NEW ZEALAND JOURNAL OF MATHEMATICS Volume 26 (1997), 67-72

A G E N E RALISED U N IF O R M C O N V E R G E N C E A N D D IN I’S T H E O R E M

I v a n K u p k a

(Received August 1996)

Abstract. We show that all hypotheses of the classical Dini’s theorem have topological analogues. A topological generalization of Dini’s theorem is proved.The notion of strong convergence of functions replaces the uniform one.

1 . Introduction

This paper is one of a series of papers dealing with the following question: How to obtain some “uniform looking results” without a uniformity or even without a quasiuniformity structure? (See e.g. [3], [4]). The notions we are working with are purely topological, with no additional structures, unless otherwise indicated. For this purpose, we shall start by recalling the definition of strong convergence.

Uniform convergence of a sequence of functions with values in a metric space Y, is defined using “e-covers” of Y , i.e. using the open covers of Y containing all open balls with radius e > 0. The use of all covers, not only the e-ones, will give us a stronger convergence, which does not need a metric structure on Y.

Definition 1 ([4]). Let X be an arbitrary set and (Y ,T ) be a topological space. Let { / 7; 7 G T} be a net of functions from X to Y. Let p be an open cover of Y. We say, that the net { / 7; 7 € T} converges to a function f : X Y p-uniformly if there exists a e F such that for every 7 > a and for every x € X there exists B e p such that / 7 (x) e B and f (x ) e B.

We say that the net { / 7 ;7 € T} converges to the function / strongly if it con­verges to / p-uniformly for every open cover p of the space Y.

If Y is uniformizable, the strong convergence implies always the uniform one. In addition, if X is compact, the converse is true. [4]

Definition 2 . Let X , Y, f and { / 7; 7 G T} be as in Definition 1. Let C be any open cover in (Y,T). We say that the net { / 7 ; 7 e r } converges to / C'-monotonically iff V 7 € T V i e X V F 6 C: if { f ( x ) , f y (x )} C V then V

> 7 fa {x) G V and { / 7} converges to / pointwise on X .

2 . A Topological Formulation o f D ini’s Theorem

In all generalizations of Dini’s theorem the role of the semicontinuity of functions / 7 and / is essential. How to reformulate this property without aid of additional structures on 7 ? The first solution would be to assume that all / 7 and / are continuous. But this hypothesis can be easily weakened. Examining several proofs

1991 A M S Mathematics Subject Classification: Primary 54A20; Secondary 54C08, 54C35, 26A15. K ey words and phrases: Dini’s theorem, strong convergence, function vanishing in infinity.

6 8 IVAN KUPKA

of Dini’s theorem one can see that the continuity or semicontinuity assumptions serve mainly one purpose: to obtain open preimages of some special open sets - such as open intervals in R, open balls in metric spaces etc. For this, it suffices to work with two-valued multifunctions x :—> { / ( x ) ; / 7 (x)} whose “upper” preimages of some special sets will be open.

Definition 3. Let (X, T) and (Y, r) be topological spaces, let C be an open cover in (Y ,r). A multifunction F : X —> Y is called C-u.s.c. (C - upper-semicontinuous) iff V O e C V x G X if F{x) C O then there exists an open neighborhood Ox of x such that F(O x) C O.

For each V C Y , let F+(V ) = {x G X ; F(x) C V }.

Example 4. Let functions f : R —> R, g : R R be defined as follows:

V k G Z V x G [k, k + 1) f (x ) = k\

V k G Z Vx e (k,k + 1} g(x) = k — 1.

We can see that / is an upper-semicontinuous function, g is a lower-semicon- tinuous one and for every x in X f (x ) > g(x) holds. Neither / nor g is continuous. The multifunction F : R —> R defined by Vx G R F (x) = { f (x ) ,g (x ) } is not upper- semicontinuous in the classical sense (a multifunction G : X —> Y is called upper- semicontinuous iff the set G+ (V ) is open for every open V in Y ). Nevertheless for every interval (a, b) C R the set F + ((a, 6)) is open. For example F + ((—1,1)) = 0 , F + ((-3 ,2 )) = / - 1(( -3 ,2 )) n ^ _ 1( ( -3 ,2 )) = [-2 ,2 ) n (-1 ,3 ] = ( -1 ,2 ). Let us denote B {(a, 6); a, b G R, a < b}. B is a base of the natural topology on R and the multifunction F is B - upper-semicontinuous.

Theorem 5. Let (X, T) be a compact topological space and let (Y ,t) be an arbi­trary topological space. Let f : X —» Y be a function and let { / 7; 7 G T} be a net of functions / 7 : X —► Y converging pointwise to f . Let

(m) There exists a base B of the topology t such that the net { / 7 ; 7 e r } converges to f , B-monotonically.

(c) V 7 G T the multifunction F1 : X —>Y, defined by F^(x) — { / 7 (x), f { x ) } , is B-u.s.c.

Then the function f is continuous and the net { / 7; 7 G T} converges to f strongly.

P roof. First, for strong convergence, let p be an arbitrary open cover of Y. We have to prove

(*) 370 such that V a > 7o V a : G X 3 V Gp such that, { f a (x ), f { x ) } C V holds.

Since B is a base of the topology r there exists an open cover c of Y such that

(1) c C B and VF G c B O G p , with V C O.Let P = (F + (F ); V G c, 7 G T}. Then, P is an open cover of X .The space X being compact, there exists a finite subcover {P 1? P2> • • • , Pn} oi P.

There exist indices /?i, # 2, . . . ,/3n and open sets B\, B2, ■.. ,B n from c such, that P1 = F + (B i) ,P 2 = F + (B 2) , . . . , P „ = F + (B „) holds.

A GENERALISED UNIFORM CONVERGENCE AND DINI’S THEOREM 69

Choose a 7 from T such, that 7 > /% for i = 1 ,2 ,... , n. Then, F .̂ (£?*) C F+(Bi) Vi = 1 ,2 ,... ,n. So the set {F+(.Bi), F+(£?2), ■ ■ . , F + (5 n)} is a finite open cover of the space X . Moreover the set {F+(£?i), F+(£?2), • • • , F^(B n)} is a finite open cover of X for every a > 7 .

But this implies Va > 7 V i € X 3 £ a,x € {f?i, B2, . . . >^n} such that { / Q(x), /(a ;)} C Ba)X G B. Hence, (1) implies that (*) holds and the net { / 7; 7 G T} converges to / strongly.

Next, for the continuity of / , let x G X and let O be an open neighborhood of f (x ) in Y. Then there exists an open set V G B satisfying f ( x ) G V C O. The pointwise convergence and the assumption (m) imply there exists (3 G T such, that {fp (x ), f ( x ) } C V. By (c), it follows that there exists an open neighborhood W of x such, that Fp(W) C V C O holds. But this implies that f (W ) C O, and the continuity of / follows. ® □

Theorem 5 implies a quite general theorem proved in [6]. Before deriving this theorem we need some definitions.

Definition 6 ([6 ]). An ordered quadruple (Y ,T ,U , £ ) will be called a space of semicontinuity if (Y, T ) is a topological space, U C T, C C T and the collection of sets U n C := {C D D\ C &U and D G £ } forms a base of the topology T.

Definition 7 ([6]). We say, that a net {y7 : 7 G T} of points of a space Y of semicontinuity converges monotonically from above to a point y G Y if the following three conditions are fulfilled:

(1) V C G C if y G C then V 7 G T : y7 G C;(u) V D e U :

1. if there exists 7 G T such that y7 G D then y G D\2. if /3 G T and yp G D then V 8 > /3 : y$ G D holds.

(c) The net (y7 : 7 G T) converges to y in the topology T.

Definition 8 ([6 ]). Let {X ,T ) be a topological space, let (Y ,T ,U ,C ) be a space of semicontinuity. A function / : X —> Y is said to be lower semicontinuous (upper semicontinuous) if VC G C : / - 1(C') G T (VD GW: f ~ 1 (D) G T) holds.

Rem ark 9. A natural model of a space of semicontinuity is the space (R, T,ZY, £), where (R ,T ) is the set of real numbers with its natural topology, U := { ( —00, a); a G M} and C := { ( 6, + 00); b G 1 } . In this case this definition of lower (upper) semicontinuity coincides with the classical one.

Theorem 10 ([6 ]). Let {X ,T ) be a compact topological space and (Y,T,U , C) be a space of semicontinuity. Let f : X —> Y be a lower semicontinuous function. Let { / 7; 7 G T} be a net of upper semicontinuous functions from X into Y. Let the net { / 7; 7 G T} converge to the function f pointwise monotonically from abovei.e. V i G I , f-y(x ) converge monotonically to f (x ) from above. Then the net { / 7; t e r } converges to the function f strongly and f is continuous.

We now show that Theorem 5 is a generalization of Theorem 10. For this, it suffices to verify that the hypothesis of Theorem 5 imply the hypotheses (m) and(c) of Theorem 10.

70 IVAN KUPKA

Proposition 11. The hypotheses of Theorem 10 imply those of Theorem 5.

Proof. Let B — U n C. Then, B is a base of the topology T. To prove (m) let x G X and V G B such that, f (x ) G V. Then, there exists a 7 G T such that, f-y(x) G V. There exist two open sets O G £, W G U such that, V = O n W. Since { / 7; 7 G T} converges to / monotonically from above (see Definition 7) it follows that fp{x) G O for each (3, and there exists a 7 G T, such that f a(x ) G W , for each a > 7 , so that (m) follows.

Now we prove that (c) holds. Let 7 G T, let a multifunction P7 : X —> Y be defined by: Vx G X Fy (x) = { / 7 (x), f (x ) } . Let t G X , P G B such, that F1 {t) C P. Let Pi G £, P2 G U such, that P = Pi fl P2. Since f ( t ) G Pi and / is lower-semicontinuous, there exists an open neighborhood 0 \ of t such, that /(O i) C Pi. Since V 2 G Oi {fp (z);@ G T} converges to f (z ) monotonically from above, / 7 (0 i) C Pi holds too. The upper semicontinuity of / 7 and the fact / 7 (t) G P2 imply there exists an open neighborhood 0 2 of t such that fy {0 2) C P2. Since V 2: G O2 fy (z ) € P2, the condition (u) of Definition 7 implies \ /z G 0 2 f (z ) G P2. Hence, (c) follows. □

3. Functions, Approaching a Point

In this section we present a generalization of Theorem 5. The result is inspired by an idea concerning real functions presented in [2]. First we shall recall the definition of functions vanishing at infinity and a Dini’s theorem, both presented in[2]-

Definition 12 ([2]). Let X be a topological space. A function / : X —> R is said to be vanishing at infinity, if for each e > 0 there exists a compact set K £ C X such that |/(x)| < e for every x G X — K e.

Theorem 13 ([2]). Let X be a topological space. Let { f n} be a decreasing sequence of upper semicontinuous real functions on X (upper semicontinuous in the classical sense), which are vanishing at infinity. Let the sequence { / n} converge pointwise to a continuous function f : X —» M, vanishing at infinity. Then the convergence is uniform.

The result presented above can be generalized. In what follows we do so replacing the space R by an arbitrary topological space. Of course, the notion of the function vanishing at infinity must be generalized. The following definition shows one of the posssible ways of doing it.

Definition 14. Let X , Y be topological spaces. Let yQ G Y. We say that a function / : X —> y is approaching yQ if for every open neighborhood O of yQ there exists a compact set K o C X such that f ( t ) G O for every t G X — Ko-

One can see, that a real function is vanishing at infinity if and only if it is approaching the point 0 in the sense of Definition 12. The following theorem is a topological Dini’s theorem for X , not necessarily compact.

Theorem 15. Let (X ,T ), (Y ,t ) be topological spaces. Let f : X —> Y be a function and let {/7 ; 7 G T} be a net of functions f 1 \ X —̂ Y converging pointwise to f . Let

A GENERALISED UNIFORM CONVERGENCE AND DINI’S THEOREM 71

(a) There exists a point yQ G Y such that the functions f and / 7 are approaching yQ for every 7 G I\

(m) There exists a base B of the topology r such, that the net { / 7; 7 G T} con­verges to f B-monotonically.

(c) V7 G T the multifunction F7 : X —» Y - defined by F7 (:r) = { / 7 (x), f ( x ) } - is B-u.s.c.

Then the function f is continuous and the net { / 7; 7 G T} converges to f strongly.

Proof. Let p be an open cover of Y .Then, there exists an open cover c of Y such that

(1) c c B and VV" G c 3 0 e p V C O.Let p = {F+(yy,Vec,7 er}.Then, P is an open cover of X . Fix an open set O G c such, that yQ G O. Since

/ is approaching y0 there exists a compact set K o C X such that f ( t ) G O for every t G X — Ko-

Since K o is compact, there exists a finite subcollection {P 1; P2, ■■■ , Pn} of P which is an open cover of Ko-

There exist indices /3\, fa, - - - ,{3n and open sets B\, B2, . . . , Bn from c such that Pi = P2 = F ^ (£ 2)>. . . , Pn = Fpn(Bn) holds. Choose a 7 from T such,that 7 > fa for i = 1 ,2 ,... , n. The assumption (m) gives us Fp.(Bi) C F+(Bi) Vi = 1 ,2 ,... ,n. So the collection {F + (B i), F + (B 2), - - - , F +(B n)} is a finite open cover of the set K o- Moreover for every a > 7 the set

{F+(B1),F+(B2),...,F+(Bn)}is a finite open cover of K o too.

The function / 7 is approaching yQ too. So there exists a compact set K~f c X such, that f 7 (t) G O for every t G X — K 7. Let C — K o^ jK ^ . The set C is compact. The function / | C and the net {fp \ C}\(3 G T}, the set C with the inherited topology from X and the space Y satisfy the conditions of Theorem 5. Therefore

(i) there exists a 6 G T such that 6 > 7 and Va > ̂ Va: G C G c such thatC Ba.

Moreover for every t G X — C { f { t ) , f 7 {t)} C O so X — C C F + {0 ) holds and from 6 > 7 and from the assumption (m) we obtain

(ii) V a > 6 X - C c F + { 0 ) .

So Vi G X — C { f a(x), f ( x ) } C O G c. But (i) and (ii) imply { / 7 ;7 G T} converges to f c - uniformly and hence, p - uniformly. Since p was an arbitrary open cover of Y it converges to / strongly.

The proof of the continuity of / is the same as the one in Theorem 5. □

Corollary 16 (Theorem 1 of [2]). Let X be a topological space. Let { f n} be a sequence of continuous real functions on X which are vanishing at infinity. Let the sequence { / n} converge pointwise to a continuous function f : X —> IR, vanishing at infinity. Let Vx G X the sequence of real numbers |f n(x) — f(x)\ converge

72 IVAN KUPKA

monotonically to 0. Then the sequence { f n} converges to f strongly, so it converges to f uniformly too.

One could formulate other statements concerning functions, number series or multifunctions, which were immediate consequences of Theorem 15. As for multi- functions we hope to present them in another paper, devoted to some problems of what we could name “topological analysis” . We note only, that Theorem 15 gener­alizes all results concerning multifunctions in [6 ] and a result of Beer (Theorem 3 of [1]), because these results were implied by Theorem 10.

References

1. G. Beer, The approximation of upper semicontinuous multifunctions by step multifunctions Pacific J. Math. 87 (1980), 11-19.

2. L. Hola and T. Neubrunn, A remark of functions vanishing at infinity, Radovi Matematicki 7 (1991), 185-189.

3. I. Kupka, A Banach fixed point theorem for topological spaces, Revista Colombiana de Matematicas X X V I (1992), 95-100.

4. I. Kupka and V. Toma, A uniform convergence for non-uniform spaces Publ. Math. Debrecen 47/3-4 (1995), 299-309.

5. K. Kuratowski, Topology, Academic Press, New York, 1966.6 . V. Toma, Strong convergence and Dini theorems for non-uniform spaces, Ann.

Math. Blaise Pascal 4 No. 2 (1997), 97-102.

Ivan KupkaMatematicko-Fyzikalna Fakulta UKMlynska dolina84215 BratislavaSLOVAKIAkupka@fmph .uniba.sk