Wilhelm, Marek

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

Commentationes Mathematicae Universitatis CarolinaeMarek WilhelmAlmost lower semicontinuous multifunctions and the Souslin-graph theoremCommentationes Mathematicae Universitatis Carolinae, Vol. 23 (1982), No. 1,147,148--149,150--158Persistent URL: http://dml.cz/dmlcz/106139Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 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 mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE 23,1 (1982) ALMOST LOWER SEMICONTINUOUS MULTIFUNCTIONS AND THE SOUSLIN-GRAPH THEOREM M. WILHELM Abstract: Almost continuous mappings and almost lower semi continuous multifunctions are investigated. A Souslin-graph theorem for multihomomorphisms with values in an ana-lytic space is proved. Kev words: Multifunction, almost lower semicontinuity, Souslin-graph. Classification: 54C60 1. Introduction. The term "almost continuity" is used here in the sense of Bradford and Goffman L4l. We show that each almost continuous mapping having the Baire property and taking values in a regular space is continuous (Theorem 4). It follows that each almost continuous mapping having a Sous-lin graph and taking values- in an analytic space is continu-ous (Theorem 6)• We define and investigate "almost lower semicontinuity" of multifunctions. Under category type assumptions certain multifunctions possess automatically this property (Theorems 1,2,3). Let F:G—» H be a multihomomorphism with F (H) = G. If G is of second category and H is separable or Lindelof, then F is lower semi continuous iff it is lower-Baire (Theorem 5). If G is inductively generated by second category groups and - 147 H is analytic, then F is lower semicontinuous provided it has a Souslin graph (Theorem 7). This version of the Souslin-graph theorem ia based on ideas due to Fro If k ETl and C6]. 2- Almost lower semicontlnuous multifunctional. Almost continuous mappings were considered first, as it seems, by Blumberg [I] and Block and Cargal [2], under unlike names. The term "almost continuity" was used by Bradford and Goffman [4]. Let X and Y be topological spaces and f a mapping of X to Y (f:X—.>Y),. Given x eX, f is said to be almost continuous at x if for each open set V in Y containing f(x), xe Int D(f (V)), Here D(E), where EcX, denotes (as in C103) the set of all points x of X that are of second category in X relative to I (i.e. U n E is of second category in X for each open U a x ). This definition of almost continuity is equivalent to those given in the above-mentioned papers, and can be extended, in a natural nayf to multifunction. By a multifunction F of X to Y (F:X—> Yl we mean a function which to every point xeX as-signs a subset F(x) of Y (not necessarily closed or nonempty). Definition. A multifunction F:X—^ Y is almost lower se-mi continuous at x if for every open set V in Y xc-F"1(V) impliea Xclnt D(F-1(V)). Here the inverse image F (V) denotes, as always, the set of all x' satisfying f(x')oVtjO. The set of all points x of X such that F is almost lower semi*-continuous at x will be denoted by L^(F); in case I- (F) - Xf F will be called almost lower semicontlnuous. Thus, F is al-most lower semicontlnuous if and only if for every open set V 148 in Y F ^ t V J c I n t D(F"1CV)). Let L(F)/ stand for the se t of a l l points x e X such that F i s lower semi continuous at x, i . e . x e l n t F~ (V) for a l l open V c Y intersect ing F(x) . Notice that LQ (F)nF," 1 (Y)cInt D(F" 1 (Y))cInt D(X) and L(F)nF" 1 (Y)nInt D(XJ)CL Q CF) , while obviously X \F~X(Y) c L(F) O L ^ C F ) . In particular, if X is a Baire space Ci.e. X = DCX)), then L(F)cLfl(F). If F ia al-most lower semicontinuous, then F (Y) is a Baire apac* (in itself). The usefulness of the property of almost lower semiconti-nuity stems from the fact that it is automatically satisfied under some category-type assumptions, while, on the other hand, it is a convenient starting point to the Souslin-graph, closed graph, open mapping and Blumberg theorems. The following theorem extends some observations from L3], r?l and C41. Theorem 1. Let F be a multifunction of X to Y. If the 8pace Y ia second-countable, then (i) The set La(F) ia residual in X; (ii) the restriction FlLa(F) :La(F.l—-* Y is almost lower aemicontinuous* More generally, for each residual set A c L (F), F|A ia almost lower semicontinuous. Proof, (i) Let *VnJ be a base for Y. A point x t X la not in L If I if and :>nly if there ia n auch that xe F~*(V ) 149 and x$Int D(F"*1(Vn)). Thus L (F) = X \ w , LF~L( Vnl\Int D(F"1(Vn))3. a ŝ'l n n Each set of the form E\ Int D(E) is of first category because DdE) \ Int D(E) is closed co-dense and E\D(E) is of first ca-tegory by the Banach category theorem (cf. L101). Henc* L (F) is residual. (ii); Let Ac LQ(F), be residual in X. Put E « F"l{\). Then AnD(E) = krs DlAoBJc Dft(AnE) and xcAnlnt D(E)c Intft(AnD(E) )JC IntADft(A oE), for x e A o E , which shows almost lower semicontinuity of F|A. By a graph of a multifunction F:X—>Y we mean the set Gr F M(x,y}:yeF(x)}cXxY, In the following the letters G, H stand for topological groups. We say that F:G - > H is a multihomomorphism if Gr F is a sub-group of GxH. For multihomomorphisms we have the following simple criterion of almost lower semicontinuity. Lemma 1. A multihomomorphism F:G —> H is almost lower semicontinuous if (and only if) for each neighbourhood V of ert- the inverse image F~ (V) is of second category in G. Proof. Let V be a symmetric neighbourhood of e K and put E =- F"1(V) and U == Int D(E). Since E\U is a first category set in G (by the Banach category theorem; see the previous proof) and E is of second category (by the hypothesis), the set U is non-empty. This implies that eQ e Int D(F"" (v )) be-cause 150 -eQCtr^cInt (LDCF",1(Vr))3^)r int D(LF"1(V)j 2), « Int DLF"1^).], Thus eG€Lfl(F). If now x<f F"1(T), where V is open in H, then eQ€ F^CVy"*1), where ypflxInV; hence eQeInt D[F""1(Ty""1).l, and so x ̂  Int B([F"1(Vy""1)J x) » Int D£F"*1(V)3 ; x£ LQ(F). Now we need a generalization of a lemma of Pettis [13] for multihomomorphisms. First a definition ([13]). A subset E of H is £-bounded in H if for every neighbourhood V of e H there exists a sequence 1y jcl such that B c^ KJ y^ o^yn* Each separable or Lindelof (in particular, ^-compact) subspa-ee E of H is &-bounded in H. If H is metrizable, the three notions ( 6"-boundedness, separability and Lindelof property of EcH) coincide. Lemma 2. Let F:G—?»H be a multihomomorphism such that F(G), is €» -bounded in H. If F (H) is a second category set in G, then so is F~ (V) for any neighbourhood V of e„. Proof. Given open V 3 e H , choose-[ ynJ c F(G> so that F(G)c <= ^ y-J^Vy-n* Choose xne=F iCyJ. Then F X(H) « L A Xr,E u oEx , where E -» F~ (V), Hence E is of second category in G. Lemma 3 (L133). If H is ^-bounded, then each set Ec B is ^-bounded in H. CO Proof. Let H ~ tj\J^ ynT»->Vyn, where V i s a neighbourhood of eH and i y n l c H , Choose h n e ! r . y n V whenever poss ible (neN^) and h n eIr»Vy n whenever poss ib le (neNp) . Then T>.cNl, « <n, e N " n ,.n, e M n <n e J\L n Since T^e H was arbitrary, E i s 6^-bounded. If now H i s 6f-bounded, then F(G) i s (^-bounded in H 151 (Lemma 3) and the lemmas Nos. 2 and 1 may be applied, provid-ed F" (H) ia of second category. Thus we get Theor•m 2. Let F be a multihomomorphism of G to H, whe-re H id a ^-bounded group (e.g. separable or Lindelof). If F \H) is of second category in G, then F is almost lower se-mi continuous. For linear multifunctions the as9umption of e'-bounded-ness of the range apace may be omitted and the proof reduced. Let S and T be topological vector apacea; F:S—-> T ia a line-ar multifunction if Gr F ia a linear aubspace of S?<¥. Theorem j. Each linear multifunction F:S—-> T such that F"**(T) is of second category in S is almost lower semi continu-ous* CO Proof. Let V be a neighbourhood of 0^. Since T ^^J* nV, F~X(T) -jSj nF^CV). Hence F~X(V) is of second category in S and we apply Lemma 1. That is all about "automatic" almost lower aemicontinui-ty. Now we will consider the question, when almost lower se-micontinuity (resp. almost continuity) implies lower semicon-tinuity (resp. continuity). For mappings we have a quite sa-tisfactory answer: Theorem 4. Let X be a Baire space, and let Y be a regu-lar space (even not neces9arily T ). A mapping f:X—* Y is continuous if (and only if) it is almost continuous and haa the Baire property. Proof* Let xef~ (V), where V is open in Y. Choose open - 152 -set WcY with f(x)c W and WcV. Since f is almost continuous at x, xeU =- Int D(f-1(w)). Let ueU; we will show that f(u)c 6 Tm Let Z be an open neighbourhood of f(u). Since f is al-most continuous at u, uelnt D(f* (Z)). Hence Unf" (Z) is a second category set in X. Since f has the Baire property, the-re exists an open set GcX such that GAf (Z) is of first category in X. Now UnG is of second category in X. It fol-lows that f~ (w)/nG is of second category in X. Hence f (w)n Of" (Z) is of second category in X, which yields W n ^ D . Thus we have proved that tin)eW* The theorem cannot be extended to multifunctions, with-out additional assumptions. Example 1. Each of the following multifunctions is al-most lower semicontinuous and lower-Baire (i.e. F~ (V) has the Baire property whenever ?cY is open), but not lower semi-continuous. (a) Fix) * 41? for xc I\ Q and Fix) * 0 for xc In Q (I =-[0,1], Q - the rationale); F:I—> I is single-valued. (b) Fix) =- Y for xeI\Q and Fix) » U? for xe In Q, where Y is the discrete spac* 40,11; F (Y) * I. (c)/ (cf. C53). Let I * P^u P2» where P* are dense and co-dense Gd^-sets in I, and let g be the natural mapping of the space Y * P-,® P2 onto I; g is continuous and almost open (i.e. g(U$c Int D(g(U)) for each open Uc Y). Define F « g"1. An analogue of Theorem 4 for multihomomorphisms holds true. To see this, let FiGs H be an almost lower semi conti-nuous lower-Baire multlhomomorphism, and consider the induced mapping f:X->Y, where X » F^CH) and Y « H/F(eQ) (Y need not 153 be a T -space of a group) . The assumptions of Theorem 4 a re s a t i s f i e d (Y i s r e g u l a r by [ 9 ; 5.19, 5 .20 ] ) . Hence f i s con-t i n u o u s . Since the quo t i en t mapping y.VL—>Y i s open (cf • L9; 5.17.1), t h i s impl ies lower semicont inu i ty of F:X —->H. X i s a second category subgroup of G having the Baire proper -t y ; by the Banach-Kuratowski-Pet t is theorem (cf. [ 1 ; Theorem 1J , [10; 13.XI] and L l3 ; Theorem 1 J ) , X i s open in G. Hence F:<3—^H i s lower semi cont inuous. Thus, in view of Theorem 2 , we get Theorem ?. Let F be a multihomomorphism of G to H such tha t F (H) i s of second category in G. ( i ) F i s lower semi-continuous i f (f*nd only i f ) i t i s almost lower semi continuous and lower-Baire . ( i i ) Suppose the group H i s ©'-bounded ( e . g . separable «?r L inde lo f ) . Then F i s lower semicontinuous i f (and only i f ) i t i s lower-Baire . For l i nea r F, ( i i ) holds with no assumption on the range vec to r space (by Theorem 3 ) . 3 . Souslin-graph theorem. A Tyspace Y io said to be an a n a l y t i c space (or a K-Souslin space) i f there e x i s t s a Po l ish space X and a compact-valued upper semicontinuous mult ifunc-t i o n $ of X onto Y (Fro l ik L6J; for some equivalent d e f i n i -t ions see 16] and [ 8 ] ) . Each ana ly t i c space i s a Lindelof space, hence paracompact and normal (cf. L 61 a n d l 8 j ) . By a Souslln s e t , in a given space, we mean the r e s u l t of per -forming the Souslin operat ion (A) (denoted a l so S) on a sy s -tem of closed s e t s in the space. Since the c o l l e c t i o n of a l l s e t s having the Baire property i s closed under the operat ion - 154 -(A), each Souslin set has the Baire property (cf. LlQD). L. Schwartz [151 proved that if S and T are locally con-vex spaces, S - ultrabornological (i.e. inductive limit of Banach spaces), T - continuous image of a Polish space, then each borel graph linear map f : S — > T is continuous and each continuous linear map g:T o n °> S is open. Frolik 171 proved that if G is a vector space which is inductively generated by second category vector spaces and H is an analytic locally convex space, then (1) each Souslin-graph homomorphism f:G—> H is conti-nuous. Martineau [11] proved, among other results, that if G is a second category analytic group and H is an analytic group, then each continuous homomorphism g:H onto> G is open; Perez Carreras [12] showed that the theorem remains true if G is not necessarily analytic. In this section we shall show that if G is inductively generated by second category groups snd H is an analytic group, then the statements (1) and (2) hold, where (2) each Souslin-graph homomorphism g:H onto> G is open. The main tools are Theorem 5 and the following lemma due to Rogers and Willmott [141 (a nice proof is given in Frolik L 7; Lemma 13 ). Lemma 4. Let F : X — ^ Y be a multifunction, where Y is an analytic space. If Gr F is a Souslin set in Xx Y, then F is upper-Souslin (i.e. F~ (A), is Souslin whenever A is clo-sed) , and hence upper-Baire. Combining the lemma with Theorem 4 we get 155 -Theorem 6. Let X be a Baire space, and let Y be an ana-lytic space. A mapping f:X—~»Y is continuous if (and only if) it is almost continuous and its graph is a Souslin set in XxY. Example 1 (c) shows that an almost lower semicontlnuous Souslin-graph multifunction F:X—> Y need not be lower semi-continuous, even if X is compact, Y Polish and Gr F closed. Lemma 5. Each upper Baire multihomomorphism F:G—>H is lower-Baire. Proof. Let y be the canonical mapping of H onto H/F(eQ). Let Y be open in H. Since <f is open and continuous, the set F~(V), * F"X(Hi\ F " 1 ^ "hu/F(eG) \ f CV)] ) has the Baire property in G. If F i s lower-Baire and F(eQ) i s compact, then g> i s c l o -sed (cf. [ 9 ; 5.18]) and, consequently, F i s upper-Baire. With-out the compactness assumption, the converse to Lemma 5 i s not true• Example 2 . Let H be a closed normal subgroup of H, G » * H/H , <$ :H —> G the canonical quotient mapping and F » y> t :G —*H; F i s even lower semicontlnuous and has a closed graph. Nevertheless F need not be upper-Baire ( i t i s upper-Baire pro-vided H i s analyt ic; see Lemma 4 ) . Take for instance H * R * R a and H * 4 0 * x R d > where Rd denotes? R (the rea l s ) endowed with the d i s cre te topo logy. Choose a set A in R which has not the Baire property and put K = -f (x,x) € H:x £ A ;̂ K i s closed and F^CK) » A. - 156 Now we are in a pos i t ion to derive the Souslin-graph theo-rem. Theorem 7. Let F:G—>H be a multihomomorphism, where H i s an analyt ic group. Assume that ( i ) F~ (H) i s of second category in G; or ( i i ) F" (H) »- G and the topology on G i s inductively gene-rated by homomorphisms h ^ iQ^—> G, where "fG^ : cc e AI i s a f a -mily of second category groups. If Gr F i s a Souslin set in G.»<H, then F i s lower semicontinu-ous. Proof, ill Follows from Lemmas 4 , 5 and Theorem 5 ( i i ) . ( i i ) Fix any e c c A. By Lemma 4 f F i s upper-Sous 1 in; hen-ce F« h^ i s upper-Sous1in, and so upper-Baire. Lemma 5 shows that F o h ^ i s lower-Baire. By Theorem 5 C i i ) f F- b^ i s lower semicontinuous. Since cO was arbitrary, the assert ion fo l lows . Clearly, Theorem 7 y i e ld s the statements mentioned i n the passage before Lemma 4 . R e f e r e n c e s Ll] S. BANACH: Uber metrische Gruppen, Studla Math. 3(1931), 101-113 . L2>] H.D. BLOCK and B. CARGAL: Arbitrary mappings, Proc. Amer. Math. Soc. 3(1952), 937-941. L3] H. BLUMBERG: New properties of a l l real funct ions, Trans. Amer. Math. Soc. 24(192?), 113-128. L4] J.C. BRADFORD and C. GOFFHAN: Metric spaces in which Blum-berg's Theorem holds, Proc. Amer. Math. Soc. 11 (1960), 667-670. 157 15] T. BYCZKOWSKI and R. POL: On the closed graph and open mapping theorems, Bull. Acad. Polon. Sci. 24 (1976), 723-726. [61 Z. j?*ROLiK: A contribution to the descriptive theory of sets and spaces, General Topology and its Rela-tions to Modern Analysis and Algebra, Proc. Symp. Prague 1961, 157-173. [7] Z. FROLIK: On the Souslin-graph theorem, Comment. Math. Univ. Carolinae 9(1968), 243-249. L8J Z. FROLIK: A survey of separable descriptive theory of sets and spaces, Czech. Math. J. 20(1970), 406-467. [91 E. HEWITT and K.A. ROSS: Abstract Harmonic Analysis, vol. I, Berlin 1963. tlOJ K. KURATOWSKI: Topology, Vol. I, New York 1966. [11] A. MARTINEAU: Sur des theorems de S. Banach et de L. Schwartz concernant le graphe ferm£, Studia Math. 30(1968), 43-51. tl?] P. PEREZ CARRERAS: Espacios K-Suslinianos y gruD03 topo-logicos, Collectanea Math. 27(1976), 105-108. [131 B.J. PETTIS: On continuity and openness of homomorphisms in topological groups, Annals Math. 52(1950), 29 3-308. £141 C.A. ROGERS and R.C. "WILLMOTT: On the projection of Sous-lin sets, Mathematika 13(1966), 147-150. C151 L. SCHWARTZ: Sur le th£oreme du graphe ferm4, C.R. Acad. Sci. Paris 263(1966), 602-605. Institute of Mathematics, Technical University, pi. Grunwaldzkl 13, 50-377 Wroclaw, Poland (Oblátům 14.8. 1981) - 158 

Almost lower semicontinuous multifunctions and the Souslin-graph theorem

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Almost lower semicontinuous multifunctions and the Souslin-graph theorem

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