Suppose X is a topological space  and Y a proximity space , 
 fn L C f (Leader Convergence) iff for each A  in X, B in Y, f(A) near B implies eventually  fn (A) is near B.  L.C.  is a generalization of
 U. C. (Uniform Convergence). In this paper we study L. C. and various generalizations and prove analogues of the classical results of Arzelà, Dini and others

DI CONCILIO, Anna

S. A. Naimpally

English

Suppose X is a topological space  and Y a proximity space , 
 fn L C f (Leader Convergence) iff for each A  in X, B in Y, f(A) near B implies eventually  fn (A) is near B.  L.C.  is a generalization of
 U. C. (Uniform Convergence). In this paper we study L. C. and various generalizations and prove analogues of the classical results of Arzelà, Dini and others

Archivio della Ricerca - Università di Salerno

Mh. Math. 103, 93-102 (1987) Malhemalik 9 by Springer-Verlag 1987 Proximal Convergence By A. Di Concilio, Salerno, and S. A. Naimpally*, Thunder  Bay (Received 20 February 1986) Abstract. Suppose 35 is a topological space, (Y,O) a proximity space, fn, L.C. fare in yX where n is in a directed set D. We sayfn ~ f(Leader Convergence) iff for each A c X, B c Y , f (A)  r B implies eventually, f n (A) ~ B. L.C. is a generalization of U.C. (Uniform Convergence). In this paper we study L.C. and various generaliza- tions and prove analogues of the classical results of Arzel~, Dini and others. 1. Introduction. It is well known that the pointwise limit f ,  of  a sequence of  continuous functions (fn : n e N), need not  be continuous. In 1841 Weierstrass discovered uniform convergence which provides a sufficient condition f o r f t o  be continuous. A search was conducted for necessary and sufficient conditions by several mathematicians Arzelfi, Dini, Hobson ,  Seidel, Stokes and others. Concerning all this informa- tion, we have "Hobson ' s  Choice" **, namely the monumental  work  of  E. W. HOBSON [3]. Later on some of  these ideas were generalized in the setting of  uniform spaces discovered by WEIL [7]. For  their importance in Functional  Analysis, especially the results of  Arzelfi and Dini, see BARTLE [ 1 ]. Our aim in this paper is to study necessary and/or  sufficient conditions for a net of  continuous functions (fn : n e D) to converge to a continuous function f i n  the setting of  proximity spaces. LEADER [4] was one of  the first mathematicians who attempted this in 1960. Leader generalized Weierstrass' uniform convergence to "conver- gence in proximity" when the range space is EF and showed that it is sufficient to preserve p-continuity as well as continuity. Uniform * This research was partially supported by an operating grant from NSERC (Canada) and a visiting professorship under CNR (Italia). ** The Concise Oxford Dictionary defines it as "to take it or leave the one offer". 7 Monatshefte ffir Mathematik, Bd. 103/2 94 A. DI CONCILIO and S. A. NAIMPALLY convergence implies convergence in proximity;  the reverse implication holds when the range space is totally bounded  or (fn) is a sequence. NJASTAD [6] generalized these results, to generalized uni form structu- res. Al though  it is possible to work in more  general spaces, we will suppose that  all our  spaces are T~. Consider  the following axioms on the relation 0 on the power  set of  a nonempty  set X: (a) A 0 B implies A 4= 0, B r 0, (b) A ~ B implies B ~ A, (c) A n B -r 0 implies A ~ B, (d) A 6 ( B u  C) iff AdB or A6C, (1.1) (e) A ~ B and b ~ C for each b ~ B implies A d C, (f) p ~ B implies there is an E c X such that  p r E and (X - E) ~ B. (g) A ~ B implies there is an E c X such that  A ~ E and (X - E) ~ B. (1.2) Definitions. A relation 0 on the power  set of  X (a) is called a LO-proximity iff it satisfies (1.1) (a)--(e);  (b) is called an R-proximity iff it satisfies 1.1 (a)--(d) and (f); (c) is called an EF-proximity iff it satisfies 1.1 ( a ) I ( d )  and (g). It is known  that  EF implies R and LO but  LO and R are independent .  (d) I f  X is a topological  space with closure - then ~ is compatible with - iff for each p~X, A c X, pdA iff p ~ A - .  Every T~-space X has a compatible  LO-proximity given by A~oB i f f A -  r i B -  r 0 (1.3) and a compatible  R-proximity given by AOrB i f f (A n B - )  u ( A -  n B )  r 0 .  (1.4) Every Tychonof f  space X has a compatible  EF-proximity  given by A Ce B iff there is a cont inuous  funct ion f :  X ~  [0, 1] (1.5) such that  f(A) = 0 and f(B) = 1. I f  ~ denotes any compatible  proximity on a T~-space X, then p ~ q implies p = q .  (1.6) In this paper  (X, 01), (Y,02) denote any of  the three types of  Proximal Convergence 95 proximity spaces unless we mention a specific one. A function f :  X ~  Y is p-continuous iff whenever A 81 B in X, f (A)  a~f(B) in Y. (1.7) If we replace A by p, we get the definition of continuity o f f  at p. Throughout this paper D is a directed set, (f,:n e D) is a net of functions on X to Y. Also f:  X ~  Y is a function. For more details regarding proximity spaces see [5]. 2. Sufficient Conditions. We begin with a condition which is strictly weaker than Leader's but which is sufficient for the limit functionfto be (p-)continuous when each f,  is. (2.1) Definition. R.C. f ,  ~ f iff for all subsets A, B o f  X, whenever f (A)  ~2f(B), then eventually f ,  (A) r (B). R.C, (2.2) Theorem. I f  f ,  ~ f and each f ,  is p-continuous (resp. Continuous), then f is p-continuous (resp. continuous). Proof We need only prove the preservation of p-continuity; the case of continuity is similar. Suppose f(A)r then eventually f~ (A) ~f~ (B) and since each f~ is p-continuous, A ~i B. Thus f is p- continuous. Although R.C. preserves (p-)continuity, unfortunately it is not stronger than pointwise convergence (P.C.) ! (2.3) Example. Suppose X= Y= N with the usual metric proxim- ity. Suppose f ( x ) = x  for each xeX. For each nEN, set f ,  (x)= x+n. R.C. P C. Here the sequencef~ ~ fbu t f~  -~J. Since it is well known that P.C. does not preserve continuity, it follows that P.C. and R.C. are inde- pendent. Since it is desirable to have convergence stronger than P.C we define: (2.4) Definition. X.C. = P.C. + R.C. Clearly, X.C. (2.5) Corollary. I f  f n ~ f and each f~ is p-continuous (continuous), then so is f 96 A. DI CONCILIO and S. A. NAIMPALLY We now recall Leader's definition and call it L.C. (Leader Convergence): (2.6) Definition. f ~ f  iff for each A ~ X and B c  Y if f(A) r B, then eventually f ,  (A) ~2 B. By choosing A = p, it follows that L.C. is stronger than P.C. We now show that L.C. is strictly stronger than X.C. when 82 if EF. L.C. X.C. (2.7) Theorem. I f  ~ 2 is EF and s ~ f,  then f~ --* f L.C. R.C. Proof. It is enough to show that f ,  ~ f implies f~ --* f. Sup- pose f(A)r Since 8 2 is EF, there is a subset E of Y such that f (A)~2E and ( Y - E ) ~ z f ( B ) .  By (L.C.), eventually f~(A)~2E and Thus f,, z_-;f. (Y - E) ~2f(B). So eventually, fn (A) ~2f~ (B). R C (2.8) Corollary, (LEADER [4]) I f  ~32 is EF, then L.C. preserves p- continuity (continuity). We now give examples to show that X.C. does not imply L.C. and L.C. in R-proximity does not imply L.C. in EF-proximity. (2.9) Examples. (a) Suppose X = Y = [ -  l, 1] with the usual metric proximities. For each x e X, n e N  we define: f ( x )  = 0 and f~(x) = nx(1 + n2x2) -1 P C  Clearly, fn -:-;f and sincef(A) = f (B)  = {0} for all subsets A, B of X, R C  X.C. f ,  z_;f. Sof~ -~f.  I f A  = {n - l "neN} and B = {2-1}, then f (A)~zB L.C. but fn (A) (~2 B for each n s N. Thus f ,  -~ f. We note here that X and Y are compact Hausdorff. Note that the obvious continuity of f follows from (2.5) but not from (2.8). (b) Suppose X = Y = N with the usual topology. For each x e Y and n ~ N we define: f ( x )  = x 2 and in (X) = (X + n-1)2 .  P C  Clearly, f ,  ~ f .  If Y has the usual metric proximity, which is EF, L.C. then f ~ - ' o f  For A = { n : n ~ N \ { 1 } } ~ X  and B =  { ( n + n - l )  2" Proximal Convergence 97 n ~ N \ { 1 } } c  Y, f (A)r  but f , (A)  n B r  for each n and so f ,  (A)02 B. Suppose now that 02 is an R-proximity defined by A (]2B iff (A c~ B ~) u (A~n B) r 0 (2.10) or both A and B are infinite. If A c X, B c Y and f (A)  r B, then f (A)  ~~ B = 0 = f (A)  ~ B ~ and either A or B is finite. We can then verify, by actual calculations which we omit, that eventually, fn (A)r B. Similarly, if A and B are both subsets of X and f (A)r  in the usual metric prox- imity or R-proximity (2.10) then we can show that eventually . X . C .  f,, (A) ~2f~ (B) 1.e. f~ --* f .  (c) Suppose X = Y = N and for each x = X, neN, f ( x )  = x x < 0  f , (x)  = + 2n -1 x > 0 L C .  U.C. If  (]2 is the usual metric proximity, then f~ ~ f (and fn ~ f )  but if (]~ is an R-proximity defined by (2.10), then f ,  - ~ f  in X.C. or L.C. To see this we choose A = { x E N : x  < 0} and B = { x e N : x  > 0}. Here f (A)  r but f~ (A) (]2f~ (B). (2.1 1) Remarks. LEADER [4] showed that if 62 is induced by a uniformity, then U.C. implies L.C. Further, Leader showed that U.C. = L.C. if Y is totally bounded or (f,) is a sequence (see Sec- tion 3). Example (2.9) (a) shows that even if X and Y are compact, and (f,) is a sequence, X.C. need not imply L.C. or U.C. From the proof of Theorem (2.2) it is obvious that X.C. is too strong for the preservation of p-continuity (continuity). It is sufficient fOrfn (A)r (B) for just one n. However, since we would likef~ to be 'near 'f ,  we requiref~ t ' c f  as well as a condition such asf ,  (A)r (B) frequently. Such considerations led Dini to define Simple Uniform Convergence (S.U.C.). Here we define an analogue of S.U.C. and call it D.C. (Dini Convergence). (2.12) Definition. D.c P.c. f~ -~ f i f f f~  ~ f a n d  for all subsets A, B of X, f (A)  ~2 f (B)  implies frequently, f~ (A) ~2f, (B). (2.13) Theorem. I f  f ,  DC f and each f ,  is p-continuous (continuous), then so is if. 98 A. D1 CONCILIO and S. A. NAIMPALLY Obviously X . C . ~ D . C .  but the converse is not  true as the following example shows. (2.14) Example. (Hobson) Here X = Y = ~, ( x )  = x [n x + (1  - n x )  - U2n(X ) = -- x[(n + 1)x 2 + [1 - (n + 1)x]2] -1 P.C. fro(X) = Un(X) ~ f ( x )  = X[X 2 + (1 -- X)2] -1 . n = l  D.C. X.C. Here fn ~ f but f~ ~ f. 3. Generalizations. In this section we suppose that ~1 and 82 are EF- proximities induced by uniformities q /and C on X, Y respectively. We generalize L.C. to Simple Leader Convergence (S.L.C.) in analogy with S.U.C. and study its relationship with S.U.C. of Dini and Q.U.C. (Quasi Uniform Convergence of Arzeia). S U C  P.C. (3.1) Definitions. ( a ) f  n 2;  f i f f f ~  ~ f a n d  for each V in ~U and for all x in X, frequently f ,  (x)e V[f(x)]. Q U C  P C .  (b) f~ 2; "f iff f~ -:-; f and for each V in ~ and for each m e D, there exists a finite set {ni: ieI} c D such that ni > m and for each x~X, f~ , (x )e  V[f(x)] for some i~I. S L C  P.C. (c) f~ :7-; f i f f f ,  ~ f a n d  for each A c X, B = Y, f (A)  r implies frequently fn (A) r B. (3.2) Theorem. S.U.C. implies S.L.C. Proof Supposefn s.m.c.~ f and f (A) r  B for A c X a n d  B c Y. Then 2 there is a V~V such that V[f(A)] ~ B = 0. S.U.C. implies that frequently for each a ~ A,f~ (a) ~ V[f(a)] and so f~ (A) c V[f(A)]. So S.L.C. V[fn(A)] c~B = 0 frequently i.e. f~ (A) r B i.e.J~ --> J. (3.3) Theorem. I f  (Y,~t r) is totally bounded, then S.L.C. implies Q.U.C. Proof. For Vs 3e- there is a finite set {y~: 1 ~< i ~< m} c Y such that Y =  ~)V[Yi]. Suppose Ai=f-l[V(y~)],  then X =  Q)Ai and i=1  i=1  Proximal Convergence 99 2 S.L.C _ f (A i )~2[Y-V[y i ] ] .  Since f~ ~ J ;  for each reeD,  there ~i > m such that 2 2 f~(A;) r  V[yiJ], i.e. f,~(Ai) c V[y~], 1 <. i<~ m .  exists For any x s X there is an i such that x e Ai and 2 2 f~i (x) 6 V[yg], f ( x )  e V[Yi] 9 4 f Q.U.C. r Hencef~,(x)e V[f(x)] and sojn ~ j .  (3.4) Corollary. ( L E A D E R  [ 4 ] .  I f  ( Y ,  ~ / ' )  is totally bounded and L.c .  U C  f~ -~ f ,  then f ,  2 ;  f (Our proof patterned after the above one is different from Leader's or Nj~stad's [6]). S.L.C Q U C (3.5) Theorem. I f  X is compact and f ,  ~ f E C (X, Y), then fn ~ " fProof  For each p ~ X and V~ ~/~, s incefis  continuous at p, there is a nbhd. Up in X such that f(Up) c V [f(p)]. 2 So f (Up)r  V[f(p)]]. For each m e D ,  there exists an np> m such that 2 2 /,~(Up)~2[Y- V[f(p)]], i .e./ ,~(Up)c V[f(p)]. q 2 Since Xis compact, 2" = U U~, Ui = Upi, np~ = ni andf~, (Ui) c V[f(p)], l <<. i <~ q, n~ > m. i=1 3 . ,0 Q.U.C. r So for each x E X, x e Ui for some i andf~ (x) e V[f(x)] a.e.j~ --. j .  L.C. U C 3.6. Corollary. I f  X is compact and f ,  ~ f then f~ 2 ;  f By imitating the proof of NJ/~STAD [6] Theorem 4, we get 3.7 Theorem. I f  D is linearly ordered (in particular i f  f n is a sequence), then S.L.C. implies Q.U.C. 4. Necessary and Sufficient Conditions. Here we investigate neces- sary and sufficient conditions for the preservation of continuity. If each fn is continuous, then it is obvious from Theorem (2.2) that for 100 A. DI CONCILIO and S. A. NAIMPALLY f to be cont inuous,  the following condi t ion - -  replacing A by p - -  is enough:  For  each p in X, B ~ X, whenever f ( p ) f z f ( B )  , (4.1) eventually (or frequently)fn (P) r 9 Our  mot iva t ion  for defining a convergence which is necessary and sufficient for the continui ty o f f ( w h e n  X is compact)  is the condi t ion Q.U.C. defined by Arzelfi (BARTLE [1]). So we call this A.C. (Arzelfi Convergence).  f ~ f i f f f ~ f  and for each p ~ X ,  B o X ,  (4.2) Definition. A.C. P.C. whenever f ( P ) ~ 2  f(B), then for each reeD,  there is a finite set q {ni: 1 <<. i <~ q} ~ D, n i > m and B = U Bi such that  fn, (p) r (Bi) for 1 ~< i~< q. i=1 (4.3) Remarks. Clearly L.C. =~ X.C. ~ A.C. Example (2.14) with X = [ -  1, 1] shows that  A.C. does not  imply X.C. We now prove an analogue of  Arzelfi's Theorem.  pc (4.4) Theorem. Suppose X is compact and 82 is EF. I f  f ,  - ~ f  and each f~ is continuous then a necessary and sufficient condition for f A C  to be continuous is that f ,  2 ; f .  Proof. Suppose f is cont inuous,  p ~ X, B c J(, and f (p )  r f (B) .  S i n c e f ( B - )  c f (B )  - , f ( p )  r - and Xis compact ,  we may suppose that  B (and f (B) )  are compact .  Since 82 is EF, there is an open set V c Y such that  f ( B )  c V and f (p )  r V. Suppose m ~ D. Since P.C. m '  f ,  ~ f ,  there is an > m such that  for each n > m', f~ (P) r V. (4.5) 9 P.C.  { f - 1  S m c e f , ~ f ,  B c  n ( V ) : n > m ' } . S i n c e B i s c o m p a c t ,  q B c U f  ~l(V),q6[N, ni > m' . i = l  q Clearly B = ~ B~, Bi = B c~fs 1 <. i<. q. By (4.5), i=1 9 A . C .  L, (P) r (Bi), I.e. fn ~ f '  Conversely, A.C, suppose f~ ~ f and each f~ is continuous9 If  Proximal Convergence 101 q f (p )  r and m ~ D, there exists ni > m, 1 ~< i ~< q, B = U Bi such that  ~= 1 Li (p) :2L, (8,.). This implies that  p 81B~ which, in turn, implies that  p 81B. So f is cont inuous.  Next  we prove a near converse of Theorem (4.4). (4.6) Theorem. P.C. = A.C. on C(X, ~) implies X is pseudocom- pact. Proof. If  X is not  pseudocompact ,  then there is a cont inuous  unbounded  function f :  X ~  ~+.  For  each x e X  and neN,  define L(x) =[ 0 P C A.C. Then f ,  -:-;f but  f~ ~ f.  n - 1  f ( x )  0 <. f ( x )  <~ n n ( 1 - n ) ( f ( x ) - n -  1) n<<.f(x)<<.n+l n + 1 <. f ( x ) .  (4.7) Corollary. I f  X is normal, then X & countably compact if  P.C. = A.C. on C(X, ~). (4.8) Remarks. Since the sequence of functions (f,) constructed in (4.6) is mono tone  increasing, it follows that  it is also a converse of  Dini 's  Theorem. That  is: if P.C. = U.C. on C(X,'N) for m o n o t o n e  nets implies Xis  pseudocompact .  The same applies to Corollary (4.7). I f  X is a metric space, then pseudocompactness  is equivalent  to compactness.  So for metric spaces X the following are equivalent: (a) X is compact  (b) P.C. = A.C. on C(X, ~) (c) P.C. = Q.U.C. on C (x, N) (d) P.C. = U.C. for mono tone  nets on C(X, ~). References [1] BARTLE, R. G. : On compactness in functional analysis. Trans. Amer. Math. Soc. 79, 35--57 (1955). [2] HARRIS, D. : Regular-closed spaces and proximities. Pacific J. Math. 34, 675-- 685 (1970). 102 A. DI CONCILI0 and S. A. NAIMPALLY: Proximal Convergence [3] HOBSON, E. W. : The Theory of  Functions of a Real Variable. Vol. II. New York: Dover. 1927. [4] LEADER, S. : On completion of proximity spaces by local clusters. Fund. Math. 48, 201--216 (1960). [5] NAIMPALLY, S. A., WARRACK, B. D. : Proximity Spaces. Cambridge: Univer- sity Press. 1970. [6] NJASTAD, O. : Some properties of proximity and generalized uniformity. Math. Scand. 12, 47--56 (1963). [7] WEIL, A. : Sur les espaces a structure uniforme et sur la topologie generale. Actual. Scient. Ind. 551, Paris: Hermann. 1937. A. DI CONCILIO Istituto di Matematica Facolta di Scienze 1-84100 Salerno Italy and S. A. NAIMPALLY Department of Mathematical Sciences Lakehead University Thunder Bay, Ontario, P7B 5El Canada 

Proximal convergence

https://www.iris.unisa.it/bitstream/11386/3122456/1/proximalconvergence.pdf

Proximal convergence

Abstract

Similar works

Full text

Available Versions

Archivio della Ricerca - Università di Salerno