Journal of Pure Mathematics, Vol-24

2007·4.6 MB·

Volume 24, 2007
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G-men &
MINIMAL STRUCTURES, m-OPEN MULTIFUNCTIONS
AND BITOPOLOGICAL SPACES
OIRI POPA
Takashi N and Valeriu
ABSTRACT : By using m-open multifunctions from a topological space into an m-space, we establish
the unified theory for several weak forms of open multifunctions between bitopological spaces.
Key words and pharases : m-structure, m-open set, (/, j)-m-open multifunction, bitopological space,
multifunction.
AMS Subject Classification : 54A 05, 5400, 54C60, 54E55.
1. INTRODUCTION
Semi-open sets, preopen sets, a-open sets and p-open sets play an important role in the
researching of generalizations of open functions and open multifunctions. By using these sets,
several authors introduced and studied various types of modifications of open functions and
open multifunctions in topological spaces and bitopological spaces. Maheshwari and Prasad
[20] and Bose [6] introduced the concepts of semi-open sets and semi-open functions in
bitopological spaces.. Jelic' [12], [14], Kar and Bhattacharyya [15] and Khedr et al. [16]
introduced and studied the concepts of preopen sets and preopen functions in bitopological
spaces. The notions of a-open sets and a-open functions in bitopological spaces were studied
in [13], [24] and [17]. Some forms of open multifunctions are studied in [5], [7] and [8],
Recently, in [30] and [31] the present authors introduced the notions of minimal structures,
CT-spaces and m-continuity.
In the present paper, we introduce the notion of an m-open multifunctions from a
topological space into an m-space and establish the unified theory for several weak forms of
open multifunetions between bitopological spaces. We obtain some characterizations of m-open
multifunctions and characterize the set of all points at which a multifunction is not m-open.
In the last part, some new modifications of open multifunctions between bitopological spaces
is introduced and investigated.
2. PRELIMINARIES
Let (X., t) be a topological space and A a subset of X. The closure of A and the interior of
A are denoted by Cl(/i) and Int(zQ, respectively.
2 TAKASHI NOfRI AND VALERIU POPA
Definition 2.1. Let (.X., x) be a topological space. A subset A of A is said to be a -open [25]
(resp. semi-open [18], preopen [22], $-open [1] or semi-preopen [3], b-open [4] or y-open [11])
if A c Int(Cl(Int(y4») (resp. A c Cl(Int(4l)), A c Int(Cl(^)j, A c CI(Int(Cl(^))),
A c Int(CI(^))uCl(Int(/f))).
The family of all semi-open (resp. preopen, a-open, P-open, semi-preopen, £-open) sets
in A is denoted by SO(A) (resp. PO(A), a(A), p(A), SPO(A), BO(A)).
Definition 2.2. The complement of a semi-open (resp. preopen, a-open, P-open, semi-preopen,
&-open) set is said to be semi-closed [9] (resp. preclosed [10], a -closed [23], $-closed [1],
semi-preclosed [3], b-closed [4]).
Definition 2.3. The intersection of all semi-closed (resp. preclosed, a-closed, p-closed, semi-
preclosed, 6-closed) sets of X containing A is called the semi-closure [9] (resp. pre-closure
[10], a-closure [23], $-closure [2], serm-preclosure [3], b-closure [4]) of A and is denoted by
sC104) (resp. pCI(^), aCl(^), pCl(/f), spCl(/f), bCI(^).
Definition 2.4. The union of all semi-open (resp. prepen, a-open, P-open, semi-preopen, £-open)
sets of Xcontained in A is called the semi-interior (resp. preinterior, a-interior, P-interior, semi­
preinterior, b-interior) of A and is denoted by slnt(<4) (resp. plnt(/l), alnt(^(), plnt(<4), splnt(^),
blntOf)).
Throught the present paper, (X, x) and (7, a) (briefly Xand T) always denote topological
spaces and F : X-» Y(resp. f:X-* Y) presents a multivalued (resp. single valued) function.
For a multifunction F : X -» Y, we shall denote the upper and lower inverse of a subset B
of a space Y by F¥{B) and F^(B), respectively, that is
F+(B) = {x e X : F(x) c B} and = {x e X : F(x) n B;* o).
Definition 2.5. A function f: (AT, x) -> (7, x) is said to be open at a point x e X if for each
open set U containing x, there exists an open set V of 7 containing /(x) such that VaflJJ).
If f is open at each point x e A, then f is said to be open.
Remark 2.1. A function f:(X, x) -> (7, a) is open if and only if HU) is open for each open
set U of X.
Definition 2.6. function f: (A, x) -» (7, a) is said to be semi-open [26] (resp. preopen [22],
a-open [23], P-open [1]) if flJJ) is semi-open (resp. preopen, a-open, P-open) for each open
set U of X.
Definition 2.7. A multifunction F : (A, x) —> (7, a) is said to be open [5] (resp. semi-open
[29], preopen [8], a-open [7], P-open) if F(U) is open (resp. semi-open, preopen, a-open,
P-open) for each open set U of A.
MINIMAL STRUCTURES, m-OPEN MULTIFUNCTIONS AND BITOPOLOGICAL SPACES 3
3. MINIMAL STRUCTURES AND m-OPEN MULTIFUNCITONS
Definition 3.1. A subfamily mx of the power set P(X) of a nonempty set X is called a minimal
structure (or briefly m-structure) [30], [31] on X if 0 e mx and X e mx.
By (X, mx) (or briefly (X, m)\ we denote a nonempty set X with a minimal structure
mx on X and call it an m-space. Each member of mx is said to be m^open (or briefly m-operi)
and the complement of an wr^-open set is said to be m^losed (or briefly m-closed).
Definition 32. Let X be a nonempty set and mx an m-structure on X. For a subset A of X,
the m ^closure of A and the m^-interior of A are defined in [21] as follows:
(1) mrCl(^) = n{F : A c F, X- F e mx),
(2) m^-MtiA) = u{U : U c A, Ue mx}.
Remark 3.1. Let (X, i;) be a topological space and A be a. subset of X. If mx = x (resp, 80(Z),
PO(-Y), a(X), P(A0, BO(Jf)), then we have
(a) mrCI(A) = C 1(A) (resp. sCM), pCI(^), aCl(A, pCl(^), bCi(/()),
(b) mrInt(^) = Int(^) (resp. slnt(^), pint(A), alnt(/f, plnt(/l), blnt(^[)).
Lemma 3.1. (Maki et aJ. [21]). Let (X, mx) be an m-space. For subsets A and B of X, the
following properties hold:
(1) m^CXiX - A) = X - m^MCA) and m^Int(X - A) = X - m^C^A),
(2) If {X - A) e mx, then mrCl(i) = A and if A 6 mx, then mrInt(^) = A,
(3) mrCl(0) = 0, mrCl(Z) = X, mrInt( 0) = 0 and mrInt(Z) = X,
(4) If A c. B, then m^-Cl(^) c m^-CXiB) and m^lntiA) c m^lntfB),
(5) A c m^CXiA) and m^lntfA) c A,
(6) mrCl(mrCl(^)) = m^CliA) and mrInt(7wrInt(yf)) = m^lntfA).
Lemma 32. (Popa and Noiri [30]). Let (X, mx) be an m-space and A a subset of X. Then
x € mrCl(/0 if and only if U n A * d for every U e mx containing x.
Definition 3.3. A minimal structure mx on a nonempty set X is said to have property 8 [21]
if the union of any family of subsets belonging to mx belongs to mx
Definition 3.3. (Popa and Noiri [32]). Let (X mx) be an m-space and mx satisfy property 8.
Then for a subset A of X, the following properties hold:
(1) A e mx if and only if m^lnXfA) = A,
(2) A is m-closed if and only if m^CliA) = A,
4 TAKASHINOIRJ AND VALERIU POPA
(3) mx-lut(A) e mx and m^C\{A) is myclosed.
Remark 3.2. Let (X, x) be a topological space and mx = SO(Z) (resp., PO(A), <x(X), (3(Z),
BO(X)\ then mx satisfies property S.
Definition 3.4. Let (Y,nty) be an m-space.
(1) A multifunction F : (X, t) -» (F, my) is said to be m-open at x e X If for each
open set U containing x, there exists V e my containing F(x) such that V c F(U). If F is
m-open at each point x e X, then F is said to be m-open.
(2) A function f: (X t) -> (Y, my) is said to be m-open at x e X if for each open set
U containing x, there exists V e my containing fx) such that V c fU). If /'is m-open at each
point x 6 X, then f is said to be m-open.
Theorem 3.1. A multifunction F: (X X) (F, my) is m-open at x e X 'where my has property
t§, if and only if for each open set U containing, x, x e F* (m y Int(F( U))).
Proof. Necessity. Let U be an open set containing x. Then, there exists V e my such that F(x)
c V c F(U) and hence F(x) c my-lnt(F(U)). Therefore, we obtain that x e F^(my-lnt(F(U))).
Sufficiency. Suppose that x e /^(mylnt^tT))) for each open set U containing x. Then
Fix) c my-lnt(F(U)). Set V = my-lnt(F(U)f then by Lemma 3.3 Ve my and F(x) cFc F(U).
Therefore, F is m-open at x.
Theorem 3.2. A multifunction F : (X, x) -» (Y, my) is m-open, where (Y,my) has property B,
if and only if F(U) is my-open for each open set U of X.
Proof. Necessity. Let U be an open set of X and x e U. Since F is m-open at x e X, by^
Theorem 3.1 we have F(x) c my-Int(F(f/)) and by Lemma 3.1 F(U) = my-\nt(.F(t/))7 By Lemma
3.3, F{U) is my-open.
Sufficiency. Let x € X and U be an open set of X containing x. Then we have F(x)
c F(U) = my-\nt(F(U)). Therefore x e Ff(mr-Int(F(f/))- By Theorem 3.1, F is m-open at arbitrary
point x e X.
Remark 33. (a) If F : (X, t) -» (Y, a) is a multifunciton and my = x (resp. SO(L), PO(7),
a(L), p(L)), we obtain Definition 2.7, that is, the definition of an open (resp. semi-open, preopen,
a-open, P-open) multifunction, where SO(7) (resp. PO(L), a(y), (3(7)) is the family of all semi­
open (resp. preopen, a-open, p-open) sets of F.
(b) If f: (X\ t) (F, o) is a function and my = a (resp. SO(F), PO(F), a(F), p(F)),
we obtain Definition 2.6.
Theorem 33. For a multifunction F: (X, x) -> (F, my), where my has property 2?, the following
properties are equivalent:
MINIMAL STRUCTURES, m-OPEN MULTIFUNCTIONS AND BITOPOLOGICAL SPACES 5
(1) F is m-open at x;
(2) If x e Int(/4) for A e P(X), then x e F+(myInt(F(A)));
(3) x e Int(F+(B)) for B e P(Y), then x e F^m rInt(F»;
(4) Ifx e FXmrC\(B)) for B e P(Y), then x e C1(J^(B)).
Proof. (1) => (2) : Let A e P(X) and x e Int(/1). Then, there exists an open set U such that
x <= U c A and hence F(x) c F(U) c F{A). Since F is m-open at x, by Theorem 3.1 and
Lemma 3.1, we obtain x e F*(mylnt(F(U))) c F+(my-Int(F(/t)))).
(2) => (3) : Let B e 7\Y) and x g Int (F+(5)). Then, x e F*(mr\nt(F(F+(B)))) c
FfyylntC*)).
(3) => (4) : Let B e P(Y) and x e Cl (F~(B)). Then x e X - Cl (F-(B)) = Int(T -
F“(5)) = Int(F^(F - B)). By (3) we have x e Ffmrlnt(Y-B)) = X - F (mrCl(B)). Hence,
x € F-(mrC\(B)). Therefore, if x e F^(mrCl(B)), then x e CI(i^(5)).
(4) => (1) : Let U be any open set of X containing x and B = Y - F(U). Since
C1(F"(5)) = Cl(F^(Y-F(U))) = C\(X-F+(F(L'))) c X- Int(f/) = X- U and x e U, we obtain
that x t Cl(F-(£)). By (4), we have x e F-{mfC\{B)) = F(mrCl(7 - F(U))) = X -
Ff(my-Int(F(L0)). Therefore, x g F* {my\n\(F(U))). By Theorem 3.1, F is m-open at x.
Theorem 3.4. For a multifunciton F: (X, %) -> (7, mY), where my has property S’, the following
properties are equivalent:
(1) F is m-open;
(2) F(Int(T)) c my-Int(F(/4)) for any subset A of X;
(3) Int(F"(J5)) c F*-(mrInt(5)) for any subset B of Y;
(4) F~(myC\(B)) c C1(F"(B)) for any subset B of Y.
Proof. (1) => (2) : Let A be any subset of X and x e Int(^). Since F is m-open at each x
e A, by Theorem 3.3 F(x) c mylnt(F(A). Hence F(Int(/4)) c mylnt(F(A)).
(2) =* (3): Let B be any subset of 7. By (2), we have F(Int(F+(5))) c mrInt(F(F+(i?)))
a my\nt{B).
(3) => (4): Let B be any subset of 7. By (3), we have X - Cl(F^(B)) = Int(7 - F~(B))
= 1 nt(F+(Y~B)) c Ff(mrInt(7 - B)) = X- F^{mrC\{B)). Hence, F^(my-C)(B)) c C1(F-(B)).
(4) => (1): Let U be any open set of X and B = Y - F(U). By (4), we have
F-(mrCl(7- F(U))) c C1(F^(7 - F(U)). Now, F~(mrC\(Y - F(U))) = F'(7 - mrInt(F(I7)))
= X-Ff(mrInt(F(L0))- And also we have Cl(F-(7- F(U))) = C\(X-F^(F(U))) c X-Int(L0
= X - U. Therefore, we obtain U c Ff(m^Int(F(17))) and hence F(U) c my-Int (F(U)). By
Lemma 3.1, F(U) = mY-lnt(F(U) and by Theorem 3.2, F is m-open.
6 TAKASHI NOIRI AND VALEREU POPA
Remark 3.4. Let F : {X, t) -> (7, a) be a multifunction and mY = SO(.¥). Then by Theorem
3.4, we obtain the result established in Lemma 5.2 of [29].
For a multifunction F : (X, a) (7, mY), we denote
D°(F) = {x e X : F is not m-open at x}.
Theorem 3.5. For a multifunction F: (X, t) -> (7 mY), where mY has property ‘S, the following
properties hold:
D\F) = uUe,{U - nmY-M(F(U)))}
= U^^Int(A) - ^(/nrInt(F(^)))}
= vBeP(y){Int(F+(B)) - F*(mrlnt(B))}
r ugeP(y){F-(mra(B) - Cl(F~(B))}.
Proof. Let x e D°(F). Then, by Theorem 3.1, there exists an open set U0 containing x such
that x e Ff(^rInt(F(f/0))). Hence x 6 UQn (X\ ^IntC i^( L/0)))) = Ua \ lX(mY-lnt(F(U0)))
c uU6T{C/ - F+(mrlnt(F(U)))}.
Conversely, let x e Uyex{U - F^mylntC/TiT)))}. Then there exists U0 e t such that
x e U0 - Fi'(my-lnt(F(U0))). Therefore, by Theorem 3.1 x 6 D°(F).
For the second equation, let x e D°(F). Then, by Theorem 3.3, there exists A{ e P(X)
such that x e Int(34,) and x £ Therefore, x e Int(/4[) - ^(myln^F^,)))
c u^e£(j)^Int(^) “ ^(/WrIntW)))}-
Conversely, x e uAep^{lnt(A) - ^(/Wylnti'F^)))}. Then there exists Ax e 7KX) such
that x e Int^fj) - JP+(/My-Int(ir(J4j))). By Theorem 3.3, x e £>°(/).
The other equations are similarly proved.
4. MINIMAL STRUCTURES AND BITOPOLOGICAL SPACES
Throughout the present paper, (X, %x, %2) and (7, a,, C2) denote bitopobgical spaces. For a
subset A of X, the closure of A and the interior of A with respect to %t are denoted by iC\(A)
and 2Int(v4), respectively, for i = 1,2. First, we shall recall some defmkions of weak forms
of open sets in a bitopobgical space.
Definition 4.1. A subset A of a bitopobgical space (X, if) is said to be
(1) (i, fy-semi-open [20] if A c /Cl(/Int(T)), where i + j, i, j - 1,2,
(2) (/, jfpreopen [12] if A c /I nt(/C!(/()), where i * j, i, j = 1,2,
(3) (i, j)-o,-open [13] if A a /Int(/Cl(/lnt(/f))), where i A j, i, j = 1, 2,
(4) (i, j)-semi-preopen [16] if there exists an (i, /Fpreopen set U such that U c A c
jC\(U), where i A j, i, j = 1,2.
MINIMAL STRUCTURES, m-OPEN MULTIFUNCTIONS AND BITOPOLOGICAL SPACES 7
The family of (/, y')-semi-open (resp. (/, y)-preopen, (/, y)-a-open (i, y)-semi-preopen) sets
of (X, x,, t2) is denoted by (i, j)SO(X) (resp. (/, j)PO(AQ, (i, j)a(X), (/, y)SPO(AO).
Remark 4.1. Let (AT, x,, x2) be a bitopological space and A a subset of X. Then (/, j)SO(X),
(/, j)PO(X), (i, j)a(X) and \i, y')SPO(A') are all m-structures on X. Hence, if mIJ = (/, j)SO(X)
(resp. (/, j)PO(X), (/, j)a(X), (/, j)SPO(AT)), then we have
(1) //yCl(A) = (i, jysCl(A) [20] (resp. («, y)-pCl(/f) [16], (/, j)-aCl(A) [24], (/, j)-
spCI(A) [16]),
(2) /ylnt(4) = (/, y)-slnt(^) (resp. (/, y>plnt(^), (i, y)-alnt(^l), (/, y>splnt(^)).
Remark 4.2. Let (X, xl5 x2) be a bitopological space.
(a) Let m = (/', y)SO(A0 (resp. (/, j)a(X)). Then, by Lemma 3.1 we obtain the result
established in Theorem 13 of [20] and Theorem 1.13 of [19] (resp. Theorem 3.6 of [24]).
(b) Let mg = (/, j)SO(X) (resp. (/, y')PO(T), (/, j)a(X), (/, j)SPO(AQ). Then, by Lemma
3.2 we obtain the result established in Theorem 1.15 of [19] (resp. Theorem 3.5 of [16], Theorem
3.5 of [24], Theorem 3.6 of [16]).
Remark 4.3. Let (X, x,, x2) be a bitopological space.
(a) It follows from Theorem 2 of [20] (resp. Theorem 4.2 of [15] or theorem 3.2 of
[16], Theorem 3.2 of [24], Theorem 3.2 of [16] that (/, /)SO(r) (resp. (/, y)PO(A[), (/, j)a(X),
(i, y')SPO(A’)) is an /n-structure on X satisfying property IS.
(b) Let mtJ = (/, y)SO(T) (resp. (i, j)PO(X), (i, j)a(X), (i, y)SPO(AO). Then, by Lemma
3.3 we obtain the result established in Theorem 1.13 of [19] (resp. Theorem 3.5 of [16], Theorem
3.6 of [24], Theorem 3.6 of [16]).
_ 5. m-OPEN MULTIFUNCTIONS IN BITOPOLOGICAL SPACES
Definition 5.1. A function f: (X x,, x2) -» (7,-0-,,-a2) is said to be (i, j)-semi-open [6] (resp.
(/, j)-preopen [15], (i, j)-a-open [17], (/, j)-semi-preopen) if for each x(-open set U of X, f(U)
is (i, y')-semi-open (resp. (i, y')-preopen, (/,_y)-a-open, (/, y')-semi-prepen) in Y.
Definition 5.2. A multifunction F : (X, Xj, x2) (7, CT,, 02) is said to be (/, j)-almost open
(cr (/, j)-preopen) [8] if for each U e F(U) is (/, y')-preopen.
Remark 5.1. (a) By Remark 4.3(a), (/, y)SO(7), (i, y‘)PO(7), (/, y')a(7) and (/, y)SPO(y) are
all m-structures on 7 satisfying property (2). Therefore, a function f : (X, x,, x2) ->
(7, 0,, 02) is (/, y')-semi-open (resp. (/, /Tpreopen, (i, y')-a-open, (/, y)-semi-preopen) if and only
if f : (X, x() -> (7, y) is m-open, where my = (i, y')SO(T) (resp. (i, y')PO(T), (/, y')a(7),
(L y)SPO(T)).
8 TAKASHINOIRI AND VALERIU POPA
(b) A multifunction F : (X, x,, %f) -> (Y, ct,, 02) is (/, y)-preopen if F : (Z, x() ->
(7, nty) is m-open, where mIJ = (/, /)PC>(7).
Definition 53. Let (7, a,, a2) be a bitopological space and mtJ = mfOj, 02) an m-structure
on Y. A multifunction F : (X, Xj, x2) —* (7, al5 02) is said to be (i, fy-m-open at x e Z if
F : (X, x() -> (7 my) is m-open at r e I
Remark 5.2. (a) A multifunction F : (Z, xl9 x2) -> (7, 0j, o2) is (/, fy-m-open at x e X if
for each x(-open set U containing x, there exists V e mtJ containing F(x) such that V c F(U).
(b) By Remark 33, it follows from that a multifunction F : (X, Xj, x2) -> (7, o1} 02)
is {i,fy-m-open, where mtJ = m(o{, 02) has property 2?, if and only if F(U) is m-open for every
Xj.-open set U of X.
(c) If f: (X, x,, x2) -> (7, Oj, 02) is a function and F : (Z, x,, x2) -> (7, 0,, 02) is
a multifunction, then by Defmiton 5.3 we obtain Definitions 5.1 and 5.2.
By Definition 5.3 and Theorems 3.1-3.5, we obtain the following theorems.
Theorem 5.1. Let (7, 0j, 02) be a bitopological space and mtj = m(0(, 02) an m-structure
on Y with property S. Then a multifunction F : (Z Xj, x2) -> (7, 0,, 02) is (/, j)-m-open at
x e X if and only if x € F+(my-Int(F(L/))) for every %-open set U containing x.
Theorem 5.2. Let (7 0(, 02) be a bitopological space and mfJ = /w(0ls 02) an m-structure
on Y with property S’. Then a multifunction F : (Z, Xj, X2) -> (7, 0j, 02) is (/, fy-m-open if
and only if F(U) is mt-open for every %-open set U of X
Theorem 5.3. Let (Y, 0,, 02) be a bitopological space and my = /w(0j, 02) an m-structure
on 7 with property S’. For a multifunction F: (Z, x,, x2) -» (7, 0,, 02), the following properties
are equivalent:
(1) F is (/, f)-m-open at x e X;
(2) Ifx e iM(A) for A e P{X), then x e Ff(wy-Int(F(T));
(3) //* e /Int(Fl~(fi)) for 5 e P(7), rten * e F^nylntfR));
(4) 1/ x e F-(ot(/C1(F)) /or B e ^(7), /few jc e /Cl(FtF)).
Theorem 5.4. Let (7, 0|, 02) be a bitopological space and mfJ = 07(0,, 02) on m-structure
on Y with property S’. For o multifunction F: (Z Xj, x2) -> (7, 0t, 02), the following properties
are equivalent:
(1) F is (/, f-m-open;
(2) F(ilntfA) c wy-Int(F(/l)) /or every subset A of X;
(3) /Int(/^(F)) c Ff(/n(/-Int(JB)) /or every subset B of Y;