Neubrunn, Tibor

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaTibor NeubrunnOn quasicontinuity of multifunctionsMathematica Slovaca, Vol. 32 (1982), No. 2, 147--154Persistent URL: http://dml.cz/dmlcz/130644Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1982Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMath. Slovaca 32, 1982, No. 2, 147—154 ON QUASICONTINUITY OF MULTIFUNCTIONS TIBOR NEUBRUNN Various definitions of continuity of multifunctions are given in many papers. They all reduce to the usual continuity if a single valued function is considered. The aim of this paper is to present two definitions of quasicontinuity of multifunctions, which reduce in case of single valued functions to the usual quasicontinuity and to prove various results for quasicontinuous multifunctions. On the other hand some examples are given, showing that there are some differences between the classical results for quasicontinuous single valued functions and those for multifunctions. Notations and preliminary results Given sets X and y, we denote S( Y) the collection of all nonempty subsets of Y. A function F: X—> S(Y) is called a multifunction. In what follows we say function instead of multifunction. If we consider a function F: X—• Y, we refer to it as to a single valued function. If F: X-*S(Y), then for A c Y we denote F~(A) = {x: F(x)nA£0} and F+(A) = {x: F(x)czA}. Now we give two definitions of quasicontinuous multifunctions. A function F: X-+ S( Y) is said to be upper quasicontinuous — briefly u-quasicontinuous (lower quasicontinuous — briefly l-quasicontinuous) at a point x0eX if for any open set V containing F(x0) (for any point z e F(x0) and for any neighbourhood V containing z) and any neighbourhood U of x0, there exists a nonempty open set G cz Usuch that F(x) cz V, (F(x)n V̂ = 0) for anyx e G. IfFis u-quasicontinuous (l-quasicontinuous) at any xeX, then it is said to be u-quasicontinuous (l-quasicontinuous). Note that if a single valued function /: y—• Y is given, then under the natural interpretation of f(x) as a one point set, both above definitions give the usual definition of a quasicontinuous function as given, e.g., in [4] (compare also the original definition in [2]). The w-quasicontinuous, as well as the /-quasicontinuous functions may be characterized by means of quasiopen sets similarly as the single valued quasicon-tinuous functions. 147 Note that a set A cX is said to be quasiopen if A czA° (E° and E denote the interior and the closure of the set E respectively). Proposition 1. A function F: X-*S(Y) is u-quasicontinuous (l-quasicontinu-ous) if and only if for any open set G the set F*(G) (F~(G)) is quasiopen. Proof. We prove the case of /-quasicontinuity, the other case is similar. Let F be /-quasicontinuous, V c Y an open set. If F"(V)=£0, choose x0eF~(V) and an arbitrary neighbourhood Uoi x0. Since F(x0)n V=£0, we obtain from /-quasicont-inuity that a nonempty open set I V c [ / exists such that F(x)n V=£ 0 for any x e W. We have Wcz (F~( V))°, hence x0 e (F~( V))°. Thus F~( V) is quasiopen. The case of F~(V)==0 is obvious. Now let F"(V) be quasiopen for any open V c Y. Let x0eX, yeF(x0), Vany neighbourhood of y and U any neighbourhood of x0. Since VnF(jc0) =£ 0, we have x0eF~(V). Since F~(V) is quasiopen and U open, we obtain that F~(V)nU is quasiopen. Thus a nonempty open set WaF~(V)nU exists. Hence F(x)n V=£0 for any xeW. Thus the /-quasicontinuity at x0 is proved. For a single valued function / : X—• y the somewhat continuity was introduced in [1]. A function is called somewhat continuous if for any open V c Y for which f~(V)4--0 we have (f~(V))°^0. In a natural way we introduce two notions of somewhat continuity for mul-tivalued functions. A function F: X-+S(Y) is said to be u-somewhat continuous (l-somewhat continuous) if for any open set V c Y for which F*( V) =£ 0(F~( V) =£0) we have (F*( V))° =£ 0(P^( V))° =£ 0. The following proposition relates the u-quasicontinuity to u -somewhat continuity and /-quasicontinuity to /-somewhat continuity. Since an analogical result with an analogical proof is known for single valued functions (see[5]), we omit the proof. Proposition 2. A function F: X-*S(Y) is u-quasicontinuous (l-quasicontinu-ous) if and only if there exists a basis 3ft of open sets in Xsuch that the restriction F\ U is u-somewhat continuous (l-somewhat continuous) for any Ue£ft. Somewhat and quasicontinuity of multifunction.? in product spaces The classical result of Kemp is ty (see [2]) asserts that a real function of two real variables, which is separately quasicontinuous, is quasicontinous as a function of two variables. In [4] and [5] it was generalized for more general topological spaces. In [6] some related results for somewhat continuous functions were obtained. In this part we give some results of somewhat continuity and quasicontinuity of multifunctions, which generalize the above mentioned. On the other hand we give 148 examples showing that a straightforward generalization result of Kempisty is not possible either for w-quasicontinuity or for /-quasicontinuity. But the results may be considered as a natural generalization of a theorem from [2] concerning the quasisemicontinuity of a real function. To obtain it one has to associate with a real function / a multivalued function x—• ( — oo, f(x)). Theorem 1. Let X be a Bake space, Y second-countable and Z a normal topological space. Let F: Xx Y-*S(Z) be a function assuming as values only closed sets. Let the sections Fx be u-somewhat continuous for every xeXand the sections Fy u-somewhat continuous and l-quasicontinuous for every yeY. Then F is u-somewhat continuous. Proof. Suppose Fnot to be w-somewhat continuous. Then there exists an open set HCLZ such that F+(H)*0 and (F*(H))°==0. Hence the set of those points (*', y') for which F(x',y')nH'£0 (1) is dense in X x Y. (E' denotes the complement of the set E.) Let (xo, y0) e F^(H). Since F(x0, yo) is a closed set and Z a normal space, there exists an open set Hi such that F(xo,yo)c:HlczH1czH. The function Fy° is w-somewhat continuous and (Fy°y(Hi) =£ 0, hence an open set G exists such that Fy°(x)czHi, for any xeG. Denote by { V„}n^i the countable base of Y and put for w = l, 2, ... An = {x:xeG, V - c K ^ ) } . We have G = U A , (2) n--i The inclusion (J A„ c G is trivial. Now let xeG. Since Fx( y0) = Fy°(x) c= Hu we n-=l have F^(Hi) =£ 0. Hence by w-somewhat continuity of Fx we obtain ( K ( H i ) ) ° * 0 . From the last it follows that a number n exists such that Vncz(Fx(H,))dFx(H1) proving that xe A,. Thus (2) holds. 149 Now we prove that any of the sets An is nowhere dense in G. Let Wcz G be a nonempty open set. According to (1) there exists a point (x', y'), x' e W for which F(x', y')nH' =£0. Let zeF(x', y')nH'. Choose a neighbourhood V of z such that VnJF/i = 0. Owing to the /-quasicontinuity of Fy' at x' we have that a nonempty set Wcz W exists such that F(x, y')nVJ=0 for any xe W. Since y ' e Vn, we have x£An for any x € VV. Thus ^ n A „ = 0, proving that the set An is nowhere dense. It follows from (2) that G is of the first cathegory, in contradiction to to the assumption that X is a Baire space. The theorem is proved. R e m a r k 1. It is easily seen from the proof that the assumption that Fx are w-somewhat continuous for all x e X may be weakened. It is sufficient to suppose that Fx are u-somewhat continuous with the exception of a set of the first category. Theorem 2. (Theorem on product u-quasicontinuity.) Let X be a Baire space. Y locally second-countable and Z normal. LetF: X—>S(Y) be closed-valued with u-quasicontinuous x-sections for every xeX and let for every yeYthe y-sections be both u-quasicontinuous and l-quasicontinuous. Then F is u-quasicontinuous. Proof. The collection of all sets UxV, where U, V are open in X and Y respectively, is a base of open sets in X x Y. Since Y is locally second-countable, we may suppose, with no loss of generality, that V is second-countable. From the assumptions of the theorem, we have immediately that Fx is w-quasicontinuous and hence w-somewhat continuous on any fixed V, for every x belonging to a fixed U, and similarly F is both w-quasicontinuous and /-quasicontinuous on U for every yeV. Thus by Theorem 2 we have that F/Ux Vis u-somewhat continuous. Since it is true for arbitrary Ux V, we obtain from Proposition 1 that F is w-quasicont-inuous. Neither Theorem 1, no Theorem 2 may be proved if we omit the assumption that Fy is /-quasicontinuous. Thus a straightforward analogy of the Kempisty Theorem for single valued functions is not true for multifunctions. E x a m p l e 1. Let us consider R2 with the usual topology. Let S be a countable subset of R2, dense in R2 and such that on any horizontal and any vertical line there is at most one point of the set 5. Such set may be constructed as follows. Denote by {Bn}n=i the sequence of all mutually distinct spheres with rational centres and rational diameters. Choose (pi9 qi)eBi. Suppose that for given n^l the points (p„ qi) were constructed for i = 1, 2, ..., n such that on every horizontal and every vertical line there is at most one of them. Now take (pn+i, qn+i) e Bn+i such that p„+i =£ pt, qn+i £ q, for i = 1,2,..., n. Then evidently on every horizontal and every vertical line there lies at most one of the points (p,, qi) (i-= 1, 2, ..., n). The set of values of such a constructed sequence {(pn, qn}n=i may be taken for S. Define F: R2-+S(R1) as follows: F(r v, = (<°>' » (x,y)*S, nX'y) l{0, 1, 2, ..., n}, if (x,y) = (pn,qn). 150 The sections Fx, Fy are w-quasicontinuous for every x and every y, respectively. Let us check it for Fx. If x±pn, where n = 1, 2, ..., then Fx(y) = {0} for every real number y and the w-quasicontinuity of Fx is evident. If for some n x=pn, then Fx(y) is equal {0}, with the exception of exactly one point y = q«, where Fx(qn) = F(pn, qn) = {0, 1, 2, ..., n}. But the w-quasicontinuity in this case is obvious too. Analogical reasoning shows that F7 is w-quasicontinuous for any real y. The function F is not w-quasicontinuous at any (x, y) e R2. Because if (x0, y0)eR2, we have F(x0, y0) = {0, 1, 2, ..., n}, where wSO. Choose the open interval (—1, « + l). We have (—1, rc + 1) ZD F(X0, y0). Now let £/ be any neighbourhood of (x0, y0) and Wcz U any nonempty set. Evidently W contains a point (p*, qk), where A:>/x. Hence F(p*, qk) = {0, 1, 2, ..., k} <t ( - 1 , H + 1). Thus Fis not w-quasicontinuous at (x0, y0). In an analogical way we may prove that F is not even w-somewhat continuous. Theorem 3. Let X be a Bake space, Y second-countable and Z a regular topological space. Let F: XxY —> S(Z) be a multifunction such that Fx is l-somewhat continuous for every xeX, Fy is both l-somewhat continuous and u-quasicontinuous for every yeY. Then F is l-somewhat continuous. Proof. Suppose F not to be /-somewhat continuous. Then there exists an open set HczZ such that F~(H)±0 and ( F " ( H ) = 0 . Hence the set of all (x',y') for which F(x', y')nH=0 is dense in Xx Y. Let (x0, y0) e F~(H) and let z e F(x0, y0)nH. Choose a neighbourhood JFJi of the point z such that HtczHrczH (3) From the fact that zeF^^x^nHi we have (FTo)"1(Hi)^0. Hence from the /-somewhat continuity of F** a nonempty open set G exists such that Fy°(x)nHi^0 for every xeG. Let { V„}*--i be a countable base of Y. We can proceed analogicaly to the proof of Theorem 1, but we put An = {x: xe G, F(x, y)nHi =?--0 for any yeVn}. Using the /-somewhat continuity of the sections Fx we prove similarly as in the 00 proof of Theorem 1 that G= [J An. n = l Now to obtain a contradiction, we have to prove that An is nowhere dense for n = 1,2, ... If W is a nonempty open set, WczG, take in Wx V„ a point (x', y') such that F(x', y')czH'. According to (3) the set Hi is an open set containing H'. Hence 151 F(x', y')aH'i. The M-quasicontinuity of F at JC' implies that a set W c W, which is nonempty and open, exists and F(x, y')czH[ for any xeW. Thus F(x,y')^Fy(x)nH1=:0. Since y'eVn, we have xiAn. Hence Wc\An=$, proving that A, is nowhere dense. The proof is finished. In the same way as Theorem 2 was proved by means of Theorem 1, we obtain from the preceding result the following Theorem 4. (Theorem on /-continuity in the product.) Let X be a Bake space, Ylocally second-countable and Z regular. Let for every xeX the sections Fx be l-quasiconiinuous and for every yeY the sections Fy are both u-quasicontinuous and l-quasicontinuous. Then F is l-quasicontinuous. R e m a r k 2. The somewhat continuity in Theorem 3 as well as the quasicon-tinuity of the sections Fx in Theorem 4 may be weakemed in a similar way as it was done for the u-somewhat continuity in Theorem 1. (See Remark 1). Since the case of a single valued function implies that w-quasicontinuity and /-quasicontinuity coincide with the usual quasicontinuity, we obtain from Theorem 4 a general variant of the Kempisty Theorem for single valued functions. Corollary. (See [5]). Let Xbe a Baire space, Ysecond countable and Z regular. Let f: Xx Y-+Z be a single valued function with fx quasicontinuous for every xeX and with f quasicontinuous for every yeY. Then f is quasicontinuous. R e m a r k 3. An analogical Corollary may be obtained from Theorem 3, guaran-teeing the somewhat continuity of the point functions f: Xx Y-+Z under the assumptions of the somewhat continuity of their jc-sections and the quasicontinuity of the y-sections (See [6]). The assumption of u -quasicontinuity of Fy in theorems 3 and 4 may not be omitted. E x a m p l e 2. Let S = {(pn, qn): w = l , 2, ...} have the same meaning as in Example 1. Define F: R2-*S(R1) as follows: F(x,y) = {\ ir=\{n}, if (x, y) = (pn, qn) (w = l , 2 , . . . ) , . {1 ,2 , . . . } , if (x,y)±(pn, qn) (/z = l , 2 , . . . ) . Fx and F7 are /-quasicontinuous for every xeR1 and y e R\ It is sufficient to prove it for Fx, because the case for F is symetrical. If x£pn (w = l , 2, . . . ) , then the /-quasicontinuity is evident, because Fx(y) — {\, 2, ...} for all ye Y. If x = pn for some n, then tx{y)-\{l,2,...}, if y±qm. But /-quasicontinuity of a function of this type may also be immediately verified. 152 F is not /-quasicontinuous at any point (x0, y0). In fact there exists at least one positive integer n0 such that n0eF(x0, y0). The interval (w0 —1, n0 + l) is a neighbourhood of n0. If we choose a neighbourhood G of (x0, y0) and any nonempty open set Wc G, then evidently Wcontains a point (pn, qn) =£ (p„o, q^). We have F(pn, qn)n(n0-l, n0 + l) = {n}n(n0-l, rio + l) = 0, hence F is not /-quasicontinuous at (x0, y0). F is even not /-somewhat continuous, because if we take for any n the set F~((—n — 1, n + 1)), it is a nonempty set which does not contain a dense set {(Pk,q*), k±n (* = 1, 2 , . . . . ) T h u s ( F - ( ( - n - l , / 2 + l)))o = 0. A remark on the topology for w-quasicontinuity Given two topological spaces X and Y, one has the possibility to define a topology on S(Y) and to consider the quasicontinuity of F: X->S(Y) as the quasicontinuity in the usual sense, taking the values of F as points of the topological space S(Y). Several topologies were considered oh various subcollections of S(Y), to define various types of continuites of multivalued functions. Some of them were defined on the subcollection F(Y) cz 5(Y) of all closed sets of Y, the others on the subcollection K(Y) of all compact subsets of Y (see e.g. [3] p. 392—393). In [7] a topology on S(Y) is defined in such a way that the base of all open sets is the collection of all families S(G)cz S(Y) where G is any open set and S(G) denotes the family of all nonempty subsets of G. By means of such a topology there is defined a type of continuity (in [7] it is called weak continuity). The following proposition shows that our notion of w-quasicontinuity may be described in this topology. Proposition 3. A function F: X—• S( Y) is u-quasicontinuous if and only if it is quasicontinuous as a single valued mapping into the topological space S(Y) with the topology defined by means of the base S(G), where S(G) is the collection of all nonempty subsets of G and G is atiy open set in Y. Proof. Let Fbe w-quasicontinuous at x0eX. Let 5(G) be element of the base containing F(x0) and U any neighbourhood of x0. Then F(JC0)CZG and by w-quasicontinuity there exists a monempty open set Wc U such that F(x) c G for every xeW. The later means that F(x) e S(G). Thus the quasicontinuity at x0 is proved. Now, let Fbe quasicontinuous at x0. If G is an open set containing F(x0) and Ua neighbourhood of x0, then there is a nonempty open set Wcz U with F(x) e S(G) for every xeW. This means F(x) c G for every xeW and the w-quasicontinuity is proved. 153 Of course the somewhat continuity of the mapping F: X-+ S( Y) considered in the usual sense as the somewhat continuity defined by means of the topology on S(Y) gives the w-somewhat continuity. Thus from the above consederations and from the known results for single valued functions we may obtain the parts of Proposition 1 and Proposition 2 concerning the w-quasicontinuous functions. Note. As the author has found out the notion of quasicontinuity for multifunc­tions was for the first time introduced in [8] in connection with different results. REFERENCES [1] GENTRY, K. R. and HOYLE, H. B.: Somewhat continuous functions. Czech. Math. J. 96, 1971, 5—12. [2] KEMPISTY, S,: Sur les fonctions quasicontinues. Fund. Math. 19, 1932, 184—197. [3] KURATOWSKI, K.—MOSTOWSKI, A.: Set theory with an introduction to descriptive set theory. PWN, Warszawa, 1976. [4] MARTIN, N. F. G.: Quasicontinuous functions on product spaces. Duke Math. J. 28, 1961, 39---44. [5] NEUBRUNN, T.: A generalized continuity and product spaces. Math. Slovaca, 26,1976,97—99. [6] NEUBRUNN, T.: Generalized continuity and separate continuity. Math. Slovaca, 27, 1977, 307—314. [7] STROTHER, W. L.: Continuous multi-valued functions. Boll, da Soc de Mat. de Sao Paolo, 1958, 87—120. [8] POP A, V.: Asupra unei decompuneri a cvasicontinuitatii multifunctilor, St. Cere. Mat. 27, 1975, 323—328. Received March 24, 1980 Katedra teorie pravděpodobnosti a matematické] statistiky MFF UK Mlýnská dolina 842 15 Bratislava О КВАЗИНЕПРЕРЫВНОСТИ МНОГОЗНАЧНЫХ ОТОБРАЖЕНИИ Тибор Нойбрун Резюме Исследуются отображения, определенные на произведении двух топологических пространств X, У, принимающие как значения непустые множества топологического пространства 2. Оп­ределены два типа квазинепрерывности таких отображений. Главным результатом работы являются теоремы, установливающие связь между отдельной квазинепрерывностью и квазинеп­рерывностью этих отображений. 154 

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On quasicontinuity of multifunctions

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On quasicontinuity of multifunctions

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