 1 ~ y, there exists O,W s t and Y ~ 0. Since X s t*, we 1\l 2 be points in/\ and S(U,cS) Then { M1 } U U [ o (,\1 1 ) such ] and , s 14 Suppose x s X and M s A. filter. (X - X and x s 1 Now X - {x} s M since M is an ultra- Set u Let x M. ~ x, we have {x} Since M does not converge to {x}) Define 8 by 8(M) = x X - {x} 8 D ( U) {x}) U {x} X. x x; if X = {x} we would have X* = X. ~ 1 (X x for each M s A. 1 Clearly, U[x 1 ] N for each nonconvergent ultrafilter N. E ~ E and S ( U 1 8) S*. E Now s ( u ' ~ = 6 ) [ M] Thus u (X 01} (X* ,U*) is T 1 DEFINITION 9. - { X } ) M. and X is a neighborhood of x that does not contain Therefore, = ./I (X,U) 1s R 3 if given x s X and u s there exists a symmetric V s U such that (V o G, V) [x] C U[x]. The above condition was introduced in [16]. It might be described as a local symmetric triangle inequality. In [16, p.41] it was shown that if 1 the Pervin structure is R 3 THEOREM 4. space. Then (X*,U*) f there exists 0 fl is regular, then • be a T 1 and R 3 quasi-uniform (X,U) is a T 1 space. It suffices to show that for each x s X and PROOF. 1IA Let (X,t) 1 , 0 7 t: t* such that x < 0 1 M s 1 02 , ~ x ~ 0 7 , and M 4 01. Let x and AI be given. does not converge to x, there exists U U[x] [x] C ~ 11A. U such M that There exists a symmetric V s U such that (V o V) M is Cauchy implies that there exists x 0 s X U[x]. with V[x 0 ] E Since f~ M. We show that x ~ V[x 0 ] . Then (x,x 0 ) E Suppose x V and (x 0 ,a) E s V. 15 Hence a c: U(x] (V V) [x] C U[x]. o But this implies that V[x J C 0 M. ~ which is impossible since U[x] There exist A o D(V) with o E A ~ X = x0 • (;\!) V[x 0 ] U {Af} Then = A S (V, o) [,\!] while M ~ X, an open neighborhood of x. THEOREi-1 5. properties of Thus Listed here are some easily verified (X*,U*). (l) X is an open dense subset of X*. (2) (X,U) (3) If is T 0 if and only if (X*,U*) (X*,U*) is T 0 • has property P and P is an open hereditary property, then (4) (X*,U*) (X,U) has property P. A is closed in X* and the subspace topology on A is the discrete topology. (5) If (X*,U*) then (X,U) is pre-compact and Hausdorff, is completely regular. 0 These properties also hold for PROOF. and S(U,o) By theorem 2, X= X*. (l) . c: U*. is open in X*. 0 (X,U). U[x] C X. Then S(U,o) [x] ( 2) • Let x r X Hence X Since T 0 is a hereditary property the sufficiency is clear. Suppose (X,U) is T 0 , then since X is open in X* i t suffices to consider the case where x* + A ,vr A E A, x*, ~~~ E X*. Let S(U,o) [ U* and we have that A M ~ S(U,cS) [ x*] . Hence X* is To. from A (l) . S(U,cS) E ( 4) • Statement ( 3) follows is closed in X* by (l) , and for any U* we have that {l.f} = f\ n S(U,cS) [:1.1] for any 16 M E A. (5). Therefore the subspace topology on A is discrete. If (X*,U*) is pre-compact and Hausdorff then since i t is complete i t must be a compact Hausdorff space. Therefore the subspace C. (X,U) must be completely regular.// A COHPLETION FOR A PRE-COMPACT STRUCTURE AND FURTHER EXAMPLES Let (X,U) be a pre-compact quasi-uniform space. ~- = U U Define X* as before and set S(U) andy E U[x] (M,y) B* = { S(U): U E U} forms a base for a LEMMA 3. quasi-uniform structure for X*. f B* Suppose S(U) U { M for some x E X}. E PROOF. ~ ~ and S(V) c s(u; ~ and 0. Denote i t by for each S(U) belong to 13*. Then S(U B*. s n V) U n B s and we show that n S(U () V) C S(U) Let (x,y) E n S(U If V). X S(V). E X, then (x,y) E V and ~ thus (x,y) which case E S(U) (x, y) such that y E (U y E V[z] B*. E E n If x = S(V). V) [z] E M. S(V). fore If x (x,z) ~ A, then y s (x,y) Thus y E U [ z] E S(U) Then there exists V E U such o M S(V)CS(U), let X then n S(V). th~t E V o V CUC S(U). = M then ( x, z) (y,z)E S(V) C and E We (y,z) (y,z) E V and there- Now if S(U). 1\l and V o V CU. (x,y)E S(V) (x,yJ E V and f: Let S(U) X= ,q ~ y ln S (U) () S (V), or there exists z "' X Hence AL show that S(V) E n If x f !\and = s: and 17 y E X, (y,z) v [ s] then there exist s E • Hence V. (s,z) o X such that y V o V[s] l V C U, that is, ,\J z , and ~ U[s] "M and we have that (x,z) = UJ,z) Thus U[s] Therefore S(V) E s ,_ S(U). and we have that B* is a base S(V) C S(U) for a quasi-uniform structure on X*.// '\, (X*,U) THEOREM 6. Suppose PROOF. is complete. (X*,~) is not complete. Then there exists a nonconvergent Cauchy ultrafilter M* on X*. Using the same argument as in the proof of theorem l we have that M C X} is an ultrafilter on X. is pre-compact i t follows that Case M converges to x (1). U[x] E M*, and U[x] E M*. Thus x E ~ M* lim is Cauchy [ 16, p. (X,U) 51] . ''v E X. such that S(V) Cu. exists S(V) Hence V[x] 1\l Since Let U U. r Then there Since x c lim ,\1, V [x] S(V) [x] ;\j. = V[x] implies that Case but this is impossible. '\~ lim ( 2) • S (V) C M = ¢. U. Now S(V)[,\1] Hence S (V) [M] M E Then :vl M E E = X. 00 U Let U c U, then there exists ( U ( V[y]: V[y] and therefore U [,\!] lim M which is a contradiction. r-: c ,'•{}). Consequently ,\l. Thus there are no ''v that is, nonconvergent Cauchy ultrafilters on (X*,U); '\, (X*,U) is complete.// '\, (X*,U) THEOREM 7. PROOF. is a completion for (X,U). '" By the previous theorem (X*,U) 1s complete. '0 It is clear that X= X*. ~ I then there exists v X there exists w E We show that U E = U . ~ such that u = v U such that S (H) C Let u X n X X X. Now V, hence W C U and we 18 '\, have that U X X X X E '\, u. :s. Let u E ux. u, then S(U) E u and u = S(U) Thus the identity mapping (X, U) n l a quasi-uniform isomorphism.// lS We note that the Pervin quasi-uniform structure lS pre-compact. A proof similar to the proof of theorem 4 shows that if (X,U) '\, THEORE.:'1 8. is T 1 and R 3 Let (X,U) then , (X*,U) is T 1 • be a pre-compact quasi-uniform space. '\, (l) (X*,U) is pre-compact if and only if A is '\, finite if and only if (2) (X*,U) is compact. X* is T 3 implies that X* is compact. '\, (l) PROOF. follows from the definition of U and the fact that completeness and pre-compactness are equivalent to compactness. Since every ultrafilter on X lS Cauchy, i t must converge to a point in X* and hence (2) follows from theorem 4.17 in EXAMPLE 4. Let N [16] .;; = {1,2,3, ... } and let U denote the quasi-uniform structure generated by the base B n c: N}, where Un = { (x,y) x = y or y ~ x ~ be an ultrafilter containing {N,{2,3,4, ... }, ... }.Then Mo EA. homeomorphic to of We show that (N 00 ,t 00 ) , A c= {,\10} and n}. { un Let ,\j 0 {3 , 4 , 5 ' . . . }, (::*,tu*> lS the one-point compactification (N, t) . Let tv! be a nonconvergent Cauchy ultrafilter on N. Note that Un [k] if k .?. n. = {k} if k -< n, and Un [k] = {k, k+', ... } It follows that Un[k] s :.1 if and only 19 Therefore, M ~ M0 and A= {M 0 }. ~ Noo be the identity on Nand i(n are continuous at each n E N. 0 = oo ) Let i N o'v and i- 1 i We show that i : N* is continu- A ous at M0 Let 0 • {oo} 0) U (X - pact, there exists k such that n E Since 0 is com- E t 0 implies n < Now k. {k,k+l V{Mol- (u k 1 8 is defined by 6(M) 8 ) [ .~~ 0 ] ) = {k 1 k+l 1 • • • } v { 00 } = k for each ME A. 0 C • • •} Now i(S .-1 '0Je show that • 00 1 l A is continuous at oo. Let S(Uk,6) be given. Now o(A1 0 ) k, .:.::. A = so let 0 {1,2, ..• ,8 (M )-l) }. and S(U 1 ,6) [M 0 ] {oo} U 0 (X 1 ) = 1, then k = 1 Let (N- 0). EXAMPLE /', U { 0 = N* and there is nothing to show.) = S(Uk,6) (N- 0) (If 8 (,"i 0 Y) Let I denote the set of integers and Un 5. : X [M 0 ] . = Y 1 Y .::. X ?_ n 1 Or Y _2 X -n} • < Let U de- note the quasi-uniform structure generated by the base { U n : n = 0,1,2, . . . }. and U is pre-compact. Then I has the discrete topology Let M be an ultrafilter containoo ing the f i 1 ter base { { 2, 3, 4, ... } , { 3, 4, 5, ... } , ... } and AI -oo be an ultrafilter containing the filter base {{-2,-3, ... }, {-3,-4, . . . }, . . . }. Then Moo and 1\l_co are nonconvergent Cauchy ultrafilters. A Hence M "I M 00 -00 Now by considering the various cases, as in example 3, we have that A= {M compact. ,M -co }. Thus Now we show that I* is Hausdorff. (I*,U*) lS Since I is discrete and open in I*, i t is clear that distinct points 20 in I are separated by disjoint open sets in I*. k I. E Choose n > k and -m such that cS 1 k. < There exists cS <~C!) = n, [Moo] = = cS 1 (tC! -co Now let 1 E D(U ) = -2. n ) A = m, and cS 2 (M -co ) There exists o - m. Now {k} (I S(Un,o {k} (\ S(Um,o 2 ) [M_oo) 1 ) D(U ) where cS(M 2 E ) 0 and = 0. = A 2 and cS(M 00 Then A S(U2,cS) [M] (l S(U2,cS) [M 1 oo -oo ({M 0 } u {2,3,4, ... }) n ({M -co } u {-2,-3,-4, ... }) =f. Hence I* is Hausdorff. D. CAUCHY FILTERS In a uniform space the adherence of a Cauchy filter equals its limit. The following example shows that this is not necessarily the case in a quasi-uniform space. EXAMPLE 6. X < Let X Since W o W y}. W we have that a quasi-uniform structure. Set F W[2) E = {X,{2,3,4,5}}. F. and~={ {1,2,3,4,5} {~} (x,y) forms a base for Denote the structure by U. Then F is a Cauchy filter since It is clear that lim F = {1,2} while adh F = X. Hence we have a cauchy filter whose limit is not equal to its adherence. It is natural to wonder if there are quasi-uniform spaces, that are not uniform spaces, in which the limit of every Cauchy filter equals its adherence. The follow- 21 ing theorem shows that such spaces do exist. THEOREM 9. Let (X,U) be a R 3 quasi-uniform space. If F is a Cauchy filter then lim F = adh F. PROOF. Let F be a Cauchy filter. prove that adh F C Since (X,U) (V a V o E E C U [x] . X such that V[a] Since X E E Vis symmetric, E V. F. E ~ill We V. Let c adh F and U (b,a) E Therefore c COROLLARY. U. E U such that E V. E Thus Let (X,U) show that V[a] C U[x]. V[x] E V [a] . E n V[a]. Hence Then (a,c) (x,b) E V, E E F. Thus x V. (b,a) (v o V o V) [x] C U[x]. V[a] C U[x], and we have U[x] (X,U) E Since F is Cauchy there exists adh F there exists b V and (a,b) (a,c) Let x is R 3 there exists a symmetric V V) [x] o lim F. It suffices to (x,b) Since E V, and Hence lim F.// E be a R 3 quasi-uniform space. is complete if and only if it is strongly complete. PROOF. The result is obvious by theorem 9. and the definitions.// In a complete uniform space if lim F follows that F is Cauchy. +¢ then it It is natural then to ask if in a complete quasi-uniform space we have a filter F such that + ¢, adh F does it follow that F is Cauchy. The follow- ing example shows that such a filter need not be Cauchy. EXAI-iPLE 7 . Set Un = { (x,y) Let X denote the set of positive integers. : x y, or x l and y 2 andy= 2n + 2k + l, where k = 1,2, ... }. u3 = 6 U{(l,S), (1,10), . . . } u {(2,9), 2n + 2k, or x For example, (2,11), . . . }. 22 Let B = { un : n = 1,2, ... }. To show B is a base for a quasi-uniform structure it suffices to prove that un cun for each n = 1,2,.0. U and n x f ( y, z) s U . n y then either x 2n + 2 Un . > 2. Hence z Let Un be given, and :£ x = y, then (x,z) = (y,z) = 1 or x = 2. Suppose x and therefore y = z. Thus (x,z) ~ = (x,y) ~ (x,y) f: u . If n = (x,y) 2n + 2 + 1 E un = 1, then y = y and we have that (x,z) On the other hand, if x = 2 then y o Un. > ~ E 3 Therefore B generates a quasi-uniform structure which we will denote by u. The topology generated by U is compact since given any open cover there exists 0 1 containing 1 and a taining 2. Clearly 0 1 U 0~ con- 0 2 contain all but at most a finite number of members of X. Hence (X,U) lS compact and therefore strongly complete. The sets of the form { n,n+1, ... } (n s X) generate a filter F and 1 s adh F, but F is not Cauchy. In a uniform space i t is well known that the neighborhood filters are minimal among the Cauchy filters. This does not hold in general for quasi-uniform spaces as the following example indicates. EXAHPLE 8. Consider the space in example 6. Let N(4) denote the neighborhood filter of 4. Clearly N(4) is the collection of super sets of {4,5}. ,~J collection of all super sets of {5}. and thus N(5) ( 5) Now N(4) is the ~ N(S) is not minimal among the Cauchy filters. 23 E. A COUNTER- EXAI:1PLE The following theorem shows that: (1) A uniformizable space can admit a strongly complete quasi-uniform structure and not admit a complete uniform structure. (2) A space can be uniformizable and admit a strongly complete quasi-uniform structure and not be real compact. (3) A countably compact space may admit a quasiuniform structure that is not pre-compact. (This can not happen with uniform structures.) Let W denote the ordinals less than Q, the first uncountable ordinal. W. t will denote the order topology for It is well known p. 74] [12, that (vl,t) is normal, countably compact, not metrizable, and not real compact. Dieudonne [5] showed that uniform structure U and (W,t) THEOREM 10. (W,t) (W,U) admits a unique compatible is not complete. admits a strongly complete quasi- uniform structure. Let P denote the Pervin quasi-uniform struc- PROOF. ture U ~ for t. n V Suppose U CU. (xI y) X I£ AsS, then A~~. P}. then U = { Set L n E P and L c S, Thus V () L ~- (U n L) n (V y}. :::. If U n n = L) Let s = L s Sand V (U nv) n L u n n L s S, £ L S. then there exists V c P such that V o V S and (V () L) o (V () L) C U () L. For 24 suppose Also (a,b) (a,b) Hence n L and that is, (b,c) ~ L implies a c c; a~ V c b and (a,c) v n c (b,c) L. c L. Then (a, c) L implies b c Therefore (a,c) Thus S is a base for a quasi-uniform structure. the generated quasi-uniform structure by U. = t u. that t Since P ~ Then there exists L U) [x] C Now U[x] t. 0. L[x] = = [I,x] respect to t. n u we have that t [x] (\ = U[x] = (W, U) exists w c W* such that w [ 1 , x] M. Case 2. c M*. E X (L E n () U) [x] (L Hence 0 '- t Let W* = [ 1 is complete. Now Therefore vv. ( x, Q] Q , o~] Let ,\! * denote the lim M*. Case l. = st. If w c vv, Since :d is Cauchy = [I,x] is a neighborhood of \2 and we have that (x, st] c (W,U) ,\!*, but this is complete. (W,U) lS complete can be M be a Cauchy ultrafilter on W. there exists x and Since W* is compact there Suppose w An alternate proof that Let such that Hence w c W and by case l, we have that is impossible. given. We will show there exists x c W such that L[x] since 1'-'l* converges to w c lim M. Denote tu. generated ultrafilter on W*. Hence L. is also a neighborhood of x with and let A1 be a Cauchy ultrafilter on and L c U, ~ tu. with respect to t. X We now show that lim u is a neighborhood of x with respect to [I,x+I) is a neighborhood of E u E c c. Thus L then w ~ U we have that t and x c 0. u. t.: E W such that [l,x] = L[x] c Since L M. t. U 25 be the trace of M on [l,x], then M1 [l,x]. Since [l,x] that z slim M1 lS an ultrafilter on is compact there exists z is R 3 Let L • n u Then there exists x s vJ such that (x,w] C Let 0 = [ 1 ,x], S = (x,w], and T = (w,rt). are open. Then V 1 , [l ,x] E u such Hence z slim M. • We now show that (X,U) W. r (L n and w U) ~ [w]. Then 0, S, and T Set V2 v1 = o x o u (W-o) x w v2 = s X s u (W-S) X w v3 X T u (W-T) X w T V 3 belong toP and hence to U. , It is easy to see that z = v 1 n v2 n v 3 (0 Thus Z s U, c (L z n U) [w] . X 0) (S is synunetric, and Therefore (W,U) theorem 9 we have that (W,U) F. u S) X z o u (T z = z. X T). Now Z[w] = (x,w] is R 3 and by the corollary to is strongly complete.// GENERAL PROPERTIES FOR A COMPLETION It is apparent that not all of the properties of a completion for a uniform space carry over for a quasi-uniform space. In this section we note that irregardless of the definitions of "Cauchy" filter and "completeness" not all of the pleasant properties of a "completion" are going to be preserved in a quasi-uniform space. We would like a definition of "Cauchy" filter and 26 "completeness" that would satisfy the following conditions. (a) The definitions would reduce to the ordinary definitions on a uniform space. (b) There exists at least one completion for every quasi-uniform space. (c) If (X,U) is (d) If (X,U) is compact, then (X,U) (e) If f (Y,V) ~ (X,U) uous where then there exists a T2 , (Y,V) T7 completion. is complete. is quasi-uniformly contin- is complete and T 2 , then there exists a quasi-uniformly continuous extension '\., f : (X*,U*) tion of (f) If (g) is any comple- (X,U). (X,U) where (Y,V) where (X*,U*) ~ is pre-compact then (X*,U*) (X*,U*) is any completion of is coP1pact (X,U). Every convergent filter is Cauchy. It is clear that each of these conditions hold 1n a uniform space. (c) THEOREM 11. spaces and (f) can not both hold (for all (X,U)). PROOF. Let (X,t) be a T 2 space that is not T 3 and U the Pervin quasi-uniform structure for t. By (c) there exists a T 2 completion (X*,U*) and by (f) (X*,U*) 1s compact. Hence THEOREM 12. space (X,U) (d) and for (X,U) is T 3 which is iP1possible.// (e) can not both hold (for all (X,U)). PROOF. Let X = ( o, 1l and U the Pervin quasi-uniform 27 structure associated with the usual topology. [-1,1] and let V be the Pervin structure associated with the usual topology on Y. (d) it is complete. Now f Let Y = (y 1 Define f v) is X~ ~ 2 and compact and by Y by f(x) =sin 1/x. is continuous and since U and V are the Pervin structures it follows that f is quasi-uniformly continuous. Now X*= [o,11 is compact with the usual topology. Let U* be the Pervin structure for X*. that (X*,U*) Now by (e) rv sion f : X* (d) we have is complete, hence it is a completion of (A,U). there exists a quasi-uniformly continuous exten~ Y, but this implies that f has a continuous extension to [o,I] which is impossible. (e) By Therefore {d) and can not both hold.// THEOREM 13. PROOF. If (d) holds then (b) holds. The trivial completion is compact and by (d) it is complete.// Considering our usual definition of Cauchy filter we obtain the following summary. Complete (or Strongly Complete) (a) holds (b) holds (c) does not hold (d) holds (e) does not hold (See theorem 12.) (f) does not hold (g) holds 28 To see that (f) does not hold consider the space in example 2. Since every filter is Cauchy in this space i t is clear that there are infinitely many nonconvergent Cauchy filters. Let A denote this collection. 0 Now the 0 construction of the strong completion (X,U) carries over 0 with A redefined as above. If 0 (X,U) is compact then i t must be pre-compact and hence A would be finite, which i t is not. 29 IV. SOME THEOREMS AND EXAMPLES REGARDING QUASI-UNIFORM SPACES A. NEIGHBORHOODS OF ~ One of the points of interest is the following. Given a quasi-uniform structure U for a set X, when is U compatible with a uniform structure? By definition U is compat- ible with a uniform structure if and only if the topology generated is uniformizable. Other sufficient conditions will be obtained. DEFINITION l. A quasi-uniform space have property P if each U X E (X,U) is said to U is a neighborhood of ~ in X with respect to the product topology. x It is well known that a compact uniform space has property P. The following example shows that this need not be the case in a quasi-uniform space. It also demonstrates that a quasi-uniform structure may be compatible with a uniform structure and not have property P. EXAMPLE l. Let X= [0,2] with the usual topology and let U denote the Pervin quasi-uniform structure. ( 1 3 2'2 ) E t SO U a neighborhood of 1 l ( --E 2 , -+E) 2 X 0 X ~, 0 u (X-0) X X E u. then there exists Now 0 Suppose that U is E > o such that Now p = Hence U is not a neighborhood of L. The following example shows that a quasi-uniform structure can have property P and not be a uniform struc- 30 ture. EXAMPLE 2. set Un = { (x,y) Let N denote the positive integers and : x s: or n y x s: y} . forms a quasi-uniform base. 1,2, ••• } Then { Un : n = Let U denote the quasi-uniform structure generated by this base. It is clear that U is not a uniform structure and the topology generated is the discrete topology. Hence 6 is open in X x X, and therefore each U s U is a neighborhood of 6. In the above example we note that U is compatible with a uniform structure. The following theorem shows that if U satisfies property P then this is always the case. Let (X,U) be a quasi-uniform space. DEFINITION 2. Then u- 1 = { u- 1 u E is a quasi-uniform structure on U} X and i t is called the conjugate quasi-uniform structure of U. U V u- 1 denotes the smallest quasi-uniform structure -1 which contains both U and U THEOREM 1. Let (X,U) . be a quasi-uniform space satis- Then U is compatible with a uniform fying property P. structure. PROOF. Let t and s denote the topologies generated by the quasi-uniform structures U and U V ly. It is clear that U that t .: .:. s. u- 1 V u uniform base for u v u- 1 • n U- 1 SUCh that n u- 1 where I u E (U n U- 1 ) It is < t. E U, form a Let 0 s s and x s 0. X respective- 1s a uniform structure and Thus it suffices to show that s clear that sets of the form exists U u- 1 , [x] C 0. rrhen there Since U is 31 a neighborhood of Q ~ t such that x b with respect to t E Q and Q x Q E Q C CU. x t, there exists Hence Q X Q c u- 1 • Therefore, X = U[X] n u- 1 [x] u- 1 ) (U {') [x] Co Hence 0 s t and thus t = s.// COROLLARY. space ~ith Let be a complete quasi-uniform (X,U) property P. Then there exists a compatible complete uniform structure. In the above proof we showed that U V u- 1 PROOF. a compatible uniform structure. u - uv u- 1 i t follows that lS Since U is complete and u- 1 is co~plete.// uv Every closed subspace of a complete quasi-uniforn A complete subspace of a space is complete. Hausdorff The following theorem gives an uniform space is closed. analogous result for a complete Hausdorff quasi-uniform space that satisfies property P. 'rHEOREM 2. Let be an arL,itrary r:ausdorff (X,U) If Y C X, and quasi-uniform space with property P. (Y,Uy) 1s strongly complete then Y is closed in X. Let a PROOF. Y : U E - l. Y. Now U[a] n U} forms a filter base on Y. ter on Y generated by B. If u u, E Y f 0, and B = { U[a]n Let F denote the fil- n then v = u y Since u is a neighborhood of A, there exists 0 s t a f 0 and 0 x 0 CU. Now 0 n y f ,0 and 0 n y X X y E uy. such that 0 f1 y c v. 32 Let b Y. 0 (\ Y. E: Then V [b] n ~ 0 Y and hence F is Cauchy on Since Y is strongly complete there exists c s y such that c s lim F. Then c c lim F' Let F' be the filter on X generated by F. and a s lim F' have that a= c s Y. DEFINITION 3. will say that Hence Y is closed in X.// Let (X,U) be a quasi-uniform space. (X,U) lection { V[x] and since X is Hausdorff we We has property S if for each x, the col- : V s U, V is symmetric} forms a fundamental system of neighborhoods for x with respect to the topology generated by U. If U has The following question arises naturally. -l property P, then does U have property P? Since this type of question will be of interest later, we make the followlng definition. DEFINITION 4. A property Q will be called a quasi-con- jugate invariant if a quasi-uniform structure has ;?roperty Ll Q implies that u-l also has property Q. The space considered in example 2, Chapter III, clearly has property P since its topology is discrete. suppose u~ 1 is a neighborhood of product conjugate topology. 2 c: with respect to the exi~ts 0 F tu-l with ox o c u- 1 • Thus taere exists positive integer -'' which Hence (2,k) s u r: such that {2,k,k+l," .. } C 0. 0 and k Then there 6 _l U ~' Now ;) ls impossible. Therefore U-l does not have property P, that is property P is not a quasi-conjugate invariant. THEOREH 3. Let (X,U) be a quasi-uniform space satis- 33 Then U- 1 satisfies property P. fying properties P and S. Let PROOF. u- 1 E U-1 Since U has property P, we • have that for each x c X there exist V(x) V(x) [x] V(x) [x] X each x c X. c u. Hence V(x) [x] for each x is symmetric i t follows that T(x) U { T (x) [x] : x s x} U such that V(x) [x] c u- 1 for By property S there exists a symmetric T(x) U such that T(x) [x] C V(x) [x] Thus X s c u- 1 • x x c T (x) [x] E U-1 X}C E Since T(x) X. for each x U { V(x) [x] Hence u- 1 is a neighborhood of A. X. f x V(x) [x] with res- pect to the product conjugate topology and therefore u- 1 has property P.// COROLLARY. Let (X,U) be a quasi-uniform satisfies properties P and S then u- 1 s~ace. If l/ is compatible with a uniform structure. The result lS an immediate consequence of PROOF. theorems l and 3.// Theorem 1.47 in [16] shows that if tu is weaker than the conjugate topology then the conjugate topology is uniformizable . . U Slnce = (U-l)-1 it follows that if tu is stronger than the conjugate topology, then tu is uniformizable. THEOREM 4. Let (X,U) be a quasi-uniform space. 1 (U- ) has property S then tu_1 PROOF. (tu) lS uniformizable. Suppose U has property S. than the conjugate topology, If U Then tu is weaker for let 0 c tu and x c 0. Since U satisfies property S we have that there exists a 34 symmetric V U such that x E E V[x] C 0. Hence V E U -l Then by theorem 1.47 in [16] we have that t V E t E u_-l U-1 with that V With X X E U and thus 0 E quently Then there exists a symmetric 0. V[x ] C 0. E u_-1 Now suppose that u-l has property s and is uniformizable. let 0 and E t u Since V is symmetric we have . ~ Therefore tu._ 1 tu. and conse- we have that tu. is uniformizable.// If COROLLARY 1. (X,t) is T 3 then it has a uniformiz- able conjugate topology. PROOF. By theorem 3.17 in [16] U, the Pervin quasi- uniform structure, has property S. tu._ 1 is uniformizable by theorem 4.// Let (X,U) be a quasi-uniform space such COROLLARY 2 • that U and U -l have property S. Then t (,(_ t and t u- 1 u is uniformizable. PROOF. since u- 1 Since U has property S we have tu. has property S we obtain tu.-1 ~ and thus t,u. = t u V t u-l = t u v u_--l • tu.. ~ tu._ 1 and Hence tu_ = Therefore tu. is uniformizable.// B. -1 U 1\ U DEFINITION 5. on X. Then U ~ THEOREM 5. on X. V = Let U and V be quasi-uniform structures { U Let U and V be quasi-uniform structures If U A V is a quasi-uniform structure on X then U A V is the greatest lower bound of the quasi-uniform 35 structures U and V. It is clear that U A V PROOF. U and U A V ~ V. ~ Now suppose that S is a quasi-uniform structure such that S and S :;: V. It s If S, then s E ul well known that if lS ul tures for X then ture. u2 " u n v. ~ Hence S ~ U U A V.// and U;o are quasi-uniform struc- need not be a quasi-uniform struc- The following example shows that even u ;\ u-J need not be a quasi-uniform structure. Let X denote the natural numbers and set EXAMPLE 3. Un = { (x,y) : x = y or x = 1 andy~ n}. Let U be the quasi-uniform structure generated by { U : n = n J ,), ... }. Now U = { (X 1 y) : = or y E E E Now (t,J) E U A U = l and c 4 } -l c V such that EX such that u-l there exists m and U which is inpossible. j_' U 1\ U- (J,k) (l,t+J) t: _:::. 1 11 • V c V r; V C X such that F . .:. 14 anr-t therefore V Hence U A U- <:;' ,_,lnce U. for each k Then t Lett= max{m,n}. V for each k .: :. m. v cu. E V U there exists n Similarly, since V X and x .: :. Suppose there exists V = y Or X ~ n. (k,l) since (t,t+!) does not form a quasi-uniform structure. If LEMMA l. U ~ u- 1 u A u-l is a quasi-uniform structure then is a uniform structure. Then 0 PROOF. u- l . Also, U E U -l -l and thus U E [; F . u and hence Therefore u- 1 E 36 DEFINITION 6. A quasi-uniform space have property * if for each symmetric U symmetric V U such that V o V CU. E (X,U) is said to U there exists a E It is apparent that each uniform structure possesses * property Not every quasi-uniform structure has property * as the space in example 1 demonstrates. It is clear that * is a quasi-conjugate invariant property. property an easy exercise to show that property It is * does not imply and lS not implied by any of the usual separation properties. However, as the next theorem shows it does characterize u- 1 those structures U for which U A is a quasi-uniform structure. Let (X,U) THEOREM 6. U A u- 1 be a quasi-uniform structure. is a quasi-uniform structure if and only if (X,U) satisfies property *· Suppose (X,U) PROOF. u " u-1 $5 and if u =t= and v::> u implies v Now U, v u (\ v u ~ u- 1 E u 1\ u E E E u ;\ u-1 u and v u /\ E U and T is symmetric. a symmetric v ric, We haVe V u- 1 • E impiies u nv Let u E E /\ u-1. u and u E u- 1 u-1 Then f_: u 1\ u-1 u- 1 • u 1\ n v E u- 1 • Hence v E E u' and T 'l'hus = By hypothesis there exists Since V is symmet- U such that V o V CT. E Now u then u ::> [',. u u-1 • Clearly satisfies property *. and thUS V E LJ !\ LJ- 1 • Hence U !\ u-l lS a quasi-uniform structure. Now suppose that U Let U E A u- 1 is a quasi-uniform structure. U and U be symmetric. Then u E u A u- 1 and there 37 u- 1 exists V s U A v s U A such that V o V Cu. u- 1 • Hence V- 1 u- 1 • Let S s and more over (X,U) Hence v n v- 1 v- 1 • s = o (V u- 1 , an ct th ere f ore v - s 1 s Then S s U and S is symmetric n v- 1 ) (v n v- 1 ) c 0 v v c u. o A U2 is a quasi-uniform structure, and denote the topology i t generates by t. 1 U and s Let U1 and U 2 be quasi-uniform structures Suppose U1 where t v satisfies property *.// THEOREM 7. on X. u, cG Now and t Then t = t 1 t 1\ 2 , are the topologies generated by U1 and U 2 2 respectively. Then there exists U s U 1 PROOF. and V s U 2 such that x s U[x] C 0 and x s V[x] C 0. Therefore 0 s t. U U V s U 1 1\ U 2 and x s (U U V) [x] C 0. Now suppose x s 0 Then there exists U s U 1 that x s U[x] C 0 E t 1 and 0 E t. E Since u 0. t E and hence 0 2 u1 and u E t 1 A t Then E 2 • ~ U2 such U2, we have that Therefore, t = L2t G denote the family of all quasi-uni- COROLLARY. form structures which generate the topology t on the set X. Then if U1 and U2 are in G and U1 A U2 is a quasi-uniform structure, then u1 Let THEOREM 8. fying property ( i) A u2 E (X,U) G. be a quasi-uniform space satis- * If U A U- 1 generates t u. then t u.- 1 is uniformiz- able. (ii) If U A U- 1 generates tu._ 1 then tu. is uniforrniz- 38 able. PROOF. ( i) ~ By hypothesis tu tu_ 1 and from theorem 1.47 in [16] we have that tu-1 is uniformizable. hypothesis tu-1 ~ ( ii) . By tu and by the remark following theorem 3 we have that tu is uniformizable.ll THEOREM 9. Let (X,U) be a quasi-uniform structure. U V u-l generates t if and only if there exists a compatu ible uniformity stronger than U. PROOF. The necessity is obvious. a compatible uniform structure E u-l then u I E u and thus u-1 Now let t denote the Suppose there exists v such that E v. u v . Let u -l u -< u v u-1 ~ v. u v u-1 . Then ~ Hence generated by topolog~ Hence U V u- 1 generate Let THEOREJVI 10. fying property *· (X,U) be a quasi-uniform space satis- Then there exists a weaker compatible un- iform structure if and only if U A u-l generates tu. patible uniform structure such that V V and hence V have that V-] E V E u-l. A u-~. U and V Suppose V is a com- The sufficiency is clear. PROOF. C Thus v c U E ~ If V U. U and V_l E U. E V, we That is, A u- 1 and we have that Denote the topo 1 ogy genera t e d b y U 1\ V ~ U U-l by t. since V ~ U A u- 1 ~ U and the hypothesis u Thus L1 1\ U- 1 generates tu. II that V generates t . u Then t C. u ~ t ~ t SATURATED QUASI-UNIFORM SPACES 39 DEFINITION 7. A topological space (X,t) is called sat- urated if for each x s X there exists a minimal open set containing x. That is, for each x s X there exists an open set Ox containing x such that if 0 s t and x s 0, then Ox C 0. It is clear that a topological space (X,t) is saturated if and only if every intersection of open sets is open. DEFINITION 8. A quasi-uniform space u : u n{ quasi-saturated if E U} is called u. E A quasi-uniform space LEivWlA 2. (X,U) (X,U) is quasi-satu- rated if and only if there exists a unique base for U con- sisting of a single set. PROOF. E U} • The sufficiency is clear. By hypoth2sis V s U. show that V o V C v. = {V} . n{ U : U It suffices to Since V s U there exists U s U such But V C U and thus V o V c U o U C V. that U o U C. V . Hence B is a base for U. LEMMA 3. Set B Let V = If (X,U) The uniqueness is obvious.// is quasi-saturated then tu is a saturated topology. PROOF. lemma 2· W[x] Let = [W} be the base for Let x s 0, where 0 s t :.L . U developed in Then x s W[x] C 0, and is the rninimal open set containing the point x.// THEOREM 11. lS B If (X,U) is quasi-saturated then (X,U) strongly complete. PROOF. Let F be a Cauchy filter. base for U consisting of a single set. Let B = {W} be the Then, since F is 40 Cauchy, there exists x each U x E ~ U we have U[x] E X such that W[x] E W[x] 1 E F. Hence for and therefore U[x] F and E lim F.// THEOREM 12. space. Let (XIt) be a saturated topological Then there exists a strongly complete compatible quasi-saturated quasi-uniform structure. PROOF. x. Set W Let Ox denote the minimal open set containing = { (x y) : y Ox 1 x E 1 X} . E Then {\'J} forms a base for a quasi-uniform structure which we will denote by u. w. It suffices to show that W o W c (b 1 c) Then b W. E Oa and c E E Ob. Let (a 1 b) Since Ob is the mini- mal open set containing b we have that c (a 1 c) If 0 VV. E Thus t W [x] • E t and X -< t u . ~ C 0 and hence tu If 0 E E 0 then X tu_ and X E E E ObC Oa· Hence = OX C 01 but OX 0 then X By theorem 11 (X 1 U) t. W and E E = W[x] OX is strongly com- plete . / / Fletcher has shown in [7] that if a topological space has property Q then i t admits a compatible strongly complete Fletcher's theorem is a clear gen- quasi-uniform structure. eralization of theorem 12. The following example shows that metacompact and paracompact are not necessary conditions for a topological space to admit a strongly complete structure. Let N denote the natural numbers and t EXAMPLE 4. {N 1 ~ 1 { 1} 1 { l ical space. 17} 1 { 1 121 Set Oi 3} 1 = {1 • • •} • 1 2 (N 1 t) 1 ••• 1 i} 1 = is a saturated topologthen { Oi : i = 1 1 2 1 • •• } 41 is an open cover for (N,t). refinement of { Oi: i = 1,2 ••• }. Let { Qa : a s Q} be an open Suppose that is con- 1 tained in only a finite number of the Oa, say Qa , ... , Qa . +N and Clearly each Q,~ "' mum element, say r. n 1 n further the set U Qa. has a maxi- Now r 1 l is contained in some QB + Qa· l fori= 1, ••• ,n. o6 Then {1,2, ••• ,r} C Q , that is 1 s 6 which is a contradiction. Hence (~,t) is a saturated space that is not metacompact. THEOREM 13. Quasi-saturated is a quasi-conjugate in- variant. PROOF. Suppose (X,U) is a quasi-saturated quasi-uni- Then there exists W s U such that if U s form structure. It is clear that {W- 1 } U then W CU. and hence by lemma 2, Let THEOREH 14. u- 1 is quasi-saturated. I I (X,U) be a quasi-saturated quasi- uniform space with a base {W}. open set in tu containing x. Let Ox denote the minimal Then W is a neighborhood of that is, U has property P if and only if W = U { Ox /',.; PROOF. The sufficiency is evident. borhood of /',. then then b X u- 1 forms a base for E E W (a] C ~-J 0a • ::::> U { Ox Hence x Ox : x (a,b) E Oa x If W is a neighIf (a,b) s X}. X oac U{ Ox X E It¥, Ox: X}.// THEOREM 15. structures on X. PROOF. Let U and V be compatible quasi-uniform If V is quasi-saturated then U Let U s U and {V} the base for V. ~ If V. (x,y) s 42 V, then y E V[x] C U[x]. This follows since V[x] smallest open set containing x. have V C U and thus U COROLLARY. If E Hence (x,y) E is the U and we V.// (X,t) is saturated then there exists one and only one compatible quasi-saturated quasi-uniform structure. PROOF. The result is an immediate consequence of theorems ll and 15.// Fletcher [6] working independently has obtained slmllar results for finite topological spaces. D. 0-COMPLETE DEFINITION 9. F will be called an open filter A filter if it has a base consisting of open sets. DEFINITION 10. Let (X,U) be a quasi-uniform space. (X,U) will be called o-complete if every open Cauchy filter (X,U) will be called strongly has a nonempty adherence. o-complete if every open Cauchy filter has a nonempty limit. DEFINITION 11. A topological space (X,t) is called generalized absolutely closed if every open cover { 0 has a finite subcollection Oa , ... ,0 l DEFINITION 12. (X,U) each u A filter an a } such that X= F in a quasi-uniform space is said to contain arbitrarily small open sets if for E U there exists x C U[x] and Ox E F. E X and Ox E tu such that x E Ox 43 It is clear that if (X,U) plete), is complete, then it is o-cornplete, LEMJI.1A 4. (strongly com- (strongly o-cornplete). Let (X,U) be a quasi-uniform space such that every Cauchy filter contains arbitrarily small open sets. Then (X,U) is strongly o-cornplete implies that (X,U) is strongly complete. PROOF. = { 0 : 0 Let F be a Cauchy filter and let B s F and 0 is open}. Let G be the filter generated by B. Since F contains arbitrarily small open sets we have that G is an open Cauchy filter. The result follows from the fact that lim G = lim F.// It is an easy observation that if (X,t) is a general- ized absolutely closed topological space and U is any cornpatible quasi-uniform structure then (X,U) is o-cornplete. This is true since in a generalized absolutely closed space every open filter has a nonempty adherence. If LEMMA 5. (X,U) is pre-compact then every open ul- trafilter is Cauchy. PROOF. Let M be CJ.r open ultrafilter on X; that is, 1\l is a maximal element in the class of all open filters on X. Let U E U, then there exists V E U such that V Since U is pre-compact there exists x 1 , ••• o V CU. ,Xn s X such that n i 1,2, •.• ,n. Hence u oi =X and so there exists ok E u 1 since M is an open ultrafilter. Consequently Now U[xk] ~ M is Cauchy.// Ok and there- 44 LEMMA 6. Let (X,U) is pre-compact and (X,U) be a quasi-uniform space. o~complete If then X is generalized absolutely closed. PROOF. It suffices to show that every open ultrafil- ter has a nonempty adherence. Let M be an open ultrafilter, then since U is pre-compact M is Cauchy by lemma 5. Since is a-complete we have that adh A1 =f fJ.// (X,U) Let (X,U) THEOREM 16. be a uniform space. generalized absolutely closed if and only if (X,U) (X,U) is is pre- compact and a-complete. Lemma 3 is a generalization of the sufficiency. PROOF. Suppose (X,U) is generalized absolutely closed, then (X,U) is a-complete by a previous comment. is pre-compact. (X,U) We now show that Let U s U, then there exists a sym- metric V s U such that V o V CU. a neighborhood cover of X. Now { V[x] Since X is generalized absolute- ly closed we have that there exists x 1 , ••• ,xn s X such that n l z f U V[xi]. X <: V[y] n : x s X} is V[xi]. 'I' hen ( y, z) E V is symmetric we have that (xi,z) fj. Let s V and since V 1 s V and (z,y) s V. Thus n (xi,y) s V o v C U or y s U[xi]. therefore (X,U) Hence X = U U[xi] and l is pre-compact.// Fletcher and Naimpally [10] working independently obtained analogous results to those found in this section. They call a-complete, almost complete and generalized absolutely closed, almost compact. They defined a quasi-un- 45 iform space to be almost pre-compact if for each u c (X,U) U there exists x , ••• 1 ,xn c X such that X= a U[x.J. l Using l these definitions they obtained the following generalization to lemma 6. A topological space is almost-compact if and only if every compatible quasi-uniform structure is almost complete and almost pre-compact. E. q-T. l SEPARATION AXIOMS Let (X,U) be a quasi-uniform space. only if given x x ~ U [ y) f if and 0 y there exists U c U such that either or y ~ U [ x] • ¢ U[y] U[y] = ~- ~ andy f larly, X is T 2 if and only if given x n f X is T 1 if and only if given x there exists U c U such that x U such that U[x] X is t U[x]. y Simi- y there exists u c The following example shows that this characterization does not carry over for an arbitrary T quasi-uniform space. 3 EXAMPLE 5. Let X denote the natural numbers and let U be the quasi-uniform structure generated by the base consisting of all sets of the form Un = x n}, for n c X. ~ { (x,y) Now tu is the discrete topology, so F = {2,4,6, ... } is closed in X. U [2n) =X, so U[F] =X. n If U c U, Thus DEFINITION 13. 3 if given x that U[x] n U[F] ¢ there exists (X,U) there does not exist U c U such that U[l] q-T : x = y or ~ A quasi-uniform space is T 3 U[F] (X,U) = , but ~- is called F, F closed, there exists U c U such = ~- 46 It is clear that if a space is g-T 3 then it is T 3 Example 1 showed that a space can be T but not q-T 3 will show that every uniform space is q-T 3 THEOREM 17. Then (X,U) Let is q-T 3 PROOF. (X,U) (V • be a R 3 quasi-uniform space. ~ Let F be closed in X and x F. n U such that U[x] E Since X - F F = 0. Since is R 3 there exists a symmetric V s U such that o V) [x] C exists f U [x] • Suppose a s V[x] F such that (f,a) s is symmetric we have (V o V) [x] C U[x]. F we • • is open there exists U (X,U) 3 • = 0. Hence V[x] COROLLARY. (x,a) s V and V[F]. (x,a) v. s But this is impossible since U[x] n V[F] V. Since V s s (a,£) Then there Thus f n V and n s = 0.// Every uniform space is q-T 3 DEFINITION 14. • A quasi-uniform space is called g-T 4 if for every pair of disjoint closed sets F and G there exists U s U such that U[F] n U[G] = 0. It is clear that every q-T 4 space every g-T 4 + T 0 space is a q-T 3 T 4 and moreover lS space; that is, quasi- normal implies quasi-regular. Let (X,U) LEMl"iA 7. F closed in X. PROOF. Then be a q-T (F,UF) 4 quasi-uniform space and is a q-T 4 space. Let G and H be closed disjoint subsets of F, then G and H are closed and disjoint in X. q-T4 there exists u v = u n F X F. Then u E v E such that U[G] UF and V[G] n n Since U[H] V[H] = 0- = 0- (X,U) Let Hence lS 47 A space can be T 4 and not q-T 4 ; consider the space 1n = example 5 with F = {2,4, ... } and G DEFINITION 15. {1,3, ... }. A quasi-uniform space (X,U) is called q-T 5 if for every pair of separated sets F and G there exists U s U such that U[F] 0 = 0. U[G] Clearly every q-T 5 space is T 5 1 is T 5 but not q-T 5 • The space 1n example • It is also evident that each space is a q-T 4 space. LEMMA 8. If (X,U) is q-T 5 then every subspace is q-Ts. PROOF. Now (clX F) Let Y C X and F and G separated in (Y,Uy). n we have that G n ~ Y ely F. (ely F) (J G Y n (ely F) Since F and G are separated 1n Y G = 0. 0. ~ n Now (clX F) Similarly n (clX G) n G C (clX F) F = 0. Thus F and G are separated in X and hence there exists U s U n such that U[G) Uy and V[G] n = 0. U[F] V[F) THEOREM 18. = 0. Let Let Hence (X,t) v = u (Y,Uy) n y (X,U) be a topological space and U If For 1 = 3 Let F, G be disjoint closed sets in X. x 0 U (X,t) is T. , l the result follows by theorem 17, since the Pervin structure of a T 3 space is R 3 0 E is q-Ti for 1 PROOF. 0, Q s t v Then is q-T 5 . / / the Pervin quasi-uniform structure for t. then Y. X such that F C 0, G C Q and 0 (X-O) x X and T = Q x Q U • = Let i Then there exists n (X-Q) Q = 0. x X. Let S Then set ~ 4. 48 u n s [0 Now U[F] (X, U) T u 0 X Q X = 0 and U[G] q-T 4 lS Q] u [X - Q) = Q, hence U[F] The proof for i • u (0 = 5 n X X] U[G] Hence follows in the same manner.// LEI1l'1A 9. (X,U) be a saturated quasi-uniform space. Let If X is Ti then i t is q-Ti f o r i = 3,4,5. PROOF. If U is quasi-saturated then there exists a w. base consisting of a single set F where F is closed. such that X W[F] C: Q. E ~ Then there exists open sets 0 and Q n 0, F C Q, and 0 (X,U) Hence Suppose X is T 3 and x is q-T 3 Q = ~. Now W[x] C 0 and The proofs for q-T 4 and • q-T 5 follow in a similar manner.// Let X denote the natural numbers. EXAMPLE 6. Let U be the uniform structure generated by the base consisting of the sets Un = { (x,y) : x = y or x ~nand y ~ topology generated is discrete and hence T 5 and T 4 is clearly not q-T 4 and hence not q-T 5 a uniform space may be T 4 Let THEOREM 19. Then space. (X,U) that for each u U exists x ly x i E U} ~ adh B. F ()G. (X, U) (q-T 5 ) . be a compact q-T 3 quasi-uniform is q-T 4 . u we have U[F] is a filter base. E The Thus we see that (T 5 ) and not q-T 4 (X,U) • Let F and G be disjoint closed sets. PROOF. U [G] • n}. n U[G] + ~- B = { Suppose U [F] () Since X is compact there We may suppose that x ~ F, since clear- By hypothesis X is q-T 3 so there exists 49 U E U such that U[x] = ~' n U[F] ~- = n Then U[x] but this is impossible since x c adh B. (U[F] n U[G]) Therefore is q-T 4 • I I (X' u) COROLLARY. Then (X,U) Let (X,U) be a compact uniform space. is q-T 4 • The following characterization of q-T 3 seems complicated but it proves a helpful tool in the following theorem. (X,U) L E i''L.'VlA 1 0 • is q-T 3 if and only if for each open set 0 and x c 0 there exists U c U such that U[x] C X - U[X- 0]. PROOF. The proof follows immediately from the defini- tion of q-T 3 .11 THEOREf--1 2 0 • is q-T3 for each l . PROOF. X 0 where 0 E E l = n ul.. Sufficiency. is x.l except for i For each i = i n.o. c o . l and 0·l u u.1 1 , ••• = i + E U[x] n that y c U[a]. t If a t ik and Then there exists a c k Let U u. E E = TIU. l Suppose X - 0 such 0 then there exists ik such that c U1· [x.1 ] and y 1· c U[ai ] . Thus Yik c k k k k k U· [x· ] C X· - U. [X. - 0 1· ] . But Y· c U[a. ] CU. lk lk lk lk lk k lk lk lk Therefore U[x] n U[X - 0] [Xi - o. ] which is impossible. k lk = ~ and hence U[x) C X - U[X - 0] and by lemma 10 we have 0 1· k • Thus y 1· U[X - 0]. 1 ' ••• ' ,ik there exists Ui such that u.1 [x·1 ] c: Xi - u.1 [X.1 - oi ] . k k k k k k k where ui =xi X xi for l il, ... ,ik. Then u that y Let Then there exists rrioi where lS open in X. lS open in t·l 0·l X Let X = nxi and 50 that (X,U) q-T 3 lS Necessity. • Let xi s Oi, where Oi is open in Xi. Let x be any point in X with the ith coordinate equal to xi. Then there exists Us U such that U[x] C X- U[XThere exists nkuk C U where uk = Xk Xk except for x ni 1 (Oi)] ~ fi- nite number of indices and Uk s Uk for each k. Then Suppose y. l Ui[xi] n Ui[Xi - Oi]. that (a.,y.) (yk) s l l -1 (Oi) each k (a.) J J +i J J J Now (a,y) Then a s X- s nkuk since for = (yk,yk) s Uk and for k = (ak,yk) Therefore Then y = (rrkuk) [X- TTi (Oi)]. + i. by a. = y. for each j + i. _l and therefore y s X- since ai s Xi- oi. (a.,y.) l l (Oi). Then there exists ai s Xi - Oi such Choosey,= x. for all j l (ITJ.;.Ykl [x] Define a= 11i s u .. s l we have EX- TT.-1 (a,y) l 1 Hence y s rrkuk[X- TTi (Oi)] which is a contra- diction. Therefore U l · [x l·] n U l· [X·l - Thus Ui [xi] C Xi - Ui [Xi -- Oi] each factor spac~ 0 l·] = ~. and by lemma 10 we have that is q-T3.// It is easy to see that the product of a family of a family of q-T 4 spaces need not be q-T 4 • family of T 4 topological spaces such that Let (Xa,ta) be a (rraxa,nata) is not 51 Let U rl' 4 • a be the Pervin structure associated with t a . (X a' Ua ) is q-T by theorem 18. 4 Then each factor space Hence if n a Ua were c-T 4 then n a t a would be T 4 which is impossible. j_ F. A COUNTER-EXAMPLE IN DEFINITION 16. spaces. Y. Let (F,0)J (X,U) and (Y,V) be quasi-uniform Let F denote the set of all functions from X to For each V s V we let W ( V) = { (f , g ) s F F : x ( f ( x) , g ( x) ) The collection of all such W(V) V s for each x s X} • forms a base for a quasi- uniform structure which we denote by W. is then a (F,(v) quasi-uniform space and W is called the quasi-unifor~ structure of quasi-uniform convergence. The following example shows that neither the set of all continuous mappings from X to Y nor the set of all quasi-uniformly continuous mappings from (Y,V) Let X and Y denote the integers. EXANPLE 7. Set 8 to (F,W). need be closed in u n = { (x,y) (X,U) c~ X = { Un: n = X x : 1,~, y or x = x l andy ~ Let n}. Then G is a base for a quasl- . . . }. uniform structure on X, which we will denote by U. 'i'he set of all vn where n = = { (x,y) 1,2, ••• , s Y x Y : = x y or x .:.: n} forms a base for a quasi-uniform struc- ture for Y, which we will denote by V. is discrete. Define f : (X,LI) ~)- (Y,V) The topology on Y by f(x) = x for each 52 X E The function f is not continuous since f- 1 (1) X. {1} ~ tx, and hence not quasi-uniformly continuous. For each natural number n, define fn : fn(x) (X,U) ~ (Y,V) by, = x for x < n = 2: n. 1 for x We will show that each fn is quasi-uniformly continuous. Let fn and Vm E V be given. (a,b) E Unthen fn(b) and thus It suffices to show that if (fn(a),fn(b)) (fn(a) ,fn(b)) EVm. E Vm. If a= b, then fn(a) If a + b, then a and b .:::. n. = 1 Thus each fn is quasi-uniformly continuous and by theorem 1.24, in [16] i t is continuous. Let F denote the set of all functions from X to Y and W the quasi-uniform structure of quasi-uniform convergence. Let U C F denote the set of all quasi-uniformly continuous functions from (X,U) to (Y,V), and let C c F denote the col- lection of all continuous functions from (X,U) to We have shown that each fn E that f c: Consider s U. U C Let Vn be given. (f(x),fn(x)); i f x < C, and f ~ C. (Y,V). We now show We claim that fn E W(Vn) [f] nthen (f(x),fn(x)) Vn' and if x.::: n then (f(x) ,fn(x)) = (x,1) E Vn. = (x,x) Hence fn E W(Vn) [f] and since Vn was arbitrary, we have that f IT c c. ~ But f C. c: Hence neither U nor C is closed in (F,W) 53 V. SUMMARY, CONCLUSIONS AND FURTHER PROBLEMS We noted that a quasi-uniform space need not have a T1 or T 2 strong completion nor a T 2 completion. It is of interest to know if an arbitrary space has a T 1 completion. Our construction showed that for certain classes a T1 completion exists. No one as yet has exhibited a topological space that does not admit a complete or strongly complete quasi-uniform structure. Our example demon- strates that a space can be uniformizable, admit a strongly complete structure and not admit a complete uniform structure. It seems that the present definition of Cauchy filter may admit too many filters. ample 2, we saw that F In Chapter III, ex- = {X} was a Cauchy filter. Since the topology on X is discrete this seems a bit unreasonable. Thus the study of other classes of "Cauchy" filters would appear to be worthwhile. Since Section F, Chapter III showed that regardless of the definition of "Cauchy" filter and "completeness" not all of the pleasant properties of the completion of a Hausdorff uniform space could be preserved for a quasi-uniform space, it seems logical for one to decide which of these properties should be preserved, if possible, before a new definition of "Cauchy" filter is proposed. On the other hand it can be noted that for special classes the completion constructed in Chapter III, Section 54 B, has many of the desired properties. In Chapter IV several topics were considered. i'1any other areas in quasi-uniform spaces also remain open to investigation. No one has as yet characterized the uni- versal quasi-uniform structure associated with a given topology, and no necessary and sufficient conditions are known for when a topology admits a minimal quasi-uniform structure. gl'ven for Necessary and sufficient conditions were u 1\ u- 1 to be a quas1-un1 . 'f orm structure. showed that U and C need not be closed in Naimpally [17] showed that if (Y,V) (F,W) We and 1s T 3 and V is the Pervin structure then U and C are closed in (F,W). More general conditions to insure that U and C be closed or complete seem desirable. The q-T. separation properties seem to this author l to at least be an interesting concept when one is more concerned about the particular quasi-uniform structure than the generated topology. 55 BIBLIOGRAPHY l. Barnhill, C. and Fletcher, P., Topological Spaces with a Unique Compatible Quasi-uniform Structure, Archiv der Math., to appear. 2. Carlson, J. W. and Hicks, T. L., On Completeness in a Quasi-uniform Space, J. Math. Anal. Appl., to appear. 3. Csaszar, A., Fondements de la Topologie Generale, Gauthier-Villars, Paris, (1960). 4. Davis, A. S., Indexed Systems of Neighborhoods for General Topological Spaces, Amer. Math. Monthly 68 (1961)' 886-893. 5. Dieudonne, J., Un Exemple d'espace Normal Non Susceptible d'une Structure Uniform d'espace Complete, C. R. Acad. Sci. Paris 209 (1939), 145-147. 6. Fletcher, P., Finite Topological Spaces and Quasi-uniform Structures, Canad. Math. Bull. 12 (1969), 771-775. 7. Fletcher, P., On Completeness of Quasi-uniform Spaces, submitted. 8. Fletcher, P., Note on Covering Quasi-unifor~lties and Hausdorff Measures, submitted. 9. Fletcher, P., Homeomorphism Groups with the Topology of Quasi-uniform Convergence, submitted. 10. Fletcher, P., and Naimpally, S. A., Almost Complete and Almost Precompact Quasi·-uniform Spaces, submitted. 11. Gaal, s. A., Point Set Topology, New York and London. 12. Gillman, L. and Jerison, M., Rings of Continuous Functions, (1960), D. Van Nostrand, Princeton, New Jersey. 13. Isbell, J., Math. Rev. 22 14. Krishnan, v. s., A Note on Semi-uniform Spaces, J. Madras Univ. B 25 (1955), 123-124. 15. Liu, (1964), Academic Press, (1961), 683. c., Absolutely Closed Spaces, Trans. Amer. Math. Soc., 130 (1968), 86-104. 56 16. Murdeshwar, M. G. and Naimpally, S. A., Quasi-uniform Topological Spaces, Noordhoff Series A, (1966). 17. Naimpally, S. A., Function Spaces of Quasi-uniform Spaces, Indag. Math. (Akad. Wetenshcappen) 68 (1965) 1 768-771. 18. Niemytzki, V. and Tychonoff, A., Beweis des Satzes, das ein Metrisierbarer Raum dann und nui dann Kompact ist, Wenn er in Jeder Metric Vollstandig ist, Fund. Math. 12 (1928), 118-120. 19. Pervin, W. J. Quasi-uniformization of Topological Spaces, Math. Ann. 14 7 (19 6 2) , 316-317 . 20. Sieber, J. L. and Pervin, W. J., Completeness in Quasiuniform Spaces, Math. Ann. 158 (1965), 79-81. 21. Sian, M. and Willmott, R. c., Hausdorff Measures on Abstract Spaces, Trans. Amer. Math. Soc., 123 (1966) 275-309. 1 22. Stoltenberg, R. A., A Completion for a Quasi-uniform Space, Proc. Amer. Math. Soc., 18 (1967), 864-867. 23. Weil, A., Sur les Espaces a Structure Uniforme et sur la Topologie Generale, Gauthier-Villars, Paris, (1937). 57 VI'l'A John Warnock Carlson was born on November 10, 1940 at Topeka, Kansas. He graduated from Blue Valley High School at Randolph, Kansas in May 1959. He entered Kansas State University that fall, where he received his Bachelor of Science degree with a major in Mathematics in June 1963, and his Master of Science degree, with a major ln Mathematics and a minor in Statistics, in August 1964. From September 1964 until June 1967 he was employed as an Instructor of Mathematics at Washburn University at Topeka, Kansas. He attended two N.S.F. Summer Institutes in Computer Science. The first was during the suwmer of 1966 at the University of Missouri-Rolla, and the second during the summer of 1967 at Texas A & 1'1 University. Since September 1967, he has been employed as an Instructor of Mathematics at the University of Missouri-Rolla, where he has been continuing his graduate training. He was married to the former Sara Ann Hollinger on September 9, 1961. and David. They have two children, Catheryn