Internat. J. Math. & Math. Sci.
VOL. 13 NO. 2 (1990) 275-280
275
IRRESOLUTE MULTIFUNCTIONS
V. POPA
Department of Mathematics
Higher Education Institute
5500
Bacau, Romania
(Received February 12, 1987)
ABSTRACT.
This paper considers a new class of multifunctions, the irresolute multi-
unctions.
For the irresolute multlfunctions we give some theorems of characterizations.
Some relations between continuous multifunctions and irresolute multifunctions are
established.
KEY WORDS AND PHRASES.
Quasicontinuous multifunction, irresolute multifunction,
strongly continuous multifunction.
1980 AMS SUBJECT CLASSIFICATION CODE.
54C60.
I.
INTRODUCTION.
In [I] Levine defines a set A in a topological space X to be semi-open if there
exists an open set U c X such that U c A c Cl U, where C1 U denotes the closure of U.
The family of all semi-open sets in X is denoted by SO(X).
if its complement is semi-open.
ing a set A is the semi-closure of A denoted by Scl A.
implies Scl A
c
A set
is semi-closed
The intersection of all the semi-closed sets contain-
Also, ScI(A)= Scl(Scl A), A c B
Scl B, A c Scl A c C1 A and that A is semi-closed iff A=Scl A [2], [3].
The notion of irresolute functions was introduced by Crossley and Hildebrand in
[4] in this way:
DEFINITION I.
Let X and Y be two topological spaces.
SO(Y), f-l(v) SO(X).
of upper (lower) irresolute multifunctions
A function f:XY
is irreso-
lute if for each V
The notion
was introduced by Ewert and
Lipski in [5].
Let X and Y be two topological spaces.
(a) A multifunction F:XY is upper irresolute (u.i.) at a point x 6 X if for any semiopen set W c y such that F(x) c W, there exists a semi-open set U c X containing x
such that F(U) c W.
DEFINITION 2.
(b)
A multifunction F:XY is lower irresolute (l.i.) at a point x X if or any semic y such that F(x) 0 W
there is a semi-open set U c X containing x
U.
such that F(y) 0
V Y
open set W
,
V. POPA
2Y6
A multifunction F:XY is upper (lower) irresolute if it has this property in any
point xX [5].
Some properties of the lower (upper) irresolute multifunctions are studied in [5].
The notion of quasicontinuous multifunctions was introduced and studied by Banzaru
(c)
and Crivat in [6].
A multifunction F:XY
Let X and Y be two topological spaces.
DEFINITION 3.
is
quasicontinuous at a point x X if for any neighborhood U of x and for any open sets
G # @ there exists a non-empty open set
G c y such that F(x) c G and F(x)
2
1
2
GI,
U such that F(G
G
c
x
X [6].
U
U)
c
G
1
and
F(y)
G
2
#
#,
VYe GU-
The multifunction F:XY is quasicontinuous if it has this property at any point
Some properties of quasicontinuous multifunctions are studied in [7], [6] and [8].
DEFINITION 4. Let X and Y two topological spaces. A multifunction F:XY is
irresolute at a point xeX if for any semi-open sets GI, G c y such that F(x) c G and
1
2
F(x)
F(y)
G
c
there exists a semi-open set U c X containing x such that F(U) c G 1 and
2
G
,Vye U.
2
The multifunction F:Y is irresolute if it has this property at any point x
X.
REMARK I. If F:Y is irresolute then F is upper and lower irresolute.
REMARK 2. By Theorem I.I [8] it follows that if F:XY is irresolute then F is quasicontinuous.
2.
CHARACTERIZATIONS.
Let X,Y be two topological spaces and let S(y) and K(y) be classes of all non-empty
and non-empty compact subsets of Y, respectively. For a multifunction F:XY we will
denote
F+(B) ={xeX:F(x)
c
F-(B)= {xe X:F(x)
B};
B
@}
for any subset B c y.
Let A be a set of a topological space X.U is a semi-neighbourhood
A # @.
which intersects A if there exists a semi-open set V c X such that V c U and V
THEOREM I. For a multifunction F:XY the following are equivalent:
I. F is irresolute at x X.
G # @, there
2. For any semi-open sets GI, G c y with F(x) c G and F(x)
1
2
2
DEFINITION 6.
results the relation
x
CI {Int
[F+(GI
For every semi-open set.
3.
F-(G2)] }.
GI,
G
2
y with F(x)
c
G
1
and F(x) N G
2
and for
any open set U c X containing x, there exists a non-empty open set Gu c U such that
F(Gu)
c
G
1
and F(y)
(I)
G
2
V
YeGU.
G e SO(Y)
with F(x) c S and F(x) N G # @. Then
2
1
2
G # @, V Ye U, thus
there is U eSO(X) containing x such that F(U) c G and F(y)
1
2
PROOF.
=>
(2). Let
,
GI,
IRRESOLUTE MULTIFUNCTIONS
F+(GI
F+(GI
F -(G ). Then x U
2
set in X, then by Theorem
[i] x6 U c C1 [Int U]
U c
x
(2)
(3).
=>
U
and x
c
c
Let G I, G e SO(Y) be with F(x)
2
Cl{Int[F+ (G I)
277
F-(G2)
[F+(GI N F-(G2)] }.
Cl{Int
G
and
Since U is a semi-open
F(x) N G 2
0.
Then
F-(G2)] }. Let U X be any open set such that x eU. Then
U
[Int F+(GI
F-(G2)] 0. Since Int[F+(Gl F-(G2) c Int F+(GI Int F-(G2)
then U 0 [Int F+(GI 0 Int F-(G2)]
0. Put G
[Int F+(GI
U, then
Int F-(G2)]
U
G
u 0, GU c U, Gu Int F+(GI c F+(GI and GU c Int F-(G2) c F-(G2) and thus
x
F(Gu)
G
c
(3)
1
=>
and
0
F(y)
G
(i). Let U
X
0,V y eGU.
2
be the system of the open sets from X containing x.
open set U c X such that x eU and for every semi-open set
and F(x)
F(y)
G
F(z)
0 G
in X,
I.
2.
G
2
2
x6
0, there exists a non-empty open set G
U
2
S, F(S)
c
G
and F(t) 0 G
1
4.
For every set
BI,
Int{CI[F-(B I)
For every sets
ScI[F-(B I)
7.
BI,
c
G
G
and
and
C1 W, thus W is a semi-open set
S, thus F is irresolute at x.
0,V t
2
S
F+(GI
B2
c
y,
BI,
slnt[F-(B I) N
B
2
y,
F+(B2 )]
V
2
2)
y, F (V I)
U
F+(v2
F-(Scl B I)
U
F+(Scl B2).
there results the relation
F-(Scl B I)
U
F+(Scl B2).
there results the relation
F-(slnt B I)
0
F+(slnt B2).
For each point x of X and for each semi-neighbourhood V
2
is a semi-closed set in X.
there results the relation
F+(B2)]}
F+(B2 )]
U
VI,
y,
B2
U
For every set
neighbourhood V
8.
GU, then W is open, xcl W, F(W)
2
For every semi-closed set
6.
U)
U such that F(G
G
THEOREM 2. For a multifunction F:XY the following are equivalent:
F is irresolute.
For every semi-open set GI, G c y,
F (G e SO(X).
3.
5.
y with F(x)
x
W U {x}, then W
Put S
W.
0, V z
U
UeU
Let W
0, Vy GU.
GI,G2
For any
which intersects
F(x),
F+(Vl
of F(x) and for each semi-
F (V2) is a semi-neighbourhood of x.
For each point x of X and for each semi-neighbourhood V of F(x) and for each semiI
neighbourhood V 2 which intersects F(x), there is a semi-neighbourhood U of x such that
F(U)
V
1
and F(y)
V
2
0, V y
U.
V. POPA
278
PROOF. (I)
F(x)
(I)
0 G
GI,
G
SO(Y)
2
F+(G1
thus F+(GI
F+(G1
F-(G2) there follows that F+(G1
0
in
F (G
I and
(G2)]
F-(G2) c
2)
0 F
and as x is choosen arbitrarily
CI{Int[F+(GI
0
F-(G2)]
and
of [6].
is a semi-open set by Theorem
(3). For if V c y, then F-(Y-V)=X-F+(V) and F+(Y-V) X-F-(V).
(4). Suppose that (3) holds and let B I, B2 two arbitrary subsets of Y, then
=>
=>
and Scl B
1
thus F(x) c G
0, then F being irresolute according to the Theorem I, implication
2
(2)
(3)
F+(GI 0 F-(G2),
and x
(2) there follows that x e Cl{Int[
ffi>
Scl B
(2). Let
ffi>
2
are semi-closed sets in Y.
Then
F-(Scl B I)
U
F+(Scl B2)
is a semi-
By Theorem 1 of [3]
closed set of X.
[F-(Scl
Int{Cl
Since we have Ac Scl
B1)
A then
+ (Scl
B2)]}c
U F
F+(A)
F+(Scl
c
F-(Scl B 1)
A) and F-(A)
F+(Scl
U
F-(A)
c
c
F-(Scl A).
Consequently,
Int{Cl
[F-(BI) U F+(B2)]}
c
(4)
F+(B2 )]
U Int{Cl
(5)
F-(Scl B 1)
X-
U
(6) X-
[F-(sInt
BI)
F+(sInt B2)
=>
0
(7).
which intersects
c
F+(B2 )]
F-(Scl B I)
U
sInt[F-(Bl) fi F+(B2 )]
2
c
F+(sInt B2)]
U
U
F+(Scl B2)
c
x X,
Let
Scl
[X-F-(B I)
8
c
F-(Y-sInt B2) (X--sInt B1)) U (X-F+(sInt
and thus sInt[F-(B I) 0
)] D F-(sInt B I)
U
F(x)
V
a seml-neighbourhood of F(x) and V
F(x), then there
V2,
U
c
1
and
exists two semi-open sets U
F(x)
0 U
2
#
0, thus xe
0
c
B2))
F+(B2
0
0
[F-(B I)
F+(B2 )]
[F+(Y-B1 U F-(Y-B2) F+(ScI(Y-BI ))
=Scl
1
a seml-nelghbourhood
2
and U
2
F+(UI F-(U2).
x F+(UI
F-(U2) F+(sInt UI) F-(sInt U2) slnt[F+(Ul F-(U2)
F-(V2)] F+(VI F-(V2). From x sInt[F+(Ul F-(U2) F+(VI)
that F+(VI
F-(V2) is a seml-neighbourhood of x.
and U
c
F+(Scl B2).
F+(B2)]} [F-(BI) U F+(B2 )]
[(X-F-(BI)) U (X-F+(B2))]
F-(ScI(Y-B2)) F+(Y-sInt BI)
(6)
U
F+(Scl B2)]}
B 1) U
F+(Scl B2).
U
=>
[F-(B I)
Scl
U
Int{Cl[F-(Scl
(5). From Scl A=A U Int C1 A follows ScI[F-(B I) U
=>
F-(Scl B I)
c
0
0
e
1
c
V
1
By hypothesis
0
c
such that U
c
0
[F+(VI
sInt
c
F-(V2)
it follows
0
(7)
=>
(8).
Let x e X,
V
1
a
hood which intersects F(x), then U
F(U)
c
V
1
and
F(y) 0V2
0, VyeU.
semi-neighbourhood of F(x) and V
F+(VI
0
F-(V 2)
is a
2
a seml-nelghbour-
semi-neiEhbourhood
of x,
IRRESOLUTE MULTIFUNCTIONS
(1). Evident.
COROLLARY I. For a single valued mapping
(8)
279
=>
I.
f is irresolute at x.
2.
For each semi-open set G
x eCl[Int
c
f:X/Y the following are equivalent:
y with f(x) e G, there results the relation
f-l(G)].
X containing x and for any semi-open set G
f(x) e G, there exists a non-empty open set G c U such that f(GU) c G.
u
3.
For any open set U
c
c
y with
COROLLARY 2. For a single valued mapping f:XY the following are equivalent:
I. f is irresolute.
2.
f-l(G) e SO(X),
3.
For each semi-closed set V
4.
For each subset B
c
y,
5.
For each subset B
c
y,
6.
For each subset B
c
y,
7.
For each point x of X and for each semi-neighbourhood V of f(x),
f-l(v)
GeSO(Y). (Definition I.I [4]).
f-l(v) is a semi-closed set. (Theorem
Int[Cl fY-I(B)] c f-l(scl B).
Scl f-l(B) c f-l(scl B). Theorem 1.6, [4])
sInt f-l(B)
f-I (sInt B).
c
y,
1.4, [4]).
D
is a semi-neighbourhood of x.
8. For each point x of X and for each semi-neighbourhood V of f(x) there is a
semi-neighbourhood U of x such that f(U) c V.
3.
CONTINUOUS MULTIFUNCTIONS AND IRRESOLUTE MULTIFUNCTIONS.
The notion of strongly continuous multifunctions was introduced in [9] as a generali-
zation of the univocal strongly continuous mapping defined by Levlne in
DEFINITION 7.
if for each subset B c
DEFINITION 8.
[I0].
The multifunction F:XY is strongly lower seml-contlnuous (s.l.s.c.)
y, F-(B)
is a open set in X [9].
The multifunction F:XY is strongly upper semi-continuous (s.u.s.c)
y, F+(B) is an open set in X.
THEOREM 3. If F:X/Y is a multifunction so that:
I. F is upper irresolute.
if for each subset B
2.
F is strongly lower semi-continuous, then F is irresolute.
). F being upper irresolute then
SO(Y). Let x
Let GI, G
2
F+(GI
PROOF.
there is a semi-open set U containing x and
by Theorem
F+(GI
of [6], x
there follows that
X by Theorem 1 of [I].
F+(GI N F-(G2)
DEFINITION 9.
we have
U
c
Cl[Int U]
F+(GI
F being
c
c
F(U)
Cl[Int
Cl[Int
c
F (G
and
2)
As x is chosen
tus F+(GI
arbitrarily in
is a semi-open set in
is an open set in X.
SO(X) and by Theorem 2, implication (2)
=>
(I).
Then
F is irresolute.
A multifunction F:XY is said to be injective if for x I,
F(X I) N F(x 2)
X, then
Since U is semi-open in
I.
F+(GI)].
F+(GI )]
s.l.s.c, then
G
x2
X, x I
x
0.
A multifunction F:XY is said to be pre-semi-open if for any semi-open set A
set F(A) is semi-open.
c
X the
2
280
V. POPA
DEFINITION I0. A set A is called regular open if A=Int[Cl A].
THEOREM 5. Let Y be a regular space and F:XY S(Y) be a pre-seml-open and irresolute
multifunction. If one of the conditions holds:
I. Int F(X)
0 for every xe X.
F is injective,
lower semi-continuous.
is
F
Then
PROOF. In a topological space (Y,T) the intersections of two regular open sets
forms a base for a topology TS on Y, called the semi-regularization of T. If the Y is
2.
a regular space then T=T
S.
The proof follows then by Remark I and by Theorems 7 and
I0 from [5].
THEOREM 6.
closed sets.
Let Y be a regular space or a space which has a basis composed of openIf F:XK(Y) is a pre-semi-open, irresolute and injective multifunctlon,
then F is continuous.
PROOF.
Follows from Remark I, Theorems 7 and II of [5] and Remark 8 from [5].
ACKNOWLEDGEMENT.
The author would like to thank the referee for his helpful suggestions
toward the improvement of this paper.
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2.
3.
4.
5.
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