Acta Math. Hungar. 64 (1) (1994), 41-54.

D-COMPLETE EXTENSIONS OF QUASI-UNIFORM SPACES* A. CSs163

(Budapest), member of the Academy

0. I n t r o d u c t i o n . Let (x,/4) be a quasi-uniform space. A pair (t, s) of filters in X is said to be a Cauchy filter pair iff, for U C/4, there are T E t and S E s such that T • S C U. A filter s is said to be a D-filter (Cauchy filter in [8]-[12], D-Cauchy-filter in [13]) iff there exists a filter t such that (t,s) is a Cauchy filter pair. The space (X,/4) (or the quasi-uniformity/4) is said to be D-complete (complete in [8]-[12]) iff every D-filter converges in (X,/4) (i.e. with respect to the topology/4tp induced by/4). The papers [8]-[12] con'~ain constructions of D-complete extensions for some classes of quasi-uniform spaces. More precisely, quiet spaces are concerned in [8]-[10]; (X,/4) and /4 are said to be quiet iff, for U E /4, there exists U0 E/4 such that, if x,y E X and Uo(x) C s, U0--l(y) e t for some Cauchy filter pair (t,s), then (x,y) E U. In [11]-[12], the construction is defined for stable spaces where (X,/4) or/4 is said to be stable iff every D-filter is stable, and the filter s is said to be stable (see e.g. [2], p.126) iff, for every U ell, N{U(S):S E s} E s. The purpose of this paper is to present further constructions of Dcomplete extensions, permitting to establish the existence of such an extension for every quasi-uniform space (while special classes of spaces are considered in [7]-[12]). 1. D - c o m p l e t e s t r i c t e x t e n s i o n s . Let (X, 7") be a topological space, Y D X, and let us be given, for a E Y, a filter s(a) in X that is open (i.e. it is generated by a filter base composed of open sets) in the topology 7-; in particular, for a E X, let s(a) denote the T-neighbourhood filter of a. It is we]l-known (see e.g. [2], p.122) that there exist then topologies T ~on Y such that s(a) is the trace in X of the T'-neighbourhood filter v'(a) of a E Y (i.e. v ' ( a ) l X = s(a)), consequently T'IX = 7-. Among these topologies, there is a coarsest one, called the strict extension of T for the trace filter system {s(a): a E Y}; for this topology, the sets s(G) = {a E Y: G E s(a)}, where G is T-open, constitute a base. There is also a finest one, the loose extension, for which

v'(a)={SU{a}:Ses(a)}

(aeY).

* Research s u p p o r t e d by Hungarian National Foundation for Scientific Research, grant no. 2114.

42

A. CSASZiR

Consider now a quasi-uniform space (X,L/) and define T = L/tp. If we look for a quasi-uniformity L/' on Y compatible with the trace filter system (i.e. such that s ( a ) i s the trace in X of the L/'tp-neighbourhood filter of a E Y) and w i t h / 4 (i.e. satisfying L/'IX = L/), then it is necessary that every s(a) should be/4-round (i.e. S E s(a) has to imply the existence of So E s(a) and U E/4 such that U(So) C S) (see [2], 1.1). In the following, we understand by a trace filter system in a quasiuniform space (X,/4) a family S = {s(a):a E Y} of L/-round filters in X such that Y D X and s(a) is the L/tP-neighbourhood filter of a if a E X (observe that a/4tP-neighbourhood filter is always L/-round and a L/-round filter is necessarily/4tP-open). If 5(' is a quasi-uniformity on Y such that L/'IX = /4, and (t,s) is a filter pair in X, then, clearly, it is /4-Cauchy iff there is a L/'-Cauchy filter pair (t',s') satisfying t = t'lX, s = s'lX. Therefore, if we look for a D-complete extension /4', it is necessary that every D-filter in (X,/4) should be convergent for/4,tp i.e. it should be finer than some trace filter s(a). These considerations motivate the following terminology: a trace filter system {s(a):a E Y} in (X,/4) is said to be admissible iff every D-filter in (X,L/) is finer than some s(a). We would like to construct D-complete compatible extensions L/' for a given admissible trace filter system S. For this purpose, we need some lemmas. LEMMA 1.1. If (Y,T') is the strict extension of the topological space (X, 7 ) for the trace filter system {s(a):a E Y}, s' is a T'-open filter and s']X -+ a E Y , then s I --+ a. PROOF. For a q-open set G such that a E s(G), there is a T'-open set G' E s' satisfying G' N X C s(G). Then G' N X C s(G) MX = G, and b E G' implies G' N X E s ( b ) , G E s ( b ) , b E s(G), hence G' C s(G). [] For a filter s in the quasi-uniform space (X,L/), let us denote by L/(s) the U-envelope of s, i.e. the filter {U(S): S E s, U E/4}, coarser than s and L/round (see [2], 4.6). By/4-1 we denote the quasi-uniformity { u - l : U E L/}, by L/-tp its topology. LEMMA 1.2. If ( t , s ) i s a Cauchy filter pair in (X,L/), then (L/-l(t), /4(s)) is a Cauchy filter pair as well. Consequently every D-filter is finer than a round D-filter. [] THEOREM 1.3. Let S = {s(a):a E Y } be an admissible trace filter system for (X,/4). Suppose that/4' is an extension of L~ compatible with S such that (a) U 'tp is the strict extension of~4 tp, (b) X is dense both for N Itp and L/I-tp. Then/4~ is D-complete. Acta Mathemat~ca Hungarica 6~, 199~

D-COMPLETE EXTENSIONS OF QUASI-UNIFORM SPACES

43

PROOF. By 1.2, it suffices to show that s ~is convergent for/,/~tp whenever (t ~, s ~) is a U~-Cauchy filter pair such that s ~ is U~-round, t ~ is/4~-l-round. Then by (b) (t,s) is a/4-Cauchy filter pair for t = t ' l X , s = s'iX. The D-filter s in (X,/4) is coarser than some trace filter s(a), hence it converges for/4 ~tp, and the same is true for s ~ by 1.1. [] Let us say that an extension/4! of/4, compatible with the trace filter system {s(a): a E Y}, is uniformly strict (strict in [2]) iff, for U' 9 there is Ug 9 such that

s( UD(a) M X) c U'(a)

(a E Y).

Then clearly /4~tp is'the strict extension of/4tL [2] 6.2 and 7.3 furnish necessary and sufficient conditions (in fact, rather complicated ones) for the existence of a uniformly strict extension for a given trace filter system; in particular, [2] 7.3 and (6.2.7) show that X is /4'-tLdense whenever/4' is uniformly strict and the trace filters s(a) are D-filters. Thus we can state: COROLLARY 1.4. If S = {s(a): a 9 Y} is an admissible trace filter system in (X,/4) such that every s(a) is a D-filter, andLl ~ is a uniformly strict extension of~4 compatible with S, then/4~ is D-complete. [] COROLLARY 1.5 (J. Des If {s(a):a 9 Y} is an admissible trace filter system in (X,/4) such that every s(a) is a stable D-filter, then a D-complete, uniformly strict extension/4~(S), compatible with /4 and this system, is generated by the entourages (1.5.1)

ur= { (a,b):U(S) 9 s(b) for s ~ s(a)}

where U 9 PROOF. By 1.4 and [2], 6.2 and 6.3, a / 4 r satisfying these conditions is generated by the subbase composed of the entourages W(U) (U E/4) defined in the following manner. Suppose x, y E X, p, q 9 Y - X, and put (1.5.2)

~u(p) = A { u ( s ) : s 9 s(p)} 9 s(p),

(1.5.3)

(x,y) e w ( u )

ix (x,v) e u,

(1.5.4)

(p,x) 9 W(U)

iff x 9 au(p),

(1.5.5)

(x,p) 9 w ( u )

ix u(x) 9 ~(p),

(1.5.6)

(p,q) C W(U)

iff av(p) e s(q). Acta Mathematica Hungarica 6]~, 1994

44

i. CSiSZiR

We show that the entourages (1.5.1) generate /~s~(S) (they constitute a uniform base because U1 C U2 implies U~ C U~). This is contained in LEMMA 1.6 ([6], 1.7). / f S is a trace filter system in (X,H) composed of stable filters, then the entourages U' given by (1.5.1) and those W(U) defined by (1.5.2)-(1.5.6) generate the same quasi-uniformity H~(S). PROOF. It is sufficient to show U~ C W(U) and W(Uo) C U' whenever V, Vo e U, Vg c u. In fact, (x,y) E U~ implies Uo(Uo(x)) E s(y), y E U(x), and (x,y) E E W(Uo) implies y E Uo(x), hence y E Uo(V) for V E s(x), Uo(y) C U(V), v ( v ) e s(y), (x,y) e u'. Similarly (p,x) E U~ implies (p,x) E U', so U(S) E s(x) for S E s(p), x E av(p). Conversely, (p,x) E W(Uo) implies x E Uo(S) for S E s(p), Vo(x) c v ( s ) e s(x), (p,x) e g'. Further (x,p) E U~ implies Uo(Uo(x)) E s(p), U(x) E s(p), (x,p) E E W(U), and (x,p) E W(Uo)implies Uo(x) E s(p), Uo(S) E s(p) for S E e ~(x), (x,p) e u; c v'. Finally (p,q) E U~ implies Uo(S) E s(q) for S E s(p), hence U(S) D D Uo(Uo(S)),i.e. U(S) D Uo(T) for some T E s(q), consequently av(p) D D auo(q) E s(q), (p,q) E W(U). Conversely (p,q) E W(Uo)implies au0(P)E E s(q), hence Uo(S) E s(q) for S E s(p), (p,q) E U; C U'. [] Let us say that (X,/d) is a D-space iff every D-filter is finer than some stable D-filter. E.g. every D-complete space is a D-space because any neighbourhood filter is a stable D-filter:

N{U(S):x E intS} D U(x)

(U E/4),

and T • S C U i f T = {x}, S = U(x), so that (t,s) is a Cauchy filter pair if s is the neighbourhood filter of x and t = ~, where ~: = ,il for A = {x} and ;~ = { s c x: s ~ A}. Now we can state: THEOREM 1.7 (J. Des If (X,H) is a D-space, then there is an admissible trace filter system S composed of stable D-filters, and then b/~(S) is a D-complete extension ofl~, compatible with S. PROOF. 1.5 applies because, if s is a stable D-filter, then b/(s) is round, stable ([2], 4.6), and a D-filter by 1.2; in particular, the utp-neighbourhood filters are round, stable D-filters, so that it suffices to choose for {s(p):p E E Y - X} the collection of all non-Utp-convergent, round, stable D-filters

in (x,u).

[]

Instead of all these filters we may use a part of them provided every non-convergent D-filter is finer than one of the trace filters selected. Acta Mathematica Hungarica 64, 199~,

45

D-COMPLETE EXTENSIONS OF QUASI-UNIFORM SPACES

The hypotheses of 1.7 are fulfilled if (X,/4) is stable; then 1.7 furnishes the D-complete extension constructed in [11] (provided/4tp is To and S is reduced, i.e. x E X , p , q E Y - X , p ~ q i m p l y s(x) # s(p) # s(q)). However, a D-space need not be stable, even if it is D-complete: EXAMPLE 1.8. For x , y E R, ~ > 0, let (x,y) E U~ hold iff x = y or x < 0 < y < x + ~. T h e n clearly U} C U:~, hence {U~:c > 0} is a base for a quasi-uniformity/4 on R. Uv(x) = {x} if x > O, U~(x) = {x} U (0,x + ~) if x = < 0 ( w h e r e ( 0 , x + r U~implies either T = {x}, 5, C U~(x), or T N ( 0 , + o c ) = {x},. 5, = {x}, x < i n f T + + s , or T C ( - o c , 0 ] , 5' C [0,+oc), s u p s < i n f T + s. Consequently, for a Cauchy filter pair (t, s), either t = ~ and s --+ x, or s = 2 (and then s -~ x again), or [0, +cr E s and s ~ 0: (R,/4) is D-complete. However, if t =

> 0},

s =

> 0},

then (t, s) is a Cauchy filter pair, but s is not stable.

[]

EXAMPLE 1.9 (cf. [4], 0.7). If X = R - {0} and we consider the subspace on X of the space (it, U) of the previous example, then any D-filter coarser t h a n the D-filter s i x (using still the same notation) must coincide with it (because it is the trace of a filter that converges to 0 in UtP), and s i X is not stable: the subspace in question is not a D-space. [] A non-D-complete, non-stable D-space is presented by EXAMPLE 1.10 (cf. [5], 7.12). For X = R - {0} again, let (x,y) E U~ hold iffx = y or x < 0 < y and - x y < s. Then U~ = U~, and {U~:s > 0) is a base for a quasi-uniformity/4. For c > 0, U~(c) = {c}, U~(-c) = { - c ) U (0, ~). Hence, if 0 ~ T x 5' C C Ur t h e n either T = {x), 5" C U~(x), or T N ( 0 , + c o ) = {x}, 5' = {x}, or T C (-a,O), 5, C (O,b), ab < s. Thus any non-convergent D-filter s necessarily contains some b o u n d e d set A C (0,-4-oc), and then A is a stable D-filter coarser t h a n s. However, the Euclidean neighbourhood filter of c > 0 is a non-stable D-filter. []

THEOREM 1.11. I f ( X , U ) is a uniform space, and every S(a) is a round, Cauchy filter in the trace filter system S = {s(a):a E Y}, then U~(S) is a uniformity. PROOF. Observe that, in a uniform space, Cauchy filters are stable and coincide with the D-filters, so that the hypotheses of 1.5 are fulfilled. For U E / 4 , select a symmetric U1 E /4, U2 C U. Then U~ C U~-I; in fact, (a,b) E U~ implies U1(5') E s(b) for some 5, E s(a) satisfying 5' • 5" C U1. For any T E s(b), we have U1(5")N T E s(b), and x E 5", y E U1(5")N T imply (z,y) e U1 for some z 9 5,, hence x E Vl(z) C V l ( U I ( B ) ) C V(y),.

5" c U(UI(5")NT) C U(T) Es(a),consequently(b,a)E U'.

[]

Acta Mathematica Hungarica 64, 1994

46

.4. cshszhn

T h u s L/~(S) is the standard completion of a uniform space (cf. [1], p.256) provided the trace filter system S composed of all round Cauchy filters is reduced (cf. [8], T h e o r e m 1, [12], Theorem 1). 2. D - c o m p l e t e

l o o s e e x t e n s i o n s . Let S be a trace filter system in a quasi-uniform space (X,/4). It is known ([2], 4.8 and 6.1) that, in general, there is no compatible extension o f / t compatible with the strict extension of /4tp for S. However, there is always an extension compatible with the loose extension ([2], 2.2). In order to construct it (see [2], 2.1), let us denote by E the collection of all maps a: Y --+ exp X such that a(x) = {x} for x E X , a(p) E s(p) for p E Y - X, and let us keep the convention that a, b, c denote points of Y, x,y,z belong to X , and p,q,r belong to Y - X. For U E/4, a E E, let W(U,a) be the entourage on Y defined by (2.1) (2.2)

(x,y) e W(U,a) (p,x) E W(U,a)

(2.z) (2.4)

iff iff

(x,y) E U, x E U(a(p)),

(x,p) r w(u,o), (p,q) E

W(U,a)

iff p = q.

T h e n { W(U,a):U E / / , a E X} is a quasi-uniform base that generates a quasi-uniformity/4~(S) compatible w i t h / 4 and with the loose extension of /4tp for the given trace filter system. Let us call it the uniformly loose extension of/4 for S. THEOREM 2.1. If an admissible trace filter system is composed of Dfilters, then the uniformly loose extension is D-complete. PROOF. Select, for every a E Y, a filter t(a) in X such that (t(a), s(a)) is a Cauchy filter pair; in particular, let t(x) = & for x E X. Denote by 6) the collection of all maps r: Y -+ exp X such that r(a) E t(a), satisfying the condition 7(x) = {x} for x E X. For A C Y, denote r(A) = U{r(a):a

E A}.

Let ( t ' , s ' ) be a Cauchy filter pair in (Y,/4~(S)). We have to show t h a t s I is convergent. By 1.2 we can suppose that s I is/4}(S)-round so t h a t s = = s'lX is a proper filter. The sets r(T') (T' E t', r e 6)) clearly constitute a filter base in X t h a t generates a filter t in X. The pair t, s) is a L/-Cauchy filter pair. In fact, given U E/4, we can choose U1 E/4 such that U12 C U, and then maps r E O, a E E such that r(a) x a ( a ) C /]1 for each a E Y. Choose T' E t', S' E s' such t h a t T' x S' C W(U1, a). Then r(T') x ( S' N X) C U. Acta Mathematica Hungarica 6~, 199~

D-COMPLETE EXTENSIONS OF QUASI-UNIFORM SPACES

47

To see this, let x E r ( T ' ) , y E S' N X. Then x C r(a) for some a C T', hence (a,y) e W(UI,O'), i.e. (a,y) E U1 if a E X or y e Ul(a(a)) if a e EY-X. In the first case x = a and (x,y) E //1 C U, in the second one r ( a ) x a ( a ) C U1 implies a(a) C Ul(x), y E UI(UI(x)) C U(x). T h e D-filter s converges with respect to U~(S) tp since the trace filter system is admissible. Then s t converges as well. This is clear if X E s t, and this is the case if ttiX is a proper filter because then T t x S t C W ( U I , a ) impliesS tCXbyT tNX?L-oand(2.3). IfY-XEt t a n d X C s t then the same inclusion implies T' - X = {p} = S' - X by (2.4). Thus {p} E t t and, for U E / / , a E s there is S t E s t such that {p} x S t C W(U,a), so t h a t s t -+p.

[]

C O R O L L A R Y 2.2. Every quasi-uniform space has a D-complete uniformly loose extension.

PROOF. By 1.2, there are admissible trace filter systems composed of D-filters. [] Instead of 2.1, we could use [3], Theorem 3.3. In fact, it is proved there t h a t L/~(S) is S P - c o m p l e t e if S is composed (of the neighbourhood filters and) of the non-convergent, round SP-filters. In (X,/4), a filter s is said to be an SP-filter iff, for U E/4, there is x E X such that U(x) E s, and (X,/4) o r / 4 is SP-complete iff every SP-filter is convergent. Now a D-filter is obviously an SP-filter, so an S P - c o m p l e t e space is D-complete. A further possible tool in proving 2.2 would be the observation that U~(S) is S P - c o m p l e t e (hence D - c o m p l e t e ) i f S is composed (of the neighb o u r h o o d filters and) of all non-convergent, round filters. In fact, the m e t h o d of proof of [3], Theorem 3.3 furnishes the following statement: if s' is an SP-filter for U~(S) and s'iX is/4~(S)tP-convergent then s', too, is convergent. However, an extension involving a narrower class of new points is more valuable, hence 2.1 can be considered, in some sense, to be a better result t h a n those yielding S P - c o m p l e t e extensions but using huge classes of trace filters. As to 2.1, it is worth-while to observe that U~(S) is not necessarily Dcomplete if S is an admissible trace filter system: EXAMPLE 2.3. Consider the upper half-plane of the Sorgenfrey plane, i.e. X = R x ( 0 , + c o ) , and, for x , y E X , x = (xl,x2), y = (yl,y2), c > > 0, (x, y) E U~ iff xi ~ yi <:. xl + ~ (i = 1,2), and let the quasi-uniformity /4 be generated by {U~:c > 0}. Then U is D-complete so that any trace filter system is admissible. Let Y - X = R - {0} and the sets S(p,c) = = (p - E,p + ~) x (0,c) (E > 0) constitute a base for s(p). Now = ( p - e,p + e +

x (o,a + Acta Mathematica Hungarica 64, 1994

48

i. CSASZAR

hence Ue(S(p,~)) C S(p, 2e), and s(p) is round. For/4' = L/~(S) (with S being composed of the given trace filters s(p) (p E Y - X) and the/4tp_ neighbourhood filters s(z) (z E X)), s' = fily s is a D-filter if s is generated by the base {Q(r162 > 0}, Q(e) = (0,c) • (0,E); in fact, we can choose t' = = fily{T(~):e > 0}, T(~) = ( - e , 0 ) C Y - X. In order to see this, choose e > 0 and a E E, then 5p > 0 such that S(p, 5p) C 0.(p), and observe that p E T(e) implies

D (pD (p,p+

p,p + •

+

(0, p +

D

D

i.e. T(s) • Q(~) c W(U2~,0.). However, s' does not bl'tp-converge.

[]

The construction of the uniformly loose extension can be slightly generalized. For this purpose, let Eo be a subset of E such that (2.5)

for a E Y, S e s(a), there is a 0. C E0 such that a(a) C S

(automatically fulfilled for a e X). Then it is easy to check (see [2], 2.1) that W ( U I , a ) 2 C W ( U , o ' ) i f U2 C U, so that (2.6)

{W(U,0.):U

U,a c Z0}

is a subbase for a quasi-uniformity U~(E0) (it is a base if, for 0.1,0"2 E E0, there is a E E0 such that a(p) C a~(p) N a2(p) for each p C Y - X). By (2.5) b/~(E0) is compatible with the given trace filter system (and still induces the corresponding loose extension of/4tp): LEMMA 2.4 (cf. [5], 6.5 and [6], Problem 32). If a trace filter system S is given in (X, bt), and Eo C E fulfils (2.5), then (2.6) is a subbase for a quasi-uniformity/4~(E0) compatible with/4 and with the loose extension of 14tp for S. [] If we look for a D-complete L/~(Eo), we suppose that the trace filter system is admissible and it is composed of D-filters, and then we can repeat the reasoning in the proof of 2.1, provided the following condition is fulfilled: (2.7) it is possible to select filters t(a) for a e Y such that ( t ( a ) , s ( a ) ) is a Cauchy filter pair, t(x) = 2 for x E X, and, for U e L/, there are r E O and 0. E E0 such that r(a) • a(a) C U for a E Y. COROLLARY 2.5. I f a n admissible trace filter system composed of Dfilters is given and Eo C E fulfils (2.5) and (2.7), then N~(Eo) is a Dcomplete extension. [] b/~(Eo) can be distinct from/4~(S) (see 3.9). Acta Mathematica Hungarica 64, 199~

D-COMPLETE EXTENSIONS OF QUASI-UNIFORM SPACES

49

If (2.5) holds without (2.7) being fulfilled, then U~(E0) may or may not be D-complete: EXAMPLE 2.6. Let X = 1 ~ - Q , / 4 be the restriction to X of the Sorgenfrey quasi-uniformity~ i.e. be generated by the collection of Ue (c > 0) where (x,y) E U~ iffx, y E X , x <=y < x + s . For Y = R, p E Y - X = = Q, let s(p) be generated by { (p,p+ ~) N X : s > 0}. Then s(p)is a round D-filter and S = {s(a): a E Y} is admissible. Choose a function f: Q ~ R such that f(p) > 0 for p E Q, and define 1

E0 = {an:n E N}.

Then Eo fulfils (2.5), and (2.7) holds iff f is bounded. a) If f ( _ 1 ) = 2n, A = (0,1) N X,

--'n n"

E N, n > k

t = fily{Tk: k E N},

then (t,~t)is a L/~(Eo)-Cauchy pair because

W(U~,ak)(p) = U~(ak(p)) D (p,p+ 2 + c) MX D A for p E Tn, E > 0, n ~ k. However, A does not L/~(V,o)tp-converge. b) If f(p) = p for p > 0, f(p) = 1 for p ~_ 0, then /J~(E0) is Dcomplete. In fact, consider the quasi-uniformity/A~(E1) obtained from the ! choice f(p) = 1 for p E Q and denote by a n E E1 the corresponding map. Now a round g/~(E0)-D-filter s ~ clearly converges if there is a flter t' such that ( t ' , s ' ) i s a C a u c h y p a i r a n d Y - X • t ' o r Y - X E s'. Suppose~ r To C Y - X , $ ~ So C X, To E t', To x So C W(U~o,ano) for some E0 > 0, no E N. Then So C W((U~o,ano)(p) for a p E To, hence So is bounded, and s u p T _<_ i n f S _<_ supSo = K whenever T E t ~, S E s~lX, T C Y - X, T x x S C W(Uh,ak) for some 5 > 0, k E N. Choose c > 0, n E N, then k E N such that ~ K < i , and T E V, S E s ~ satisfying T C Y - X , Tx S C C W(Ue, a;). Then p E T implies

w(u , ak)(p) c w(u,, a')(p) i.e.

(1)(1) p,p+~p+c

C

p,p+-+g n

,

so that s ~ is a D-filter with respect to L/~(E1). The latter being D-complete by 2.5, s' converges f o r l,,[~(~o) tp : Z,~(~I) tp. [] Acta Mathematica Hungarica 64, 1994

50

~ csisz~a

3. E x t e n s i o n o f m a p s . A reasonable concept of completeness involves an extension theorem of uniformly continuous maps into complete spaces to a complete extension of the space. A theorem of this kind corresponds to the extension/4s~(S) described in 1.5. In order to formulate it, we recall that a quasi-uniform space (X,/4) or a quasi-uniformity/4 is said to be uniformly regular (regular in [2]), iff, given U e / 4 , there is U0 E/,/ such that Uo(x) C U(x) for every x C X (and for the closure with respect to

Utp).

THEOREM 3.1. If (X,H) is a D-space, S is an admissible trace filter system composed of stable D-filters, f: X ---+Z is (H, bt")-continuous for a D-complete, uniformly regular quasi-uniformity H" on Z, then f admits a (/4~(S),L/") -continuous.extension g: Y --+ Z. PROOF. If (t,s) is a b/-Cauchy filter pair in X , then (filz f(t),filz f(s)) is a H"-Cauchy filter pair in Z. Hence the/4"-D-filters filz f(s(p)) (and the filter bases f(s(p)) are convergent in/4,,tp (p e Y - X). Define g:Y ~ Z such that g(x) = f(x) for x E X, and f(s(p)) ~ g(p) for p e Y - X. Then, by the (/4tP,bl"tP)-continuity of f, f(s(a)) ~ g(a) holds for every a e Y. Now, for a given U" E/4", choose U~p e/4" such that U~(z) C U"(z) for z E Z, then U~' E / / " such that U~'2 C U~p, finally U E/4 such that (x, y) E U implies ( f ( x ) , f ( y ) ) e U;' for x, y E X. Define U ' b e (1.5.1). If a,b E Y, (a,b) E U', then we can choose S C s(a) such that f(S) C C U~'(g(a)), and clearly

f(U(S))c

U~'(UI'(g(a)) ) C U~P(g(a)).

As U ( S ) E s(b)and f(s(b)) ~ g(b), necessarily g(b) E f ( U ( S ) ) C U~P(g(a)) C U"(g(a)).

[]

[11], Theorem 2 corresponds to 3.1 in the case of stable T0-spaces and reduced trace filter systems. An essentially better extension theorem can be proved for D-complete, uniformly loose extensions: THEOREM 3.2. If S is an admissible trace filter system composed of D-filters in a space (X, bl), (Z, li") is D-complete and f: X - , Z is (bI,Ll")continuous, then f admits a (/~/~(S),L/")-continuous extension g: Y ~ Z. PROOF. We can again define g satisfying glX = f, f(s(a)) -~ g(a) for a E Y . Given U" E/4", choose U~I E / 4 " such that U~12 C U", and U E U such that (x,y) E U implies ( f ( x ) , f ( y ) ) e U~', finally define a : Y + exp X Acta Mathematica Hungarica 64, 199~

D - C O M P L E T E EXTENSIONS OF QUASI-UNIFORM SPACES

such that

51

(a(x) = {x} for x E X and) a(p) E s(p) for p E Y - X, satisfying

v;'(g(p)).

c Now (a,b) E W(U,a) E X, or a E X, b c Y - X , b E U(a(a)), hence

implies (g(a),g(b)) E V". This is clear if a,b e or a, bE Y - X .

f(b) c U~'(f(a(a)))

C

Ifac

U~'(U~'(g(a)))

C

Y-X,

b E X, then

UH(g(a)),

so that g(b)= f(b) E U"(g(a)). [] It is easy to show that the condition of uniform regularity cannot be dropped in 3.1: EXAMPLE 3.3. Let Y = R, X = Q, U be the restriction to X of the Euclidean uniformity of R, s(p) be generated by { ( p - s , p + s)M X:~ > > 0} for p ~ Y - X. Now ?,/~(S) is the Euclidean uniformity of R by 1.11. For the same trace filter system S, let L/I~ =/J~(S), then U H is D-complete by 2.1. However, the (/4,/J~)-continuous map f = idx cannot be extended in a (/J~(S)tP, L/"~P)-continuous manner, because such an extension would coincide with idR (since U ~tp is T2), and the loose extension is strictly finer than the strict one. [] In order to obtain a similar extension theorem for the extensions U~(E0), let us introduce the following terminology: a quasi-uniform space (X, Lt) or a quasi-uniformity/4 is weakly quiet iff, for U C L/, there is U0 E L/such that if ( t , s ) is a Cauchy filter pair, s ~ x for/J tp, T C t, S C s, and T x S C U0, then S C U(x). The terminology is justified by LEMMA 3.4.

A quiet space is weakly quiet.

PltOOF. For U E L/, choose U0 C b / s u c h that if (t,s) is a Cauchy filter pair, Uo(x) e s, U0--](y) E t, then (x,y) E U. Now if (t,s) is a Cauchy filter pair, s ~ x , T e t , S E s , a n d T • U0, thenU0(x) E s , Uo~(y) D T E t for y E S, hence S C U(x). [] Conversely, a weakly quiet space need not be quiet: EXAMPLE 3.5. Let X be a regular topological space and/.4 be its Pervin quasi-uniformity (see [14]), Le. the entourages

uo = ( a x a) u ( ( x - a) x x )

(c c x open)

constitute a subbase for/.4. It suffices to show that U0 = U = Uc in the sense of the weak quietness. In fact, let filter pair, s ~ x, T E t, S E s, T x S C UG. Now T C X_ _- G is impossible whenever x E G. In and x E Go C Go C G, then T C X - G would imply

Uc corresponds to (t,s) be a Cauchy fact, if Go i__ssopen T C X-G0 = H

Acta Mathematica Hungarica 64, 1994

52

s cshszhR

and T o • So C UH for suitable To E t, So E s. But y E T N T o implies So C UH(y) = H in-contradiction with s --+ x. Therefore T N G ~ ~ and z E T N G implies S C Uc(z) = G = Ua(x), while S C Ua(x) is obvious if xEG. We know from [2], 8.2 that/4 is not uniformly regular in general (e.g. if X = It), while a quiet space is necessarily uniformly regular ([13], Proposition 1.2). [] There are also weakly quiet, uniformly regular spaces that are not quiet: EXAMPLE 3.6 (J. Des Let X = ( 0 , + ~ ) x { - 1 , 0 , 1 } , and, for c > > 0, let (zl,z2) E U~ iff zl = z2 or zi = (xi,yi), xl + x2 < c, Yl < Y2. Then {Ue: s > 0} is a base for a quasi-uniformity/4. Clearly b o t h / 4 tp a n d / 4 - t p are discrete, hence/4 is uniformly regular and weakly quiet (in fact, U(z) = = U(z) for z E X, and (t, s) can be Cauchy filter pair satisfying s ~ z only i f t = s = 4, so that T E t , S C s, T x S c U~imply z E T , S C U~(z)). However, if/4 were quiet and U6 (6 > 0) were suitable for U1 in the sense of the quietness, then zi = (xi,0), xi < 6, Xl ~ x2 would imply (zl, z2) E U1 because (t, s) with

t = fil{ (0,s)

x

> 0},

s = ill( (0, c) X {1}:c > 0} is a Cauchy filter pair and g6(zl) E s, u~-l(z2) e t, while (Zl, z2) r Vl. [] On the other hand, there are uniformly regular spaces that are not weakly quiet: EXAMPLE 3.7 (J. Des Let X = It, and, for c > 0, U~(x) = {x} U U [0,+oc) if - c < x < 0, U ~ ( 0 ) = [0,c), U~(x)= {x} otherwise. Then U~ C U~, hence {U~: s > 0} is a base for a quasi-uniformity L/. For/4*P, each U~(x) is closed so that L/is uniformly regular. However, any U~ is unsuitable for U1 in the sense of the weak quietness, because (t, s) is a Cauchy filter pair if t = i l l { ( - c , 0 ) : c > 0}, s = ill{[0,c):c > 0},

further s --+ O, and ( - c , O ) x [O,+oc) C U~, ( - r E t, [0, +oc) E s, but the latter set is not contained in UI(0). [] Now we can prove: TItEOREM 3.8. Let S be an admissible trace filter system composed of D-filters in a quasi-uniSorm space (X,/4), and ~o C S satisfy (2.5) and (2.7). If (Z,14") is' D-complete and weakly quiet, and f: X ~ Z is (/4,bl")continuous, then there is a (Lt~(So),bl")-continuous extension g: Y ~ Z

off. PROOF. We can define g satisfying gIX = f and f ( s ( a ) ) --+ g(a) for a E Y. Given U II EL/II, select U~I C/4/I such that U(~t2 C U I~, then U~I E E /4" such that S '~ C U~(z) whenever (t ~, #~) is a/g"-Cauchy filter pair, Acta Ma~hematica Hungarica 64, 199.~

D-COMPLETE EXTENSIONS OF QUASI-UNIFORM SPACES

53

such that S" C U~(z) whenever (t", s") is a/4"-Cauchy filter pair, s" --, --+ z, T" E t", S" E s", T" x r C U~'. We can suppose U~' C U~~. For U~', choose first U E/4 such that (x,y) E U implies (f(x),f(y)) E UI', then t(a), r E O, a E E0 according to (2.7). We show that (a,b) 9 W(V,a) implies (g(a),g(b)) e U". It suffices to consider the case a 9 Y - X, b 9 X. Then b 9 U ( a ( a ) ) , r(a) x a(a) C U, hence f(b) 9 U~'(f(a(a)), f ( r ( a ) ) x f(a(a)) C U~',

f(r(a)) E t " = filz f ( t ( a ) ) ,

f(a(a)) E s" = filz f(s(a)),

and (t", s") is a/,/"-Cauchy filter pair, s" -+ g(a). Thus

f(a(a)) C Ug~(g(a)),

U~'(f(a(a))) C U"(g(a)),

and g(b)= f(b) E U"(g(a)). [] The condition of weak quietness cannot be dropped in 3.8: EXAMPLE 3.9 (cf. 1.9). Let X = ( R - { 0 } ) x N , ( z l , z 2 ) E U~ for~ > > 0 iffeither Zl = z2 or z / = (xi,ni), nl = n:, Xl < 0 < x: < xl + E. Then {U~:v > 0} is a base for a quasi-uniformity 3/. Define Y = R x N, and let s(p,~) be generated, for p~ = (0, n), by {(0,(5) x {n}:(5 > 0}, finally ~60'~) = (0,(5) • {~}. It is easy to see that the trace filter system S = {s(a): a E Y} is composed of (round) D-filters: U~(a6(pn)) = a~(p~), (t(pn),s(p,0) is ~ Cauchy filter pair if t(pn) is generated by { (-(5,0) x {n}:(5 > 0} because

((-(5,0) x {n)) x ((0,(5)x {hi) c u26 Clearly (2.5) and (2.7) are fulfilled for E0 = {a~: (5 > 0}. This trace filter system is admissible because, if ( t, s) is a Cauchy filter pair distinct from those of the form (4, ~ ), then necessarily ( - o c , 0) • {n} E t, (0, + co ) • {n} E E s for some n, and s is finer than s(pn). Thus L/~(S) and g/~(E0) both are D-complete extensions of/g, they induce the loose extension of/,/tp (= the discrete topology of X). The only (g/~(E0)tp,/g~(S)tP)-continuous extension of idx is idy (because the loose extension is T2). However, idy is not (/g~(E0),/,/~(S))continuous. In fact, let a(p~) = (0,1) x {n}, then W(Ue,a6) C W(Ul,a) does not hold for any ~ > 0, (5 > 0 because (~,n) e W(U~,~)(pn), while (~,n) r W(Vl,~)(p~) if 1 < ~. [] The author is thankful to Dr. J. Des for a lot of useful remarks and ingenious counter-examples. Acta Mathematica Hungarica 6~, 199~

54

~. CS.~SZ.~R:D-COMPLETE EXTENSIONS OF QUASI-UNIFORMSPACES

References [1]/~. Css163 General Topology (Budapest - Bristol, 1978). [2] A. Css163 Extensions of quasi-uniformities, Aeta. Math. Acad. Sci. Hung., 37 (1981), 121-145. [3] ~. Cs~szs Complete extensions of quasi-uniform spaces, in General Topology and its Relations to Modern Analysis and Algebra V, Proe. Fifth Topol. Syrup. 1981 (Berlin, 1983), pp. 104-113. [4]-[5] J. Des Extensions of quasi-uniformities for prescribed bitopologies I-II, Studia Sci. Math. Hung., 25 (1990), 45-67; 69-91. [6] J. Des A survey of compatible extensions (presenting 77 unsolved problems), in ~ Topology, Theory and Applications II, Colloquia Math. Soc. J. Bolyai 55 (Amsterdam, 1993), pp. 127-175. [7] J. Des Extending and completing quiet quasi-uniformities, Studia. Sci. Math. Hung., in print. [8] D. Doitchinov, On completeness of quasi-uniform spaces, C. R. Acad. Bulgare Sci., 41 (1988), 5-8. [9] D. Doitchinov, On completeness in quasi-metric spaces, Topology and its Appl., 30 (1988), 127-148. [10] D. Doitchinov, A concept of completeness of quasi-uniform spaces (in print). [11] D. Doitchinov, Another class of completable quasi-uniform spaces, C. R. Acad. Bulgare Sci. (in print). [12] D. Doitchinov, Stable quasi-uniform spaces and their completions, in print. [13] P. Fletcher and W. Hunsaker, Uniformly regular quasi-uniformities, Topology and its Appl. 37 (1990), 285-291. [14] W. J. Pervin, Quasi-uniformization of topological spaces, Math. Ann., 147 (1962), 316-317. (Received March 29, 1991) DEPARTMENT OF ANALYSIS E()TVOS LORAND UNIVERSITY H-1088 BUDAPEST, M(IZEUM KRT, 6-8.

Acta Maihematica Hungarica 64, 199g