Acta Math. Hungar. 65 (4) (1994), 365-377.

SIMULTANEOUS EXTENSIONS OF C A U C H Y S T R U C T U R E S A. CSASZAR (Budapest)," member of the Academy

O. I n t r o d u c t i o n . The papers [3] deal, among others, with the following problem: in a closure space (X, c), let a (possibly empty) family of subsets Xi (i E I) be given, and, for each i E I, a merotopy Mi on Xi; look for an extension of {c; M i ) , i.e. for a merotopy M on X such that M induces the closure c and its restriction to Xi coincides with Mi. The author considered in [2] the same problem in the case when Mi and M are filter merotopies or (equivalently) Si and S are screens on Xi and X, respectively (for the terminology, see Chapter 1 below). The present paper intends to investigate from this point of view a still more special kind of structures, namely Cauchy structures; in fact, our results will concern two classes of Cauchy structures only and questions related with Cauchy structures in general remain open until future publications. 1. P r e l i m i n a r i e s . For a set X, let us denote by F i l X the collection of all filters in X (including the improper filter e x p X ) . For a C exp X, denote by filx a = fil a the smallest filter containing a, by secx a = sec a the collection of all subsets of X that meet each element of a. In particular, we write A = filx{A} for A C X and ~: = / t for x E X, A = {x}. If a C e x p X and X0 C X, we denote by a]Xo the collection of all intersections A M Xo where A E a. If s is a filter in X then slXo is a filter in Xo. Conversely, if So E FilXo, then Sox = filx so is the finest filter s in X such that s i X o = so. For a , b C e x p X , let us introduce the notation a A b to denote the situation that each A E a meets each B E b; we write a A b in the opposite case.

A screen on X (see [2])is a set 0 # S C F i l X such that (1.1)

x EX

(1.2)

implies that there is sES

and

sCs'E

sES FilX

such that imply

x E Ns,

s~ES.

*Research supported by Hungarian National Foundation for Scientific Research, grant no. 2114.

0236-5294/94/$4.00

(D 1994 Akad~miai Kiad6, Budapest

366

s CSiSZiR

If tb r S C Fil X fulfils (1.1), we say that S is a screen base on X; it generates the screen S ~ composed of all filters in X finer than some element of S. We also say that S is a screen base for the screen SC If S is a screen oll X and Xo C X then SIXo = {s]X0:s E S} is a screen on X0. The screen S on X induces a closure (see [3], 0.1) c = c(S) oll X defined by x E c(A) i f f t h e r e i s s E S such that { x } , A E sees. Clearly if s E S and x E Cl s then s ---, x for c(S). Observe that the free filters in S do not have any influence on c(S). If X0 C X and S is a screen on X, we have c(SlX0) = e(S)lXo. A screen S on X is said to be Riesz iff vc(x) E S for x E X, where vc(x) denotes the c-neighbourhood filter of x for the closure c = c(S); S is said to be Lodato iff v~(s) E S for each s E S where c = c(S) again and, for an arbitrary s E F i l X , v~(s) is composed of all sets V C X such that there is S E s satisfying V E v~(x) for x E S. If c is a topology then vc(s) is the filter generated by the filter base composed of the c-open elements of s. If S is a Lodato screen then c(S) is a topology; if c is a topology and e = c(S) then S is Lodato iff it is generated by a screen base composed of c-open filters. If S is a l~iesz or Lodato screen on X then so is SIX0 on Xo C X. A Cauchy structure S on X (see e.g. [4], p. 12) is a screen satisfying (1.3)

sl, s 2 E S ,

slAs2

imply

SlNs2ES.

We shall call Cauchy screen or briefly C-screen on X a Cauchy structure on X in the above sense. A CR-scT~en or CL-screen is a C-screen that is Riesz or Lodato, respectively. If a screen base S fulfils (1.3) then the screen generated by S is a C-screen i S2 C St2, SlAS2 / i imply szAs2. This is the case in particular when since s 1 C Sl,

slAs2 for

Sl,S 2

E

S, 81 r S2"

If S is a C-screen on X and Xo C X then S]Xo is a C-screen on Xo because sx[XoAs2[Xo implies sxAs2, further Sl [~s2[Xo = (slIXo) N (s2]Xo). Let us agree in saying that ( s o , . . . , s~) is a Cauchy chain on X iff si E E F i l X for i = 0 , . . . , n and s i ' l A s i for i = 1 , . . . , n . An easy induction based on the observation

{

for

(1.4)

i

s i , t j E FilX, [")slA N t j 0 0 and

j

such that

iff there are

siAtj

n

shows t h a t [-1 si E S whenever S is a C-screen and ( s o , . . . ,sn) is a Cauchy 0

chMn such t h a t si E S for i = 0 , . . . , n . For screens Sl and $2 on X, $1 is said to be coarser than $2, S2 finer than $1, i f f S l D S2. If S 1 is coarser than $2 then c(S1) is coarser than c($2). Acta Mathematica Hungarlca 65, 199~

SIMULTANEOUS EXTENSIONS

367

OF CAUCHY STRUCTURES

For a screen S on X, the finest C-screen S c coarser than S is generated by n

the screen base S / composed of all intersections ~ s l where ( s 0 , . . . , s ~ ) is a 0

Cauchy chain such t h a t s~ E S (i = 0 , . . . , n ) ; S' fulfils (1.3) in consequence n

of (1.4). If B is a screen base for S, it suffices to take intersections A s i such 0

t h a t the elements of the Cauchy chain (So,..., sn) belong to B. We are now able to formulate more precisely the purpose of the present paper. Let ( X , c ) be a closure space, Xi C X for i E I (I = ~ can happen), and Si a given screen on Xi. We look for CR.- or CL-extensions of {c;Si}, i.e. for a CR-screen or CL-screen S on X such that (1.5)

c(S):e,

SlXi=Si

for

i@I.

In Sections 2 and 3, we always assume the following standard hypotheses: (X, c) is a closure space, Xi C X , Si is a screen on Xi for i E I, and extension will m e a n a screen S on X fulfilling (1.5). We write ci = c(Si) and Xij = = Xi N X j for i , j E I. 2. C R - e x t e n s i o n s . obtained:

A simple necessary condition can very easily be

LEMMA 2.1. if S is a CR-screen then c = c(S) fulfils the condition (2.1.1)

for

x , y E X,

v~(x) r vc(y)

implies

vr

PROOF. If vc(x)Avc(y) then s = vc(x)N re(y) E S since the CR-screen S contains both vc(x) and v~(y). Now s E S, x E M s imply s ~ x for c and vc(x) C s. Similarly vr C s so that s = re(x) = vr [] For topological spaces, condition (2.1.1) is often called axiom ($2) (see e.g. [1], p. 95). Therefore we shall say that the closure c (or the closure space (X, c)) is $2 if[" (2.1.1) holds. This is the case of course if e is Hausdorff (or T2), i.e. if x ~ y implies vr According to [5], Definition 3.1, the C-screen S is said to be Hausdorff iff c(S) is T1 (i.e. separated). It is easy to prove the converse of 2.1 in the following form: LEMMA 2.2. If C is an $2 closure then B = { v c ( x ) : x E X } is a screen base for a CR-screen S such that e = c(S). PROOF. By ($2) the screen generated by B is Cauchy. If x E c(A), v o ( x ) e s satisfies x e n A e sec,,dz). Ifs S, z e n s , A e s e c t , then s D vc(y) for some y E X; by x E ~ vc(y) and ($2), v~(y) = vc(x), hence A E secvc(x) and x E c(A). Thus c = c(S) and S is Riesz. [] Observe t h a t a C-screen need not be a Cl%-screen although it induces a T2 closure: Acta Matheraatica Hungarica 65, 199~

~. csiszia

368

EXAMPLE 2.3. In a set X, let us say that s C F i l X is an elementary n

filter if s = N u l where ul is an ultrafilter for i = 0 , . . . , n . Then by (1.4) 0 u A s for an ultrafilter u implies u = ui for some i; hence s is an elementary filter iff there are finitely many ultrafilters only finer than s. Therefore a filter finer than an elementary filter is elementary itself. Let S be a screen on X and denote by S ~ the collection of e x p X and of M1 elementary filters belonging to S. Then S ~ is a screen since ~ is an elementary filter for x E X. If S is a C-screen then the same holds for S e. We have c(S e) = c(S) because S ~ C S implies that c(S e) is finer than c(S), andifsES,

x E Ns, A E secs, t h e n k D s ,

(slA) x D s , so that by taking

an ultrafilter u finer t h a n (siA) x , we obtain an elementary filter s r = k M u finer t h a n s and satisfying {x}, d E sees'. Hence c ( S ) i s finer than c(S~). Consider now a non-discrete T2 topology c that is first countable and let S be a C-screen such that c = c(S) (by 2.2, S may be chosen to be a CR-screen). T h e n c = c(S ~) and S e is a C-screen; however, S e cannot contain vc(x) for a point x that is the limit of a sequence (xn) such that x ~ xn ~ Xm for n ~ m. In fact, there are infinitely many subsequences of (Xn) each two of which correspond to disjoint sets of indices; and by choosing ultrafilters finer t h a n the corresponding F%chet filters, it turns out that there are infinitely m a n y ultrufilters finer than vc(x), and the latter cannot belong to S ~. [] LEMMA 2.4 (cf. [2], (2.7.1)). /f S is a niesz extension then vc(x)IXi E E Si for x E X , i E I. [] LEMMA 2.5. If S is an extension and si E Si then s X i E S. PROOF. There is s E S such that siXi = si and s x is finer than s.

[]

COgOLLAKY 2.6. If S is a Cauchy extension and ( t o , . . . ,tn) is a Cauchy chain such that tj = s X, sj E Sij, ij E I, then

(2.6.1)

tj

IXk E Sk

(k E I).

[]

COROLLARY 2.7. If S is a CP~-extension, si E Si, and x E X is a cluster point for c of s X, then s x ---* x with respect to c. PROOF. v c ( x ) A s X by hypothesis and both filters belong to S, hence s - - - v c ( x ) Ms x E S a n d x e Ms, so s--* x for c, a n d s x is finer than s. [] Now we can prove: TttEOREM 2.8. There is a Cl~-extension of {c; Si} iff the following conditions hold: (a) c is an $2 closure, (b) vc(x)lXi E Si for x E X , i E I, Acta Matheraatica Hungarica 65, 199,~

SIMULTANEOUS

EXTENSIONS

OF CAUCHY STRUCTURES

~6(0

(c) /f si E Si and x E X is a cluster point for c of s(\" then s:v --* x for c, (d) if ( t o , . . . , t ~ ) is a Cauchy chain such that tj = s f , sj E Sij, ij E I,

If these conditions are fulfilled then the filters vc(x) (x E X ) and the filters n

(2.8.1)

fitj

(tj

as in

(d))

0

constitute a screen base for the finest CR-extension S~R = S~R(e; Si). PROOF. Necessity: 2.1, 2.4, 2.7, 2.6. Sufficiency: The collection B of the neighbourhood filters vc(x) and the filters (2.8.1) is obviously a screen base. It generates a Cauchy screen because s', s" E B, s'As" imply s' N s" E B. This is a consequence of (a) if both s' and s" are neighbourhood filters, and of (d) if both have the form (2.8.1) (cf. n

(1.4)). If s ' = vc(x), s " =

["]tj as in (2.8.1), then by (1.4) one of the filters 0

tj has x for cluster point (with respect to e) and then, by (c), tj --+ x. Now t / _ l A t / (except for j = 0) and t j A t j + l (except for j = n) imply, again by (c), tj-1 --+ x, tj+l -+ x. After a finite number of steps we obtain tj --+ x for each j , hence s" --+ x, and s' gl s" = s'. By 2.2 c(S) is coarser than c for the screen S generated by B. If s E S, {x}, A E secs, then x E c(A) (and c ( S ) i s finer than c). In fact, we may' suppose that s E B, and the case s = vc(x) is settled by 2.2. If s is of the form (2.8.1) then {x} E sectj = s~: for a j, hence by (c) s f --+ x for e, and a successive application of (c) as above furnishes s --+ x, x E c(A). Therefore S is a Riesz screen. (b) and (d) show that slXi E Si (i E E I ) i f s E B, and then f o r s E S, too. On the other hand, sl E S i i m p l i e s s~" E B C S and s~'lXi = si. By this, S = S~a is a CR-extension. If S' is another CR-extension then B C S' by 2.5, so that S C S'. [] Observe that 2.8 (c) and (d) hold as soon as they are fulfilled for filters si or sj taken from screen bases generating Si or Sj, respectively. Further necessary conditions can be easily formulated for the existence of a CR-extension: LEMMA 2.9. If S is a CR-extension then (a) ci = clXi for i E I, (b) SiiXij = S j l X i j for i , j E I, (c) Si is a CR-screen for i E I. PROOF. (a) and (b) hold for every extension ([2], (1.19) and (1.20)).

[]

Acta Mathematica Hungarica 65, 1094

370

s csaszaR

We show that each of the conditions 2.8 (a) to (d) is independent of the others even if 2.9 (a) to (c) hold. For (a), this is shown by the case I = ~. EXAMPLE 2.10. Let x = It, c be the Euclidean topology, Xo = (0, +oc), Co = c[Xo, and let So be generated by the screen base composed of all Coneighbourhood filters. By 2.2 So is a CR-screen and c(So) = co; 2.9 (b) is obvious since II[ = 1. 2.8 (c) holds because Vco(X)x has the only c-cluster point x E Xo. 2.8 (d) is always fulflled if I = {0} and So is a C-screen by (2.10.1)

s/XzXsf

(2.10.2)

ifl" s i A s j

Ns x =

(0)si x

for

si, s j E

FilXo,

for s l E FilXo.

0

However, v~(O)[Xo ~ So.

[]

E X A M P L E 2.11. Consider X, c, Xo, co as in 2.10, and let So be generated by the filters Vco(X) (x E Xo) and by

So = (vc(0)tXo) N filxor where r = { ( a , + o o ) : a E Xo}. We have co = c(So) as above because so is a free filter. Hence So is a R.iesz screen again and 2.9 (b) holds for the same reason as in 2.10. So is Cauchy since so does not have any co-cluster point. Now vc(x)lXo

=

Vco(X) E So

for

v c ( x ) [ X 0 = e x p X o E So

x E Xo,

for

x <0,

(o)lXo D So, so that 2.8 (b) holds. However, s~" has the c-cluster point 0 without coilverging to 0. [] , 2

EXAMPLE 2.12.

Let 1~ = It x {i} for i = 0,1,2, X = UY/, c the o Euclidean topology of R 2 restricted to X , X i = X - Yi, ci = c]Xi, ri = = { (a, +oo) x {i}:a E It}. Let S / b e generated by the screen base composed of the filters vc(x)[X/ (x E X/) and, for i = 0, of sm =filxora

and

So2=filxor2,

for i = 1 of

sl = (filx~ ro) f~ (filx~ r2), Acta h4afhematica ffungarlca 65, 199~

SIMULTANEOUSEXTENSIONSOF CAUCHYSTRUCTURES

371

for i = 2 of :

( lx.

n (filx.

Now ci = c(Si) since SOl, so2, Sl, s2 ~I'e fi'ee filters, hence the screens Si are Riesz. They are Cauchy because Sox, s02, sl, s2 do not have any c-cluster points and SolAs02. For the same reason, 2.8 (c) holds, and 2.8 (b) is obviously valid. 2.9 (b) follows fi'om the formulae

sol[Xol = exp Xm = exp Xl[Xol , Sol[Xo2 --- fily1 rl = s2]Xo2, so2lXol = fil)5 r2 = SllXol,

so2]X02 = exp X02 = exp X2]X02, Sl[Xl2 = filyo ro = s2[X12 and the obvious ones concerning the neighbourhood filters. However, (siX,@") is a Cauchy chain and

IXo =

(filXorl) ~ (filxor2) r So.

[]

From a certain point of view, Example 2.12 is the best one; in fact, we can

show:

LEMMA 2.13. If II] =< 2, each Si is a Cauehy screen, and 2.9 (b) is fulfilled, then 2.8 (d) holds. PROOF. This is obvious for I = 0 and we have shown it in 2.10 for [I I = 1. Let I = {0,1} and consider a Cauchy chain ( t 0 , . . . , t ~ ) such that tj = s f , sj E So or $1 for each j. Suppose sj E So for Jl =< J =< J~- Then by (2.10.2) J2

j2

N sX = sX J=Jl

for

s=

N sj J=jl

and by (2.10.1) (sj~,... ,sj2 ) is a Cauchy chain in Xo. Hence s E So and sXj~-IAsX

(except for

j, = 0 ) ,

SX AsX.t_I (except for

j2 = n)

clearly imply

sXj1-1AsX,

sX AsX+I" Acta J~4athematlca Hungarica 65, 1994

372

h cshszha

A similar statement is valid if sj E Sl for jl <= j _-
(z13.1)

soXlX,= (solXo,) x'

for

soE FilXo.

Namely, so is a base in X for sox, hence soIX 1 -~- solXol is a base in X1 for s xlX1. On the other hand, solXm is a base in X1 for (soIXol)X~, too. From so E So and (2.13.1) we obtain by 2.9 (b)

s~o'lX1 =

(sllXo,)x,

for some s, E S,, and (s,IXo,) x, D s, implies s~'iX1 E S, as stated. Suppose the statement holds for some n and consider a Cauchy chain (tO,... ,tn+l) such that tj = sy, sj E So if j = 2h, sj E $1 if j --- 2k + 1. Assume n + 1 is even (the other case is established by interchanging the roles n

of So and $1). By the induction hypothesis, s = N t j satisfies slXo E So, X siX1 E $1. Now t,~+l = s,~+l, sn+l E So implies

(2.13.2)

tj

IXo = ( s n s . + , ) I X o

0

= (slXo)ns,+,,

X X further s C t~ and tnAs,+l imply SASh+ 1 and then slXoAsn+l by Xo E X E s,~+l so that the right hand side of (2.13.2) belongs to So. On the other hand,

(2.13.3)

tj

13/'1= (sglsn+l)IX 1 = (siX1) N (~+11X1)

where sX+llX1 E S1 by the reasoning applied above for n = 0. Now t~Atn+l and tn = s~X, s,~ E $1, X1 E s~X imply t~]X1At,~+l]X1, hence, by s C t~, slX1AsX+IlX1, so that the right hand side of (2.13.3) belongs to $1. [] COROLLARY 2.14. If (2.8) (a), (b), (c), (2.9) (b), (c) are fulfilled and If I <=2, then S~R constructed as in 2.8 is the finest CR-extension. []

Observe that 2.9 (c) can be weakened to: Si is a Cauchy screen for each i. It is shown in [2], 2.8 that if c is $2 (or even satisfies a weaker hypothesis) and 2.8 (b), 2.9 (a) and (b) hold, then the finest Riesz extension S}~ is Ac~a Mar

Hungarlca 65, 1994

SIMULTANEOUS EXTENSIONS OF CAUCHY STRUCTURES

373

generated by the screen base composed of the filters v~(x) (x C X) and sff" (s, E Si, i C I). From this, we easily deduce THEOREM 2.15. Under the hypotheses of 2.8, we have

s~.

:

(s~,) ~

PROOF. If S is a CR-extension then S~ C S, consequently (S]~) c C S. From S h C (Sh)C C S we deduce that (Sh) r is an extension, namely a CR one; therefore it coincides with the finest CR-extension S~R by 2.8. [] SIR can be distinct from Sh: EXAMPLE 2.16. Consider X , Xi, c, Si as in 2.12 but for i = 1,2 only. Then by 2.14 and 2.15 (whose hypotheses are fulfilled now by 2.13) there exists S~R "- (S}~) c. Now clearly @',@" C S}~ but @" r3 s~" E (S}~) c does not belong to S}~. [] In contrast to [2], 2.7, according to which there exists a coarsest Riesz extension (if there exist any), we can show that a coarsest CR-extension need not exist: EXAMPLE 2.17. Let X = It x {0, 1}, let c be the restriction to X of the Euclidean topology of the plane, ,'o+ :

{ ( a , + ~ ) • {0):a e It},

ro-

{(-~,a)

:

•

{0):a e It},

rl= {(a,+oo) X {1}:a E It}, and so+

=

(filxro+) I"1 (filxrl) ,

So_

=

(filxro-) FI (filxr,).

For Xo = t t x {0}, let So be generated by the screen base composed of the filters vc(z)IXo (x E Xo) and So+lXo, s0_lX0. 011 the set X, consider the screens S and S' generated by the screen bases composed of the filters Vc(X) (x E X) and, in the case of S, of so+ and ill),- ro-, in the case of S', of so_ and ill• ro+. As the elements of both screen bases are pairwise in relation A, and clearly c(S) = c(S') = c, SIX0 = S'IXo = So, both S and S' are CR-extensions of So. However, any Cauchy screen coarser than both S and S ~ necessarily contains so+ Cl so- whose trace on Xo does not belong to So. [] Acta Mathematica Hungarica 65~ 1994

374

h cshszhR

9 Observe that this example shows the lack of a coarsest Cauchy extension in general, in contrast to [2], 2.6, according to which there is a coarsest extension of {c; Si} whenever an extension exists at all. 3. C L - e x t e n s i o n s ,

A great deal of questions arising on this field can be treated with methods applied for CR-extensions. If S is a Lodato screen and x E c ( c ( A ) ) , c = c(S), then there is s E S such that {x}, c(A) E secs and clearly A E sec vc(s), vc(s) E S, so that z E c(A) and c is a topology. If So = = SIXo, so E So, then Sox E S by 2.5 and vc(Sox) E S. By this, taking into account that a Lodato screen is Riesz (since Vc(X) = vc(~)), we immediately obtain, using 2.8, the necessity part of THEOREM 3.1. There are satisfied: (a) c is an $2 topology, (b) vc( )lX e si for x (c) if si E Si and x E X x for e, (d) if ( t 0 , . . . , t ~ ) is a ij E I, then

exists a CL-extension iff the following conditions

e x , i e I, is a cluster point for c of vc ( six') then v c (six') ---+ Cauchy chain such that tj = v c ( s f ) ,

tj

]Xk E Sic

sj E S,~,

(k e I).

If these conditions are fulfilled then the screen base B composed of all filters x ) and of all intersections (3.1.1)

Ntj

(tj

as in

(d))

0

generates the finest CL-extension SOL = SOL(C; Si). PROOF. We only have to check the sufficiency. Similarly to the proof of 2.8, (a) and (e) show that B is a screen base for a Cauchy screen S such that e(S) = c. Thus S is a CL-screen because c is a topology and the elements of B are c-open filters. Now (b) and (d) show SIXi C Si for i E I and Si C C SIXi follows from s~x D Vc(Sff'). If S' is an arbitrary CL-extension then B C S', hence S C S'. [] If e is a topology and Xo is e-open, so is a co = clXo-open filter in Xo, then clearly v c ( s X) - s0x. Hence, if each Xi is e,open and Si is Lodato, then 2.8 (c) and (d) coincide with 3.1 (c) and (d), respectively. For the existence of a CL-extension, further necessary conditions are, of course, 2.9 (a) and (b), and the one that the screens Si have to be CL-screens. Examples 2.10, 2.11, 2.12 show that each of the conditions 3.1 (a) to (d) is independent of the others even if the above necessary conditions Acfa Mafhemafica Hungarica 65, 199~

SIMULTANEOUS EXTENSIONS OF CAUCHYSTRUCTURES

375

are satisfied; in fact, in 2.10, 2.11, 2.12, c is always a T~ topology, the sets

Xi are c-open and the filters in the screen bases generating Si are c-open, too, so t h a t all screens Si are CL-screens and the conditions 3.1 (b), (c), (d) coincide with the respective conditions in 2.8. Moreover, it can happen that all conditions in 2.8 and M1 but one conditions in 3.1 are fulfilled and the exceptional condition in 3.1 fails to be true. EXAMPLE 3.2. Let X = R , c* be the Euclidean topology on X , c the Hausdorff topology for which v~(x) = v ~ . ( x ) i f x r 0, and a base for vc(0) is composed of the sets V - N where V is a c*-neighbourhood of 0 and

Define Xo = N so that co = clXo is discrete, and let the screen base generating So be composed of the filters 21Xo (x E Xo) and so = vc.(0)lXo. T h e n c(So) = Co since so is fi'ee, so So is a CL-screen and 2.9 (a), (b) are fulfilled. T h e same holds for 3.1 (a), (b), (d)since vc(SoX) does not have any c-cluster point in X0 and v c ( s X) lX0 = so. 2.8 (c)is satisfied, too, since s~" does not have any c-cluster points. However, 3.1 (c) fails to hold since 0 is a c-cluster point of vc(s0x ) but s x D vc(s0x ) does not c-converge to 0. [] EXAMPLE 3.3. Let c* denote the Euclidean topology on R 2, X = 1~ x x [0,+co), and c be the Niemytzki topology on X ( v c ( p ) = vc*(p)lX for p = (x,y), y > 0, and, for p = (x,0), a base for v~(p) is composed of c*-closed disks contained in X and containing p). Then c is T2 and, on Xo = R x {0}, c induces the discrete topology co = cIXo. Define Q = Q x x {0}, P = x o - Q, and let se (sQ) be composed of those sets S C Xo for which P - S (Q - S) is finite. Let a screen base for So be composed .~lXo (x E Xo) and of Sp and SQ. Then co = c(S0) (sp and sQ are fl'ee filters), and S0 is a CL-screen (spAsQ by P E sp, Q E SQ). Consequently 2.9 (a), (b), 3.1 (a), (b), 2.8 (d) are fulfilled (for tlie l a t t e r , c o n s i d e r I = {0}). 3.1 (c) holds since v c ( @ ' ) clearly does not have any cluster points in X - X0 and, if p E X0, choose S E sp such that p r S, a disk D C X such that p E D, and disks Ds C X for s E S such that s E D~, Ds N D = 0, so that V = U D~ E v ~ ( s X ) , V Cl D = ~. A similar reasoning shows that v c ( s ~ ) sES

does not have any cluster points at all for c. However, v c ( s p X ) A v c ( s ~ ) . In fact, for S E sp and a c-open V D S, let P~ denote the set of all p E P for which the disk D C X containing p and of diameter •r~ is contained in V. Then P is the union of the sets P~ and of the finite set P - S. Since P is a G6 subset of Xo for the topology c*lXo , the Baire Category T h e o r e m furnishes a P~ that is C-dense in an open interval (a,b) C Xo. T h e n (a,b) x (0, 88 C V, hence, for an arbitrary S' E s o and a A cla Mathemalic~ HuT~garica 65, ! 99~

376

h. CSs163

c-open V' D S', there is a q E S' whose first coordinate belongs to (a,b) and then necessarily V n V' ~ 0. Now vc(sff') N vc(s~) has a trace on X0 that is composed of all sets S C X0 for which Xo - S is finite; this filter does not belong to So, and 3.1 (d) fails to hold. [] This example shows that an analogue of 2.14 (by substituting a.1 (a), (b), (c) to the respective conditions in 2.8) cannot be true for CL-extensions and III= 1. It also shows that an analogue of [2], 2.14 fails to be valid for I = {0} and a c-closed X0. On the other hand: TtIEOREM 3.4. The conditions 3.1 (b) to (d) follow from 3.1 (a) and 2.8 (b) to (d) provided Xi is c-open and Si is a Lodato screen for i ~ [. Therefore, under these hypotheses, there exists a CL-extension. PROOF. By 2.8 there exists an extension, so 2.9 (a) holds and then 3.1 (c) and (d) coincide with 2.8 (c) and (d), respectively. [] According to [2], 2.17, the finest Lodato extension S~ is generated by the screen base composed of the filters vc(x)(x e X) and v~(s~.~') (si e Si, i e I) (whenever a Lodato extension exists at all). Hence 2.16 (in which c is a T2 topology, the sets Xi are c-open, and the screen bases for Si are composed of c-open filters) furnishes an example where S}~ = S~, S~I~ = S~L , consequently S~L ~ S]5. In the following example S)~ = s lcP~, S~ = $1CL, but

sh

sL and so

#

EXAMPLE 3.5. Let X = It, c be the Euclidean topology on X, X0 = = N, co = c[Xo, and So be composed of exp X0 and of all ultrafilters in X0. Then 3.1 (a) to (d) are fulfilled: vc(s0x) does not have any c-cluster points for a free ultrafilter so ill X0, vc(slX) ~Vc(S X) for two distinct free ultrafilters sl,s2 E S0, and vc(s0x) IX0 = so. However, s(}" is an ultrafilter in X, whereas v~(s0x) does not contain any of the complementary sets X0 and X - X0. []. Example 2.17 (in which e is a T2 topology and the screen base defining S ~nd S' are composed of c-open filters) shows that CL-extensions may exist without existing a coarsest one among them (in contrast to the case of Lodato extensions, see [2], 2.13). The following anMogue of 2.15 can be proved in the same way: THEOREM 3.6. Under the conditions of 3.1, we have

PROOF. (S~) c is Lodato since it admits a base composed of open filters. [] The author thanks Dr. J. Degk for valuable remarks. Acta Mathematica Hungarica 65, 199,~

SIMULTANEOUS EXTENSIONS OF CAUCHY STRUCTURES

377

References [1] J~. Css163 General Topology (Budapest-Bristol, 1978). [2] A. Css163 Simultaneous extensions of screens, Topology, Theory and Applications II (Amsterdam, 1993), 107-126. [3] /~. Css163 and J. Des Simultaneous extensions of proximities, semi-uniformities, contiguities and merotopies I-IV, Math. Pannon., 1/2 (1990), 67-90; 2/1 (1991), 19-35; 2/2 (1991), 3-23, 3/1 (1992), 57-76. [4] E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps (New YorkBasel, 1989). [5] E. Lowen-Colebunders, Hausdorff separation for Cauchy spaces (in print). (Received October 21, 1992)

EOTV()S LOR~.NDUNIVEKSITY DEPARTMENTOF ANALYSIS BUDAPEST, MOZEUMKRT. 6-8 H-1088

Acta Mathematica Hungarica 65, 199~