Kyungpook Math. J.
Volumc 24, Num야r 2 December. 1984
ON MULTIFUNC1'IONS SATISFYING CERTAIN SEMICON1'IN1JITY TYPE CONDI1'IONS
By Jesse P. Clay and James E. Joseph
The purpose of this paper is to improve a number of reccnl rcsults for multi- funetions from topologicaI spaces lo topological spaces (hercafter refcrrerr 10 as spaces). to provide paraIleIs to severaI known results for multifunctions in varied settings. to extend to mu1tifunctions somc known results for functions and to provide further applications of multifunctions.
Research on multifunctions in general and semicontinuÎty o( multifunctìons in particular has been much in evidence (see c. g. [1], [7], t11], [12], [1외 ,
[2이， [21], [24], [25], [28], [27). [31], [32] -[35] and rhe Iist of references for [35]). The notions of Iαver- and upper-semicontinuity of multifunctions extend the notion of continuity of functions ro multiIunctions and recently Joseph [21] and Smithson [32] have cxtcncled rcspectively the notions of Ð- continuity 01 functions of Fomin [9] ancl weak continuity of functions of Levine
[26j to multifunctions. Thcy have aIso provided an array 01" applications of these concepts. \'Ve wiU [urrher investigate these extensions, introducc new extensions of continuity conditions as wcll as SOme graph conditions and offer somc nc、、 applications.
Many of the results in lhis paper are obtained lhrough the use of thc Ð-c1osure opcrator, c1
e .
Sincc the introduction of c1e
by \'eliεko in [38] this operator hasbeen stuclied and utilized by many authors (sco [1], [2J, [3), [4], [5) ‘ [6].
[13]. [14]. [15]- [17). [2이 - [24). [36]. [37). [3일 [4이 ).
1. Semicon.~l1 uity type conditio
’ ‘
s and graph properties for muliifunctions and presentεtion of somc charv.cterization results. Let X and Y bc !onempty sets. A multifunction from X to Y is a function from X to P(Y)-[이 \vhere P(Y) is the family of subscts 01 Y (if α is a multifunction from X to Y 、，ve\YiU
,
vritc αeM(X ， Y)). 11 X is a space and Aζ X we denote by c1 (A) , int(A),ε(A) and F(A) (E(x) and r(x) if A= [xl) the closure, interior, collection of opcn S8tS containing A and collection of closed sets containing some element of 1:(.4), rcspectively; cle(A) is the intersection of the elements of F(A) [2이 . I f
180 Jesse P. Clay al1d James E. joseph
αeM(X， Y), Aζ X ， BCY, QCP(X), QlζP(Y)， wc denote
Q
α (x) by a(A),{xeX: α(x) nB;;60} by a• 1(B), (xeX: α(x)CB) by a + (B), (a(F) : FeQ) by a(Q) and α I(F): FeQl} by a-1(QI); it i5 、veìl-known and lairly obvious that X-α+(B) =a-1(Y-B). Lct X and Y be spaces and aeM(X, Y); a is upper-semicontinuoZls (u. s. c.) at xeX if for each WeE(a(x)) 50me Veε (x) sati5- fie5 a(V)CW. and α is uþþer-semico1l-titLUOllS (u. s. c.) if a is u. s. c. at cach XêX; a is weakly uþþer-semiC01zU1ZUOUS (ι‘. U. s. c. ) at xεX if for each WerCa(x)) some VEε(x) 5ati5fie5 α (V)CW， and α is weakly uþþer-semicontz'nuous if α 15
W. U. S. c. at each xeX; a is ()-uþþer-semicontim/,oz/,s ({J
•
e. s. c.) at xeX il for each Wer(α (x)) 50me Ver(x) satisfies a(V)ζW and a is I}-upper-semicoltli-’1αo“s (1) κ. s. c.) if a is I}-u. s. c. at each xeX.
We come now to our first two results. We givc the proof of Theorem 1. 2 only. Recall that a subset A of a space is regular-closed if A=cl(int(A)).
THEOREy! 1. 1. The following stale11lenls are eqαtνalellt for spaces X, Y aηd
aeM(X, Y):
(1) Tlre "
‘’
tl tifuncJion a is lO. tt. s. C.(2) Each BCY satisfies cl [a -1(int(clo(B)))) ζa-l(cl8(B))-
(3) Each open BCY sa#sfies cl [a ‘(jnt(cI(B)))) Ca '(cl(B)) (4) Each regular-closed BCY satisfies cl [α-'(int(B))) Ca• 'cB) (5) Each opeμ BCY salisfies cl(a -1(B))Cα l(cI(B)).
1n the following we define the I}-
…
terior of a subset A (into(A)) of a space X to be X -clo(X -A).THEOREM 1.2. The following slaletnenls are equivalent for spaces X, Y and
aεM(X， Y)
(1) The '"ιItifuncti on a
‘
s f)-u. S. C.(2) Each BCY salisfies clO [.α l(cl8(B))] Ca-l(cla(B)).
(3) Each open BζY salisfies clo[a -1(cI (B))]ζa-1(cl(B)).
(4) Each regular-closed BCY sa#sfies cl
o
[α l (i nt(B))]ζa l(B)-(5) Each open BCY satisfies a+(B)Cinto(a+(cl(B))).
(6) Each open BCY satisfies cl
O
(α→ 1(B))ζα l(cI(B)).PROOF_ Since clo(B)=cl(B) for each open subset B of a space it follows that
01: .~fu/UfuncUolIs Salisfyilzg Certain Se
’
nicolIU, ’“‘
ty TyÞe C011diti01ls 181 (3) is implied by (2). It is c1ear that (4) is implied by (3). Suppose that (1) holds let BIY and let xX - a-l(cl8(B)). Then a(x)ζY -c10(B) and, conse.quently, 50me Vel: (x) satisfies a(cl(V))CcI(Y -clO(B)). For such a V we have c1(V)na-1(int(clO(B)))=O. Hencc x흥cl6(a-lGrit(c1。(B)))) and the proof that (1) implies (2) is complete. AS5ume (4), let BCY be opcn and let xeX - into (α-l(cl(B))). Thcn xeclo(X -a+(c1 (B)))=c1
o
(α 1 (Y -cI(B))) =clo [α-1(111t(Yint(cl(B))))1 and Y -int(cl(B))is regular.cIoscd. From (4) wc obtain xeα l(y_
int(c1(B)))CX -a+ (B). Thus (4) implics (5). Now. aS5umc (5), let BζY be
。pen and let xeX-α l(cl(B)). Then xea + (Y -c1(B)) and. consequently. xeinto [a + (C1(Y - c1(B)))J =X -clo[X -a + (clα -cl(B)))J =X -c1e<a -1(B)). Thus (6) holds. Finally, assumc (6), let xeX and Wel:(a(x)) in Y. Then α(x)ncl (Y
cl(W))=O and x흔a-l(cl(Y-cl(W))). It f。llows from (6) lha[ 1=cl6(a l(Y Cl
(W))) and that somc Vel: (x) satisfies c1(V)na -1(Y -c1(W))=O. For 5uch a V we have a(c1(V))ζCI (W). This establishes that (1) holds. Q. E. 0
Thc notion of w. u. 5. c. multifunction ‘쩌s introduccù and studied in some dctail by Smith50n (32j; it extended th", notion of wcakly.conlinuou5 function.
duc to Levinc [261, to multifunctiofl3: w. u. s. c. multifunctions have been stu.
died morc recently by Popa [311. The '.1Ijuivalence of (1) and (2) in Theorcm 1.1 has been established by Popa in [3IJ. This cquivalencc 、vas kno\'\
,
n tO and produced independently by the authon; prior to the publication 01 Popa’s paper.The fact that (1) implies (5) in Theorem 1.1 was provcd by Smithson in (32J.
It was al50 shown in (32J that (5) is equivalcnt to (1) under ccrtain additional hypothcses. In fact. Noiri showed in (3이 that (1) and (5) are 여 uiyalent íor functions which arc almost contiüuu:Js in thc scnsc of Husain. ReccntIy, Espclie and Joseph (8J have estabìished for function5 that thc requirement of alm05t continuity in the sen5e of HusaiD. or indeed any additional bypotbesis. is superflous for this equivalence. The equivalence 01 (1) and (5) in Theorem 1.1 extend5 the result of Espelie and Joseph to multifunctions. The notion of O-u.
s. c. multifunction wa5 introduced by Joscph in (21J and employed to exrend the U niform Boundc이 n c-ss Principle of analysis to l11ultifunctions. This notion extends the notion of θ contbuous fUl1ctiO;l of Fomin (9] to multifunctions.
It is shown in [81 :hat a function α : X-•Y i5 wcakly.continuou5 if and only if any one of the equivalent statcments in Theorem 1.1 is satisfied. and that a is O-continuous if and only if any one of the 0quivalent statements in Thc-
182 jesse P. Ctay alld james E. Joseph orem 1. 2 is satisfied
Ler X, Y be spaces and ler αEM(X， Y); a is lower-semic011linπo"s (1. s. c, )
-l ，~~ .. , - 1
at xeX ií for each W open in Y with XEα (W) somc Vel'(x) satisfies VCa . (W), and a is lower-semicontinU01!S (1. s. c.) if α is 1. s. c. at each xeX; a is weakly lowersem t"C01z.ti1Zα OUS (t.ι 1.5. c.) at xeX if for cach W open in Y 、.v ith
xα-l(W) some Vsε (z) satisfies Vζcr-l(cl(W))， and α is weakly lower-semt"c
。lltt"nttous (w.l. s. c.) if α is W. 1, 5, C, at each XEX [32]; a is IJ-lower-semicont inuous (IJ-I. 5. c.) at XEX if for each W open in Y with xea -l(W) somc Vel'(x) satisfies CI (V)ζa-I(c1(W))， and a is e-Iower-semiconlinu01's (IJ-I.s.c,) if a is ()-l. s. c. at each xeX. 1n our next two results we provide characterization theorems for these two c1asses of multifunctions. We prove 1. 4 only. We recaII that the I}-adherelκe o! a jilteròase Q(ad
tP)
on a space is the intersection of the 1}-c1osures of its members; Q e-converge5 !o" (Q• ur) if Q is finer than r (x) [38];‘
"e denotc the adherence of Q by adQ.THEOREM 1. 3. Tlze !ollowing s!a!eme"!5 are equivaletl! !or spaces X, Y and aeM(X, Y)‘
(1) Tlze "，ulti!1αItction a is w.l_ s. c
(2) Each filteròase Q on X salis!ies a (adQ) Cadaα(Q).
(3) Eoch ACX So!i5!;es a(c1 (A)) ζc1
0
(α (A))(4) Each !'αtIlil Y Q o! subsel5 o! Y 5ali.!ie5
n
CI (α + (F))Ca+ ([IcIA(F)).Q Q
(5) Each BCY satis!ies CI(，α+ (B))Cα+ (cl8(B)).
(6) Each ope1l
BC二Y
sa!;s!iescI
(a~(B))Ca+(c1(B)
).
(7) Each oprJ1l BCY sa!is!ie5 a -l(B)Cint(a -1(c1(B))).
(8) Eacll cl05ed BCY sa!is!ie5 c1(α+(int(B)))ζa -(8).
(9) Eacll regκlar-closed BCY sa!is!ies c1(a + (int(B)))Ca + (B).
THEOREM 1. 4. The followillg 5Ia!e" ,e,,!s are eqιzνalrm! !or spaces X, Y at여 αEM(X， Y):
(1) Tlle IIlUlti!1
“’
ZCtl01Z a is ()-l. s. c.(2) Eaclz !ilteròase Q 011 X 5alis!ies a(ad{j'Q)ζadoa(Q).
(3) Each ACX 5atis!ie5 α(cI，cA))ζc10Cα (A)).
(4) Eocμ family Q o! sub5e!5 o! Y 50!
“
!ies nclA
(α +CF))Cα+ (nclA(F)).(} .... D ν
(5) Eac/z BCY satis!i
‘
es cloCa ~ (B))Ca • (c1eC
B))。n Mullij
“’ ,
lctiOllS Satisfying Certain Semiconti’”“
Iy Tyþe COlldiliollS 183(6) Each_open BCY satisfies clO(a+(B))Ca+(cI (B)).
(7) Each open BCY satisfies a - l(B)CintO(a -l(cl(B))). (8) Each closed BCY satisfies clO(a+(int(B)))Ca~(B).
(9) Each regtelar-.closed BCY sat
‘
sfies clO
(αT (int(B)))Ca + (B).PROOF. lt is fairly immediate that (3). (5). (6) and (9) arc implied by (2). (4). (5) and (8) respectively. We now provc the other implications which are necessary to complete a cycle of implications beginning at (1) and procce- ding numericaIIy. Assume (1). let Q be a filterhase on X and let xeadoQ. Choose yeα(x) and let Wεε (y). Some Veε (x) satisfies cl(V)ζα l(cl(W)). "ince each FtQ satisfies Fncl(V)7"ß. each FêQ also satisfies a(F)ncl(W)7"O. Hencc yeado a(Q) and (2) holds. Supposc Q is a family of subsels of Y which fails tO 잃I
isfy the inequality in (4). Let xel)cIo(a + (F))- α~ (l)clo(F)). There is an FeQ
G
such that a(x)<tclo(F). lt follows that a(x)<tcIo(a(，α+ (F))). and this impIies that a(cloCa+CF)))<tclo(a(a+(F))). Thus (3) fails if (4) fails. Assume (6). let BEY be open and let XEX lnt8(a-l(cl(B))). Then xecl6(X a-l(cl(B)))=
cl8(af(Y-cl(B))) and, from (6), we have xa+(cl(Y-cI(B)))Ca+(Y-B)=x - a -1(B). Therefore (6) implies (7). Complementation and the relarionsbip be tween a - 1 and a + may be used to show that (7) impIies (8). Finally. assumc (9). let xεX and let W bc open in Y with yeα (x)nW. Then :J줄cl(Y -cl(W)) a rcgular•closed subset of Y. Thus"중a + T (cl(Y - cl(W))) and from (9). x"';c!o [α+ (int(cI(Y -cI(W))))]. Consequcntly. SOme Veε(x) satisfics cl(V)CX α+
(int(cl(Y -cl(W)))) =a -1(cI (W)). Hence a i5 0-1. s. c. Q. E. D.
To the authors’ knowledge. 0-1. s. c. multifunctions ιre introduced for the first time in this paper. We note that Levine [26] has sho\Vn that a function a:X• Y is weakly-continuous if and only if Ü satisfics statcment (7) of The orem l. 3. Popa [311 has shown lhat statements (1), (7) and (8) are equi\'alcnt. These equivalences were also independently provcd by the authors prior [Q the publication of Popa’S paper. Smithson [32] introduccd α. tl. s. c. multifunctions and establishcd that statement (6) 01 Theorem 1.3 is valid whcn α i8 z，ι 1. s. c.
The equivalcnce of (1) and (6) was also shown in [32] under additional hypo theses on a. The fact that weakly-continuous functions satisfy statemcnts (2) and (3) of Theorcm 1. 3 and the fact that Ii-continuous functions satisly (2)
184 Jesse P. Clay and Ja
’
lleS E. J oseþhand (3) of Theorem 1. 4 are due to Dickman and Porter [히 • For functions.
the equivalencc of each of statements (2), (3), (4) and (5) in Theorem 1.3 to
weak~continuity and thc 어 uivalence of each of statemcnts (2), (3), (4) and (5) in Theorem 1. 4 to O-continuity are due to Joseph [21]. [22J. For functions, thc cquivalence of cach of statements (6) and (9) of TheJ)rem 1. 3 to weak continuity and thc eq uivalcnce 01 each of statements (5)- (9) to O-continuity arc duc lo Espelic ιnd ] oseph [8J
lt is immediatc that O-u. s. c. multifunctions are ω. μ• s. c. and that 0-1. s. c.
multifunctions arc w.l. s. c. Sincc it is well-known that 8-continuous functions are nOt always continuous and that weakly-continuous functions arc not neces.
sarily O-continuous, these implications cannot bc reversed. It is also well- known that thc collection of continuous functions into a regular space coincides with the collection of I:I-continuous functions and thc collection of 、、ea입y
continuOllS functions into the space. \'~le \vill nO\"~ providc other results which show that the relationships between thesc notions stated in thc lirst sentcncc 01 this paragraph arc the only ones which hold without addiLional hypothesis.
If X is a nonempty sct and xocX and Q is a filtcrbasc on X, X (xo' Q) will rcprescnt X along with the topology (VCX: xoE7'Y 01' FCV fo1' soη" FεQ)
EXAMPLE 1. 5. A 0 μ• s. c. multifunction from a compact mctric space to a compact metric spacc need not be
“.
s. c. Let N rcpresent thc sct of positive integers and let X=N XN. Lct Ql and Q2 bc thc filters of finite complcments on N and X, rcspcctivcly. Lct Y =N(1, Ql) xN(1, 01), Then X((I, 1), Q2) and Y are both compact metric spaces. Dcfine aeM(X((1,l), Q2)' Y) by a(:r, y)= [(r, y)) if (x, y)낯(1.1) and a(I,l)=Y- {(:r, y)eX: {x, y)nU)7얘); a is 0-κ. s. c. at each (r, y)7"'(1. 1) and, sincc cl(a(1,1))=Y in Y, we see that a is 0-". S. c. at (1,1). Ho“
rever a(1，l) eE(α(1，1)) in Y and if Weε((1， X1)) in((1,1), Q2)' X -W must bc finitc; so Lhere is a positivc intcger 1n7ε1， with (1，끼)eW. For such an m we have α(1. 111)=(1. …)<;1"α (1，1). Thus α .s ιS. C.
aL (1.1).
EXAMPLE 1. 6. A multifuIlction which is ι s. c. need not bc 0 κ.s.c. Let Ql bc thc filter of finite complements on N and lct Q2 be thc filter of supersets 01 {1, 2) on N. Let αeM(N(I， Q2)' N(1, 0.)) bc dcfincd by α (1)
=
U,2) and a(r)=
{x) otherwise; a is α. s. c. but α is not Ð-u. s. c. at x=2.It is obv-ious that if Y is normal and αεM(X， Y) is ，ι κ. S. C.
‘
.vith point-closed0" Multi/:ω11clio11S Sa!isíy.
’
llg Cerla’
11 Semicoll“지 uily Tyþe COJldiliotls 185 images then a is u. s. c. The Íollowing three results which we state without proof are also fairly immediateTHEOREM 1. 7. If X is regαlar and aEM(X, Y) is 1. s. c. I"en a is O-I.s.c.
THEOREM 1. 8. If Y is reg씨ar and aEM (X, Y) is 10.1. s. c. I"en a is 1. s. c.
COROLLARY 1. 9. If X and Y are reg1tlar I"en Ihe following slalemenls are eqttivalen/ for aεM(X， Y):
(1) Tl!e lI!ulliJnnclio1l a is 1. s. c (2) The multifunctioJ.
‘
a is 8-1. s. c.(3) The ""‘ltiflι nciiolL a t"s w.!. s. c
Examples (1) and (2) of [35] provide compact Hausdorff spaces X, Y and a, }.eM(X, Y) with point.closcd images; a is 1t. S. c. and not 1. s. c. whereas À is 1. s. c. and is not u. s. c.
EXAMPLE 1.10. A 1. s. c. αd{(X， Y) need not be 0-1. s. c. when Y is compact metric. Let N be the set of positive integers and let X =
{n + 움 : n> l, k>l}
UN with the topology generated by the base of open sets consisting of the usual opcn scts in X-[I} along
wit뼈t s of the form (1) U{n+ + : ε1α
k>l}
where joeN. Let Y = [2, 3] with thc usual topology. Define αeM(X， Y) by
α싼 (2 + 쉰
for "> 1 and a(x)=Y otherwise. Let W야
anon따lem 때 n때1ψpty 0야pe
nn s뼈 u뼈b않았t 0아f Y. T 돼he야n a카\W)←=(α X-(N-카(1) )씨 ))η 씨)씨끼 U이 ~keεN: 2깐 + 1 움~E:W 쩌 W 찌} 、w빼 v얘h피lκ때 c야마h
is op뼈 e당앉n
X. Thus a is 1. s. c. On the other hand, leclo(N - (1)), so a(clo(N - (1))) =Y.
Since clo(a(N - (1)))
=
(2) U{2+소 : k> 아 we scc that a(clo(N - (1) ))(/:.clo(N- (1))). Hence a is not 0-1. s. c.
It is shown in [21] that a function
a :
X• Y is O-continuous if and only if cle
(α → I(B))Cα l(cl e(B)) for each BεY and it is establishcd in [22] that α is weakly.continuous if and only if cl(a -l(B))Cclo
(α -1(B)) for each BCY. Ex.ample 1. 11 shows that the equivalenccs do not extend to O-u. s. c. and w. u. s. c.
multifunctions
EXA.\1PLE 1.11. Let Y be the set 01 reals with the usual topology and let X represent the set of nonnegative integers with (AcX: α졸A 01' X-A is finile) as the topology. Define aeM(X, Y) by a(n)
=
(n) if n# O and a(O)=Y-(X - [0)). Since cl(X - (0)) = Y in Yand (씨 is open in X for each neX - (이 we
186 Jesse P‘ Clay alld James E. joseþh
sec casily that α IS θ ι s. c. Howevcr, cl(α← '(X-{O)))=X and a- 1(cJO(X (0))) =X• (0). Hence cl(α-1(X _ (이 ))<tα 'ccJθ (X-{Q)))
A subset A of a s;ncc X is fJ-rigid if for each COvcr Q of A by open subsets of X thcre is a finitc
.o,
C.o
satisfying AClnt(U cl(V)) [5]. It is Imov‘
n [23)Qj
that a subset of a space X is l1-rigid if and only if AnadO
.o"o ø
whenever .0is a fi1tcrbasc on X satisfying Fncl(A)
"o
O for all Fε.0 and VeE(A). The fol.Jowing results are offered for multifunctions
, ....
ith fJ-rigid point images. \Ve state Tlleorem 1.12 without proofTHEOil.E!v1 1.12. The follo，νing statements are equiνalm! for spaces X. Y and aeM(X. Y) with Ð-rigid poiη t i1nages
(J) The multijuη ction a is w α . s. c.
(2) Eacι filterbase .0 on α (X) sa!is!ies adα 1(
.o
)Ca-1(ado.o).
(3) Each BCY so!is!ies c1(a -1(B))ζα-1(c1aB)).
THEORE:-1 1.13. The !ollowing s!aletnenls are equivalen! !or spaces X. Y and αξM(X. Y) ωμh Ð-rigid point inzages
(1) The 1Ilultifunclion α 's θ ι. s. C.
(2) Each !ilterôase .0 0α α(X) sa!is!ies ad
e
α→1(.o)Cα l(ad6Q)- (3) Each BCY salis!ies cle
(α-1(B))Cα l(clo(B))·PROOF. It is evident that (2) implies (3). Assume (1). let .0 be a filterbase on α (X) and let xead
o
α 1(.0). For cach Wel"(a(x)) there is a Vel"(x) such that α (cl(V))Ccl(W). For such a V each Fe.o
satisfies α -1 (F)ncl(V)잊O.
80 Fncl(W)"oO for each Weε(α (x)) and Fe.o.
It follows that a(x)nade.Q"oØ
sincc cr(x) is 8-rigid. Hcnce xw -1 (ado.Q) and (1) implies (2). The fact that (3) im plies (1) follows from equivalence (6) 01 Theorem 1. 2.r\'luch attention has been given to functions and multifunctions satisfying certain graph conditions (see [101. [181. [151- [24]. [29]. [32]. [33]). In scction 2 we ívil1 improve a number of the results on such multifunctions and give a number of new results. Let X. Y be spaces. lf xeX. let A(x) = (V - (x) : Veε
(x)). Let αeM(X. Y); α has a subcl osed graph if adα (A(x))Cα(，，) for each xsX for which A(x) is a filterbase on X [211; α has a strongly.subclosed grα'P"
if adaα(A(x))Cα (x) for cach xeX for which A(x) js a filterbase on X [21]; it is shα\'n in [21] that αeM(X. Y) has a closed graph if and only jf α has a
On Mμlti/lιIlctiOIl S Satisfyillg Certaiu Semicolltinuity TyÞe COllditiollS 187
subcloscd graph and a(x) is closed in Y for each XêX; a bas a slrongly-closed --gYilph if a has a strongly-subclosed graph and a(x) is Ð-c1oscd in Y for cach xeX. Functions \vith strongly-closed graphs werc introduced by Herrington and .Long in [15] and have bcen studied intensely since that timc. Wc menlion :m application of strongl)'-subclosed graphs. A subsct A of a spacc X is quasi H closed (QHC) relative 10 Ilze space if each filterbasc f.I on A satisfics AnadO
f.I",
ø .
A space is qιas; C-compact (QCC) if each c10sed subset is QHC rclativeto the space, a Hausdorff QCC space is caIled C-co11
,
pacl. It has been shown recently in [2] that a Hausdorff space Y is C-compact if and only if for cach Hausdorff space X, each aêM (X, Y) with a strongly-su bclosed graph is ι S. C.Finally, in this section. we recall definitions of some variations on the not ion of subcontinuous multifunctions. Subcontinuity of functions was intronduccd by FuIler [10] and was extcnded to multifunctions by Smithson [32]. Let X , Y be spaces and let aeM(X, Y); a is s"bconlinuous wit" respect 10 AcY if for each con vergen t net (x(η)] on X and net (y(n)) in Y with y(μ)eα (x(n)) for cach n, some subnet of (y(n)) converges tO SOme YêA; a is r-st'ÒcoηIi잉Wtts
CÐ-s1tbcolltinμoαs) ，ιith respecl 10 ACY if for cach convergcnt (Ð-convergent) net (x(n)) in X and net (y(n)) in Y with y(n)ôa(x(n)) for each n, some subnet of (y(n)) Ð-converges to somc YêA [21]. We say that a is subco찌 imtous [r-su- bCOll
“”“
OIlS) (fI-s1tbco써lκι01(5) if a is subcontinuous fr-subcontinuousl (Ð-subcon• tinuous) with respect to Y. The nOlion 01 r•subcontinuity for functions was introduced by l-Ierrington in [14]. The following thrcc characterization rcsults are interesting. ，;νe prove Thcorem L 16 only,THEOREM 1.14. T"e follozα ng statements are equivaleκt for spaces X, Y,
α<-11 (X, Y) and ACY:
(1) Tlze lItκItijttncl‘ on a is stlbcontinuou.s witll respect to A.
(2) For each convergent ft"llerbase Q on X. eaclz jilte1-base ['1 an Y fincr tlwn a(Q) sa!;sfies Anadf.l1"'O.
(3) Eaclz ttltrafilter 'ì/ 011 Y wlzicli
‘
is ji1ter Ihan α (Q) for a conνergen! filter òase Q 01Z X converges Ï1~ Y [0 some yεATHEORE:\1 L 15. Tlte following slalemenls are eqαivaleη1 for sþaces X, Y, ad1(X, Y) aηd ACY
(1) The 1mtltiftmctiOt~ a i s 1"-5;ιbcontinteous 1.Oit" resþect 10 A.
(2) For eaclz convergent filterbase Q 0" X, each filterbase Ql on Y finer IlzlJn
188 J esse P. Clay alld J ameS E. J oseþh
a(Q) salisfies AnadeDl잊 ø.
(3) Eaclz 싸Irafiller V on Y whiclz is finer thatL a(.Q) for a convergenl Jiltet'- base 0 on X O-cooνerges in Y 10 some yeA
THEOREM 1.16. The follawing slalemenls are eqn;valenl for sþaces X. Y aeM(X, y) ond AcY:
(1) The mullij'<nctioo a is 0-SUbC011litLuons
‘
0;1" resþecl 10 A.(2) For eaclz O-coovergenl filterbase 0 on X eacll filterbose Ql on Y finer Ihon α(Q) sol;sfies AOadeDl해.
(3) Eacll ultrafjlter V 00 Y which is finer Ihan a(O) for 0 0-converge1l1 filterbase Q on X I}-convergas ;n Y 10 so" ,e yeA.
PROOF. lt is clear that (3) implies (2) and since an ultrafilter must O-con- verge to each point of its O-adherence we see that (2) implies (3). For thc proof that (1) implies (2), let :r.X and let 0 be a filterbase on X with Q•
ø
X'Let 01 be any filter뼈se on Y finer than α (.Q). Let Q2= IMnα (N): MeD1’
NeQ). Then Q2 is a filterbase on Y finer than 01 Ua(Q). Define드011 QlxQby (Mo, No)드 (M， N) il and only if MCMo and NCNo' We see easily that (Ql xQ, 드) is a directed set. For each MeD1 and NeD choosc x(M, N)eN with
Mnα (x(M， N)) 낯 ø and choose y(M, N)eMna(M, N)). The11 y(M, N)eα
(X(M, N)) for all (M, N) Illld if VεI: (x) therc is an NoeQ with NoCcl(V).
Choose any MOeQl and let (Mo' N，이드 (M， N). Thcn x(M, N)eNCNoCcl(V).
Hence x(M, N)• .OX and. by the O-subcontinuity of a with respect lG A, (Y(M. N)) has a subnct which θ-converges in Y to some yeA. Let WeI: (y), let MoeQ ancl NoeQ. Some (M. N) satisfies (Mo' No)드(M'， N) and y(M, N)εcl(W).
Thus y(M, N)ecl (W)O(MOa(N))ζcl (W)n (Mona(No))' Thus yeade
찌
'2Cacl6Ql and (2) holds. Finally. we show that (2) implies (1). Let xeX and Iet (x(n)),(y(n)) be nets in X, Y, respectivcly,
“
ith x(n)• OX and y(n)eα (x(n)) lor cach n, Then the filterbase Q generated on Y by (y(n)) is finer than the filterbasc a(Q) wherc Q is the 용∞nvergent filterbase generated by (x(n)). Hencc An adeQ;éO and, consequently, (y(n)) has a subnet which O-converges in Y to some yeA2. Utilization of the conditions on mnltifunctions discussed in section 1.
In this scction the conditions on multifunctions discussed in section 1 are employcd to improve various knowD results for multifunctions and functions
0. M
“
Itif, “
ncHons SaUsfyiηg Certain Se…Z.COlltiηu:ïy Type Conditions 189and to obtain new results. Our first several results shO\". that regularit:y and normality may be characterized in terms of 8-closed subsets of producls、
Ð-continuous functions and θ upper-semicontinuous multifunctions. Let X. Y and Z be spaces. We usc the following notations and definition:
(a) R(α， μ• XxY. Z)= [(x. y) εXxY ‘ α (x)ecl(μ (y))) for μιν (Y. Z) and [""clio" aeM(X, Z).
(b) R(μ， α， XXY, Z)= [(x, y)eXXY: a(y)ecl(μ (z))) [or μêM(X ， Z) and
fιηction αeM(Y， Z).
(c) R(α， μ， X, Y)= {zeX: α(z)ecl(μ (x))) [or μεM(X ， Z) aηd ["ncti 011 αeM
(X, Z).
(d) N(a, μ， XXY, Z)= ((z, y)eXXY: cl(α (x)) ncl(μ (y)) 잊이 [or αeM(X， Z) and uεν (Y ， Z).
(e) N(α， μ， X, Z)= {zeX: cl(α(x) >n cl(μ(z))옹이 for α， μeM(X， Z).
(f) It HζXxY we will denole {(y, x): (x, y)
,
H) by H-1• It is well-known that H is a closed subset of XXY if and only if H-1 is a closed subset oí Y xX.DEFINITION 2.1. A subset A of the product. X xY, of two spaces X and Y has (x,y)eXXY in its (1) I}-clos"re ((2) I}-closμ re) if (cl(VjXW) nA줌 q， ((Vxcl
(W))nA 속 q，) whenever VeZ(X) in X and WeZ(Y) in Y. In this case wc 、，vrite
(X, Y)e (1) Cle(A) ((X, Y)e (2) cle(A)). A is (1) θ←c1 0sed ((2)θ-closed)
if
A= (l)c1θ(A) (A = (2)clθ(A))
Evidently, if X and Y arc spaces and HCXxY then (x, y)e(1)cle(H) if and only if (y, x)e(2)cle(H-1) and H is (1) 1}-c1oscd in XxY if and only if H-1 is (2) O-closed in Y xX; if McXxY is nonempty and H is (1) l1-closed ((2)
θ-closecl) in XxY then HnM is (1) I}-closed ((2) I}-closed) in M.
THEOREM 2. 2. nκ following stalemeηts are eqαivalenl for a space Z: (a) Z is regular.
(b) R(α， μ， XXY, Z) is a (2) I}-c/osed sαbset o[ XXY [or alt spaces X, Y,
contl ηuous f.αnctioηs αeM(X， Z) aπd 1} ι. s, c, μeM(Y， Z).
(c) R(μ， α， XxY. Z) is a (1)8•cl05ed sαbsel o[ XxY [or all spaces X, Y, continuous ft
…
ctions αeM(Y， Z) aη:d 0-κ， s， c μεM(X. Z).(d) R(α， μ， XxX, Z) is a (2) 1}•closed sχbsel of XxX for all space X, con- tinuous func#ons αεM(X， Z) aη:d 1)-μ， 5， C, μelvI (X， Z).
(e) R(μ， α， XxX, Z) is a (1) I}-c/osed subset of XxX [or all sþaces X,
190 Jesse P. Clay alld Ja
’
'ues E. ] oseþJ‘ ,
cOlliÎnuous fiιnctions aeM(X, Z) attd {}-κ. S. C. μeM(X， Z).
(f) R(a, μ， X, Z) is a {}-closed subset of X fOT all spaces X, continuous fu.
ttclions aeM (X, Z) and Ð μ.5. C. μeM(X， Z).
PROOF. To see that (a) implies (b), let Z be regular, let X, Y be any spaces, let a e M(X, Y) be a continuous function, Jet μeM (X, Y) be {} α. S. C.
and let (p, q)e(XXY) -R(a, μ， XXY, Z). Then a(p) 중cl CJt (q) so there are sets VeE(a(p)), We ε(cl(μ (q))) in Z satisfying cl(V)ncl(w)=ß. Since a and
μ are continuous and θu. s. c. at p and q respectively, there is an AeT (p) in X and a BeT(q) in Y with a(A)Ccl(V) and μ(B)ccl(W). AxB e T(p, q) and AxBnR(a, μ， XxY, Z)=ø. Thus R(a, μ， XxY, Z) is8-d∞ed in XXY. The proof that (b) implies (c) is immediate since (R(ι a. XXY. Z])-l=R(a, μ，
YxX , Z) , which is a (2) Ð-closed subset of YxX from (b). The proof that (c) implies (d) is clear sinæ R(a, μ， XxX, Z)=(R(μ， a, XXX, Z)) l which is (1) (}-closed in XxX from (c). The proof that (d) implies (e) is sim !,u to the proof of (c) implies (d). We now establish that (e) implies (f). With the hypothesis of (f) in effect, let x e X - R(a, μ， X, Z). Then (x, ，，)졸R(μ， a, XXX, Z) and there is a Ve ε (x) in X satisfying (cl(V)x V)nR(μ• a .. XXX.
Z)=ø. This gives cl(V)nR(a, μ， X， Z)=ø: soz줄clo(R(α， μ， X, Z)) and R(a,
μ， X, Z) is l1-closed. The proof that (e) implies (f) is complete. '1'0 effect the proof that (f) implies (a), suppose Z is not regular. There is a point "'0 e Z and closed set FζZ with "oeZ -F such that f.}= {VnW: Ve ε ("'01， WeE(F)) is a filterbase on Z. Choose Z。ε F and define a, μ eM(Z(;-o
m
, Z) bya(X)=z if z;Æ zα a(z~=zo' and μ (z)=FU (z) for all z. We soe readily that α IS con~tinuous. Now, if We ε(μ (z)) in Z then W is open and closed about x in Z(z(j>
Q): furthermore, μ (W)=FUWCWζcl(W) so μ is fj..u. s. c. It is obvious that
R(α， μ， Z(zrr Q), Z)=Z - {zo}. If V e E(x~ and WeE(F) in Z then (V -F)n W;ÆØ. lf pe(V-F)nw then a(p)=p and μ (p)=FU{p). so a(p)ecl(μ (p)).
Thus zoêcl(R(a, μ， Z(z()> Q), Z)). Hence Z does not satisfy condition (f). The proof that (f) implies (a) is complete.
Theorems 2.3 and 2.4 are stated without proof. The proof of Theorem 2.4 is similar to that of Theorem 2. 5.
THEOREM 2.3. The follolU써g slalem.:ls are equivolent fOT a space Z (0) Z is regαlar.
(b) R(α， μ• XxY. Z) is 0 (1) {}-closed sκbsel of XXY for all spoces X, Y,
On M UU
,
ïullctio’‘
5 Satisfyillg Certaill Semicolltill“’
ty Type COllditiollS 191 O-conlinuous fz…
clions a e M(X, Z) aná w. ι s. c. μeM(Y ， Z).(c) R(μ， a, XxY, Z) is a (2) O-closeá subset of XxY for all spaces X, Y, (J-contin1l0u.s fU7Zcti01zS a e λf(Y， Z) aná 10. u. S. c. μ eM(X, Z).
(á) R(.α， μ， XxX, Z) is a (1) O-closeá S11bset of XxX for all spaces X, 0-
cont씨u. ous fzmcUons α e M(X, Z) aná w. u. s. c. μ eM(X, Z).
(e) R(μ， α， XxX. Z) is a (2) O-closeá sαbset of XXX foγ all spaces X, (J conUηU Ot~s f;ιncUons α e1ν (X， Z) and w. α . S. c. μ S ιM(X ， Z)
(f) R(a, ι， X, Z) is a (J-closed Slκbset of X for all sρaces X, ()-continz
‘
OUSfunctiotzs α ε M(X, Z) and w. κ.S， C. μ eM(X, Z)
THEOREM 2.4. The foll owing slalemenls are eqιivaleη1 for a sρace Z (0) Z is regulor.
(b) R(a, .u, XXY, Z) is 0 (J-closed S11bset of XXY for all spaces X, Y, 0-
cα，1;1I11011S fm
,
clions aeM(X, Z) ond (J μ• s. c. .u e M(Y. Z).(c) R(μ， a, XxY, Z) ;s 0 (J-closed subsel of XxY fOl" cl/ spoces X, Y, (J- co’“t1lU Oιs fU1lclions a e M (Y, Z) and (}-u. s. c. μ e M(X, Z).
(d) R(α， μ， XxX, Z) is 0 (}-closed subsel of XxX for afl spaces X, 0-C01l' tinuous funclions aeM(X
,
Z) and {}-κ• S. C. μeM(X， Z)(e) R(μ， α， XXX, Z) is a (}-closed s1lbset of XxX for all spaces X, O-C01lt- inuous ju1tcfio1ts α e M(X, Z) and 0-11. s. C. μ eM(X ， Z).
(f) R(α， μ， X, Z) is a O-c/osed subset of X for all spaces X. 0-co/lli1l,,01ls
fπ 1tctioηs αeM (X, Z) and O-u. s. c. μ eM(X, Z).
THEOREM 2.5 The following stalements are equivaletlt for a sþace Z (0) Z is 1l0rmal.
(b) N(α， μ， XXY, Z) is a O-c/osed subset of XxY far 011 spaces, X, Y olld O-u. s. c. a e M(X, Z) alld μeM(Y， Z).
(c) N(α. μ， XxX, Z) is a O-closed subsel of X xX for all spaces X alld ()-
11-. S. C. α， μ eM(X. Z).
(d) N(α， μ， X, Z.) is a (}-c/osed subseJ of X for all sþaces X o1td (}-II. s. C.
a. μ eM(X. Z).
PROOF. 1'0 prove 미lat (a) implies (b) let Z be normal, X , Y be spaces and
αeM(X ， Zi 야M(Y， Z) be (J-u. s. c. Let (p, q) 티XxYi 一 N(α， μ， XxY, Z). Then
c1(α(p)ncl(μ(이 ) =ø 50 there are 5et5 V e S(cl(α (p))) and WeS(cl(μ (q))) in Z sati5fying c1(V)ncl(W)=Ø. Since α and μ are
e
ι. s. c. there are 5ets A e S(P)in X and B e S(q) in Y sati5fying a(cl(A))cl(V) and α (cl(B))ζcl(Wi. A xBeS ((þ,q)) in XXY and cl(AxBJnN(a, μ， XxY, Z)=ø; so (p, q)Ëi;c:Q(N(a, μ，
192 ]esse P. Clay and ]a
…
es E. ]osephXxY. Z). The pr∞f that (a) implies (b) is complete. The proof that (b) impJies (c) is obvious. Under the assumption of (c) and hypothcsis of (d) we scc that N(a. μ， XxX, Z) is O-c1oaed in XxX; it follow sthat N(a, μ ， XxX, z)nL1 is O-c1osed in Ll, the diagonal of XxX, and that N(α， μ， X, Z)=πx(N(.α， μ， X
xX, Z)
n
Ll) is O-c1oscd in X (π'x is the projection). The proof that (c) implics (d) is complete. We complete the proof of thc thcorem by showing that (d) implies (a). Supposc Z is not normal. There are disjoint ciosed subsets F,
’Fz 01 Z such that .Q={Vnw: VeZ(F,), WeZ(F2)) isafilterbascon Z. Cho ose x1sF1 and x2eF2 : define a, μ e M(Z(xj . .Q), Z) by α (x，)=F" μ (x，)=F2
’
a(x)=F
,
U {x}, and μ (x)=FzU (x) otherwise. lf W e Z(a(x)) (W e Z(μ(x)) in Z, then W(WU (x,)) is open and cJosed about x in Z(x" .Q): furthermore,a (W)CWζcl (W) (μ (WU(x， ))CWζcl(W)) ‘ thus a and μ are 0-
“
.s.c. It is c1ear that cl(a(x，))ncl(μ (.t))=ø， so x jf5;N(a, μ， Z(xl' Q), Z). We show by arguments of the type cmployed in the proof of Theorem 2.2 that"',
ecl(N(a,!J., Z("'" .Q), Z)). The proof that (d) implies (a) is then complete.
We state our Theorem 2.6 without proof.
THEOREM 2.6. Tl!e follozuing statements are eqμivalent for a space Z (a) Z is normal.
(b) N(a, μ， XxY, Z) is a (2) O-closed sιbset of XxY for all spaces X, Y,
W.U.S.C. αeM (X, Z) and O-u. s. c. μ eMCY, Z).
(c) N(α， μ， X xY, Z) is a (1) O-dosed subsel of XxY for all spaces X, Y, 6 κ• s. c. a e M(X, Z) and /0. U. S. c. μ eM(Y, Z).
(d) N(a, μ， XxX, Z) is a (2) O-closed subset of XXX for all spaces X,
W. ι s.c. a ε M(X， Z) a..d O-u.s.c. μ eM(X， Z).
(e) N(a, μ， XxX, Z) is a (1) O-c/osed subset of XXX for alJ .. paces X, 0-
U. s. c. a e M(X, Z) and 10. ι.5. C. μ eM(X， Z).
(j) N(a, μ ， X, Z) is a {}-c/osed subsel of X 껴r alJ spaces X, 0 κ• S. C. α eM (X, Z) and 10. U. s. c. μeM(X， Z).
lf X, Y are spaces and a e M(X, Y) we define the cltlsler set of α at xeX to be ad α (Z(x)) ， the strong c/tlster set of a at xe X to be ade，α (1' (x)) ， and the O-clιster set of a at x e X to be adoa(r(x)). 、rVe denote thesc three cluster sets by C(a ; x), S(a ; x) and T(a ; X) respectively. Recently, Smithson (34) has proved that if X and Y are spaceswith Y regular(not necessarily T,) then
Oη lvlultifuμ ctions Satisfying Certain Semicoutim
“
ty Tyþe COllditions 193C(α : x)= α (x) for each x e X if α is u. s. c. and, moreover, if Y is compact and C(α ; x)= α (x) [or each xεX then α is u. s. c. (This is all phrased in our terminology). ur next theorem improves on this result.
THEORE;VI 2.7. Lel X and Y be spaces wilh Y QCC and let α s M(X. Y) have O-closed point images. Then a is α. s. c. at xeX il αnd only i j S(α ;X)= α (x).
PRODF. It follows from Theorem 3. G of [20] that any u. s. c. "with O-closcd point ìmages satisfies S(α ; x)= α (x) for each x e X For thc conve1'se. if x εX
and [01' some Woe 1.'(α (x)). Q= (α (V) • W o: Veε (x)) is a filtcrbase. then 여""
adÐQ-WO sincc Y is QCC. Howeγe 1' this implies that "，，，，， adoα(1.'("，)) -Wo=S
(α ; x)-Wo= α (x) -Wo' This is a contradiction.
III [32]. α e M(X. A) is called a relractian in casc AcX and xe.α (x) for each x e A. It is established in [32] that if α e M(X. A) is a point-compact w. u. s. c.
ret1'action on the Hausdorff space X. then A is closed. Theorem 2.8 represcnts an improvement on this result.
THEO RE'I1 2. 8.
11
α eM(X. A) is a w ι. s. c. retraction ι’ith O-closed pointzmαges the1Z A is closed in X.
PROOF. Lel x e Cl(A) and suppose that x용α (x). There is a Wε1.' (x) in X with α (x)CX -cl(W) We may choose a Ve1.'(x) with a(V)Ccl(X -cl(W)).
Let yeVnWnA. Then y ε α (y)Cα (V)ccl(X -cl(W))CX - W. This is a cOntra diction.
Our next result extends Theorem 3 of Noiri [29] to multifunctions with sub- closed graphs.
THEOREM 2.9. Let αεM(X. Y) be w. μ• s. c. wtÏh a sαbclosed graρh and sιp
pose thal a(x) has a base
01
neighborhoods ，νith compact boundaries for each xeX. Thenaisu.s.c.PROOF. Let W be a neighborhood of α (x) with bd(W) compact. There is a Ve ε (x) with α (V) ζcl (JV). If α (A- [x})nbá(W)7얘 is satisfied for each ACV with A e ε (x) then Q= fA- [x): Ae ε (x)] is a filterbase on X - (x) and Q-.x.
Hence. since bd(W) is compact and α has a subclosed graph. we have 껴낯 adα (Q) nbd(W)Cα (x) nbd(W). This is a contradiction.
The following corollary improves on Corollary 3 of [29] and extends it to multifunctions.
l?-l jesse P. Clay Qlld james E. Joseph
COROLLARY 2.10. Let α ç M(X. Y) bc lU, U, s, c with O-closed point images and suppose that α(X) has a basc of neighborhoods with compact boundaries for each x e X. Then α 15 ι. s. C.
PRO::JF. It í01l0
“
s f ro:n Cor01larn 3. 11 of [2이 that α has a closed graph. Hence the proof of thc prosent corollary may bc obtained by uso of Theorsm 2.9.aO。、 c
ιOROLLARY 2.11. Lel X be locally conuecled aud lel a E M(X, Y) have a su!Jclosed graþll. If α 11laþs oþe't c01Zllected sκbsets onlo con1l.ected subsets a1ld
α (z) Itas a base 01 neigilborhoods witil comþacl boulldaries lor each x e X tlzeu α tS u. S. c.
PROOF. We will show that a is w. u. s. c. and then the desired concIusion will follow from Theorem 2.9. Let W be a neighborhood of a(x) with bd(W) compact. If a(V - (x}) -c1(W)#Ø is satisfied for all connected V e 1:( .. ), then a(V)nbd(W)#Ø is satisfied for aJl such V since a(V) is connected. Hence, as in the proof Of Theorem 2.9, we have ()#ada(E'(x)) nbd(W)Ca(x) nbd(W).
;J. con[radiction‘
We say that a spacc X is rim-compact (141J, p.276) if each point in the space has a base of neighborhoods with compact boundaries. We wilJ utilizc thc fo Jlowing proposition to obtain Our next corollary tO Theorem 2.9. We omit the
pl ∞f.
PROPOSITlON 2.12. 11 X is ri끼 compact Qtzd ACX is C01J
‘
þacl 111m A Ilas 0 base 01 1zeighborJlOods witll comþact bozendaries.Corollary 2.13 is a result due to Hrycay (18J which f01l0ws from Corollary 2. 11 and proposition 2. 12.
COROLLARY 2.13. Let X be locally c01l1lccled mui lel Y be rim-comþacl. If
αEιM(X， Y) maps co1tllecled ope1t subsets 01lto cmznected sets a1zd α has compact )Ol
’ “
i11lages. llwn a is 1(. s. c. if α has a closed graþilThe ncxt two results are similar in naturc to those results just complcted.
THEOREYI 2.14. Lel aεM(X. Y) be ψ. α. s. c. with a stroμgly-st‘bclosed graþl
, ,
nd suþposc that a(x) IIas a base 01 1leigilborllOods with QHC relalivc 10 Yboιιdaries for eacll xeX. The1t a t's ι. s. c.
PRO::JF_ Let \Y bc a ncighborhood of a(x) with ad(W) QHC relative to Y.
