 Open Access
 Total Downloads : 26
 Authors : R.Chitra, J. Sophers, O.Ravi
 Paper ID : IJERTCONV5IS04017
 Volume & Issue : NCETCPM – 2017 (Volume 5 – Issue 04)
 Published (First Online): 24042018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Slightly mPrecontinuous Multifunctions
(1)R.Chitra, (3)J. Sophers (1&3) Assistant Professor, Department of mathematics,
Kongunadu College of Engineering and Technology, Trichy, Namakkal (D.t).
Abstract – In this paper, we introduce a new class of functions namely slightly m precontinuous multifunction. Further we obtain characterizations and relationships between upper/ lower slightly m precontinuous multifunctions and other related multifunctions.
Keywords – Upper slightly precontinuous, lower slightly precontinuous, upper slightly mprecontinuous, lower slightly mprecontinuous, upper mprecontinuous, lower m precontinuous.
AMS Subject classification : 54C10, 54C08

INTRODUCTION
The notion of slightly continuous functions was introduced by R.C. Jain [2]. Nour [12] defined slightly semicontinuous functions as a weak form of slight continuous functions and obtained their properties. Further this concept is extended to develop the class of slightly m continuous functions. Popa [14] and Smithson [23] studied the concept of weakly continuous multifunctions. In this paper, we introduce the notion of slightly mprecontinuous multifunction and investigate the relationships among m precontinuity, weak mprecontinuity and slight m precontinuity for multifunctions.

P RELIMINARIES
Throughout this paper, all spaces (X,) and (Y,) are always topological spaces. A subset A of a space X is said to be regular open (resp.regular closed), if A=Int(Cl(A))(resp. A=Cl (Int(A))), where Cl(A) and Int(A) denote the closure and interior of A . A subset A of a space X is called preopen if AInt(Cl(A)).The complement of a preopen set is said to be preclosed. The family of all regular open (resp. regular closed, preopen, preclosed, clopen) sets of X is denoted by RO(X) (resp. RC(X), PO(X), PC(X), CO(X)). A subset A of a space X is said to be semiopen if ACl(Int(A)).The complement of a semi open set is called semiclosed. The union of all pre open sets of X contained in A is called the preinterior of A and is denoted by pInt(A).A subset A of a space X is said to be clopen if it is both open and closed.
Throughout this paper, the spaces (X,) and (Y,)(or simply X and Y) denote topological spaces and F:(X,)(Y,) represents a multivalued function. For a multifunction F:(X,)(Y,), we shall denote the upper and lower inverse of a set B of a space Y by F+(B) and F

respectively.
(2)O.Ravi,
(2) Head & Professor, Department of mathematics,
P. M Thevar College, Usilampatti, Madurai (D.t).
i.e. F+(B) = {xX: F(x)B},F(B) = {xX: F(x)B}.
Definition: 2.1[20]
A subfamily mX of the power set (X) of a nonempty set X is called a minimal structure (briefly m structure) on X, if mX and XmX.
Definition: 2.2 [20]
By (X, mX), we denote a nonempty set X with a minimal structure mX on X and call it an mspace. Each member of mX is said to be mX open (or briefly mopen). The complement of an mXopen set is said to be mXclosed (or briefly mclosed).
Remark: 2.3 [20]
Let (X,) be a topological space. Then the families
, SO(X), PO(X), (X), (X), are all m structures on X.
Definition: 2.4 [20]
A function f:(X,)(Y,) is said to be slightly continuous if for each point xX and each clopen set V containing f(x), there exists an open set U containing x such that f (U)V.
Definition: 2.5 [20]
A topological space (X,) is said to be extremally disconnected (briefly E.D), if the closure of each open set of X is open in X.
SLIGHTLY mPRECONTINUOUS MULTIFUNCTIONS
Definition:3.1
A multifunction F:(X,)(Y,) is said to be

Upper slightly precontinuous (resp. upper slightly continuous, upper slightly semicontinuous or upper faintly precontinuous, upper slightly continuous), if for each point xX and each clopen set V of Y containing F(x), there exists an preopen (resp. open, semiopen, open) set U of X containing x such that F(U)V.

Lower slightly precontinuous (resp. lower slightly continuous, lower slightly semicontinuous or lower faintly precontinuous, lower slightly continuous), if for each point xX and each clopen set V of Y such that F(x)V, there exists an preopen (resp. open, semiopen, open) set U of X containing x such that F(u)V, for each uU. Definition: 3.2

A function f:(X,mX)(Y,), where (X,mX) is a nonempty X with an minimal structure mX and (Y,) is a topological space, is said to be slightly mpre continuous, if for each xX and each clopen set V of Y containing f(x), there exist a preopen set UmX containing x such that f(U)V.
Definition: 3.3
A multifunction F:(X,mX) (Y,) is said to be

Upper mprecontinuous( resp. upper almost m precontinuous, upper weakly mprecontinuous), if for each point xX and each open set V of Y
containing F(x), there exists a preopen set UmX containing x such that F(U)V( resp. F(U) Int (Cl
(V)), F(U)Cl(V)).

Lower mprecontinuous (lower almost m precontinuous, lower weakly mprecontinuous), if
for each point xX and each open set V of Y such that F(x)V , there exists a pre open set UmX containing x such that F(u)V ( resp. F(u)Int(Cl(V)), F(u)Cl(V) ) for each
uU.
Definition: 3.4
Let X be a nonempty set and mX an mstructure on

For a subset A of X, the mXpreclosure of A and the mXpreinterior of A are defined as follows.

mXpCl(A) = {F:AF, XFmX}.

mXpInt(A) = {U:UA, UmX}.

Remark: 3.5
Let (X,) be a topological space and A be a subset of

If mX = (resp.SO(X), PO(X), (X)), then we have

mXCl(A) = Cl(A)(resp.sCl(A),pCl(A),
Cl(A)).

mXInt (A) = Int(A)(resp. sInt(A), pInt(A),

Int(A)).
Definition: 3.6
A multifunction F:(X,mX) (Y,), is said to be

Upper slightly mprecontinuous, if for each point xX and each colpen set V of Y containing F(x), there exist preopen set UmX containing x such that F(U)V.

Lower slightly mprecontinuous, if for each point xX and each colpen set V of Y such that F(x)V
there exists a preopen set UmX containing x such that F(u)V for each uU
Theorem: 3.7
For a multifunction F:(X, mX) (Y,) the following are equivalent.

F is upper slightly mprecontinuous

F+(V) = mXpInt(F+(V)) for each VCO(Y)

F(V) = mXpCl( F (V)) for each VCO(Y) Proof:

(1)(2): Let V be any clopen set of Y and xF+(V) then
F(x)V (1).By (1), there exists a preopen set UmX containing x such that F(U) V. Thus xU F+(V)
xmXpInt(F+(V)) (2).From eqn (1) & (2) we get,F+(V) mXpInt(F+(V)).i.e. mXpInt (F+(V)) F+(V),
by lemma (3.1) of [20] F+(V) = mXpInt(F+(V)) for each VCO(Y).

(3): Let K be any clopen set of Y. Then YK is clopen in Y.By (2), F+(V) = mXpInt(F+(V)).By lemma (3.1) of [20] we have, X F(K) = F+(YK)= mXpInt(F+(Y K)) = X[mXpCl(F(K)].Therefore F(K) = mXpCl(FK)).

(2): Let B be any clopen set of Y. Then Y B is clopen in Y.By (3) & lemma (3.1) of [20] we have, X
F+(B) = F(YB) = mXpCl(F+(V)) = X[mX
pInt(F+(V)).Therefore F+(B) = mXpInt(F+(V)).

(1): Let xX and V be any clopen set of Y containing F(x). Then xF+(V)= mXpInt(F+(V)),there exists a preopen set UmX containing x such that xUF+(V). Therefore we have, xU and UmX and f(U)V. Hence F is upper slightly mprecontinuous.
Theorem: 3.8
For a multi function F:(X, mX) (Y,) the following are equivalent.

F is lower slightly mprecontinuous.

F(V) = mXpInt((F(V)) for each VCO(Y).

F+(V) = mXpCl(F+(V)) for each VCO(Y)

Proof:

(2): Let VCO(Y) and xF(V). Then F(x)V
.By (1), there exists a preopen set UmX containing x such that F(u)V for each uU.Therefore
we have, U F(V) xU mXpInt(F(V)) xmX pInt(F(V)) F(V) mXpInt(F(V)) and by lemma (3.1)
of [20] we have F (V)= mXpInt(F(V)).

(3): Let VCO(Y).Then YVCO(Y),by (2), we have,X F+(V) = F(V)=
F(YV)= mXpInt(F(YV))= X[mXpCl(F+(V)).Hence F+(V) = mXpCl(F+(V))
(3) (1): Let xX and VCO(Y) such that
F(x)V,then xF(V) and
xX F(V) = F+(YV).By (3) we have,xmX
pCl(F+(V)).By lemma (3.2) of [20], there exists preopen set UmX containing x such that UF+(YV) = .Thus U
F(V).Therefore F(u)V for each uU. Hence F is lower
slightly mprecontinuous.
Theorem: 3.9
Let (Y,) be E.D. For a multifunction F:(X, mX)(Y,), the following are equivalent.

F is upper slightly mprecontinuous.

mXpCl(F(V)) F(Cl(V)) for every open set V of (Y,)

F+(Int(C)) mXpInt(F+(C)) for every closed set C of (Y,)
Proof:

(2): Let V be any open set of Y. Then Cl(V)CO(Y),by theorem (3.7), we have,F(Cl(V)) = mX pCl(F(Cl(V))) and F(V) F(Cl(V)),by lemma (3.1) of [20], we have, mXpCl(F(V)) mXpCl(F(Cl(V)))=F (Cl(V)) mXpCl(F(V))
F(Cl(V))

(3): Let C be any closed set of (Y,) and V = YC. Then V is open in (Y,). By lemma (3.1) of [20], we have, X [mXpInt(F+(C))] = mXpCl(xF+(C))= mXpCl(F(Y
C)) F(YInt(C)). Therefore X[mXpInt(F+(C))] = X
F+(Int(C)) F+(Int(C)) mXpInt(F+(C))

(1): Let xX and let VCO(Y) containing F(x). Then By (3) we have, xF+(V)= F+(Int(V)) mX pInt(F+(V)).There exists a preopen set UmX such that xUF+(V). Thus xU, UmX and F(U)V. Hence F is upper slightly mprecontinuous.
Theorem: 3.10
Let (Y,) be E .D. For a multifunction F:(X, mX)
(Y,), the following are equivalent.

F is lower slightly mprecontinuous.

mXpCl(F+(V)) F+(Cl(V)) for every open set V of (Y,)

F(Int(C)) mXpInt(F(C)) for every closed set C of (Y,)
Proof:

(2): Let V be any open set of Y. Then Cl(V)CO(Y).By theorem (3.8), F+(Cl(V)) = mX pCl(F+(Cl(V))) and F+(V) (F+(Cl(V))).By lemma (3.1) of [20], we have mXpCl(F+(V)) mX pCl(F+(Cl(V)))=F+(Cl(V))mXpCl(F+(V)) F+(Cl(V))

(3): Let C be any closed set of (Y,) and V= YC,
then V is open in (Y,)
By lemma (3.1) of [20], we have, X [mXpInt (F(C))] =
mXpCl(X(F(C)))= mXpCl(F+(YC)) F+(Cl(YC)) (by
given (2))=F+(YInt(C)).Therefore X [mXpInt
(F (C))] = X F(Int(C)) F(Int(C)) mXpInt(F(C))

(1): Let xX and VCO(Y) containing F(x)V
.Let V = YC is open in Y. Let xF+(C) and xX F+(C)F(YC).By (3) we have, xmXpInt (F(YC)),by theorem (3.8) we have, xmXpCl(F+(YC)).By lemma (3.2) of [20], there exists a preopen set UmX containing x such that U(F+(YC)) = .Hence U
F(V)and F (x)V for each uU. Hence F is lower slightly mprecontinuous.
Slight mprecontinuity and other forms mprecontinuity.
Theorem: 4.1
If multifunction F:(X, mX)(Y,) is upper weakly mprecontinuous, then it is upper slightly mprecontinuous.
Proof:
Let xX and VCO(Y) containing F(x).Since F is upper weakly mprecontinuous. There exists a preopen set UmX containing x such that F(U)Cl(V) = V. Hence F is upper slightly mprecontinuous.
Theorem: 4.2
If a multi function F:(X, mX)(Y,) is lower weakly mprecontinuous then it is lower slightly m precontinuous.
Proof:
Let xX and VCO(Y) such that F(x)V .Since F is lower weakly mprecontinuous there exist a preopen set UmX containing x such that F(U)Cl(V) , for each
uU F(U)(V) , for each uU. Hence F is lower slightly mprecontinuous.
Lemma: 4.3
A multifunction F:(X, mX)(Y,) is upper almost mprecontinuous (resp.lower almost mprecontinuous) iff for each regular open set V containing F(x) (resp. meeting F(x)), there exist a preopen set UmX containing x such that F(U)V (resp. F(u)(V) , for every uU)
Proof:
Let xX and V be regular open set of Y containing F(x).Since F is upper almost mprecontinuous there exist a
preopen set UmX containing x such that F(U) Int(Cl(V)) = V. Hence for each regular open set V
containing F(x), there exist a preopen set UmX containing x such that un F(U) V. Conversely, there exist a preopen set UmX containing x such that F(U) V Cl(V) IntF(U) Int(Cl(V)) F(U) Int(Cl(V)).Hence F is
upper almost mprecontinuous.
Theorem: 4.4
If a multi function F:(X, mX) (Y,) is upper slightly m precontinuous and (Y,) is E.D, then F is upper almost m precontinuous.
Proof:
Let xX and V be any regular open set of (Y,) containing F (x). Then By lemma 5.6 of [13] we have,
V CO(X).Since (Y,) is E.D and F is upper
almost m precontinuous, there exist a preopen set U mX containing x such that F(U) V. By lemma (4.3), F is
upper almost m precontinuous.
Theorem: 4.5
If a multi function F: (X, mX)(Y,) is lower slightly m precontinuous and (Y,) is E.D , then F is lower almost m precontinuous.
Proof:
Let xX and V be any regular open set of (Y,) containing F(x). Then By lemma 5.6 of [13] we have, VCO(X).Since (Y,) is E.D. Since F is lower slightly m precontinuous, there exist a pre open set UmX containing x such that F(U)V for each uU. By lemma (4.4), F is lower almost m pre continuous.
REFERENCES

T.Banzaru, More remarks on some forms of continuity for multifunctions, Bul.Stiint.Univ.Politeh.Timisser Mat.Fiz., 44(58) (1999), 3439.

R.C.Jain, The role of regularly open sets in general topology Ph.D. Thesis. Meerut Univ., 1980.

A.S.Mashhour, I.A.Hasanein and S.N.EIDeeb, a note on semi continuity and precontinuity, Indian J.Pure Appl.Math. 13 (1982), 11191123.

A.S.Mashhour.M.E.Abd EI.monsef and S.N.EIDeep, on precontinuous and weak precontinuous mappings, Proc.Math.Phys.Soc.Egypt 53(1982), 4753.

A.S.Mashhour. M.E.A.EImonsef, I.A.Hasanein, on pretopological spaces, Bull. Math. Soc.Sci. RSR.28 (1984), 3945.

H.Maki, K.C.Rao and A.Nagoor Gani, On generalizing Semiopen sets and preopen sets. Pure Appl.Math. Sci. 49(1999), 1729.

T.Noiri, Characterizations of extremally disconnected spaces, Indian J.Pure Appl.Math., 19 (1988), 325329.

T.Noiri and V.Popa, On upper and lower weakly quasi continuous multifunctions, Rev.Roumaine Math.Pures Appl., 37(1992), 499 508.

T.Noiri and V.Popa, almost weakly continuous multifunctions, Demonstratio Math, 26(1993), 363368.

A.A.Nasef and T.Noini, some weak forms of almost continuity, Acta. Math. Hungar, 74(1997), 211219.

T.Noiri, On slightly continuous functions, internet, J.Math. Math. Sci., 28(2001), 469 478.

T.M.Nour, slightly semicontinuous function, Bull.Calcutta Math.Soc., 87 (1995), 187190.

M.C.Pal and P.Bhattacharyya, Faint precontinuous functions, Soochow J.Math., 21(1995), 273289.

V.Popa, weakly continuous Multifunctions, Boll.Un.Math.Ital. (5), 15 (1978), 379388.

V.Popa, On some properties of weakly continuous multifunctions, Univ.Bacau, Stud.Cerc.Math., 1980, 121127.

V.Popa and T.Noiri, Properties of upper and lower weakly quasi cntinuous Multifunctions, univ.Bacau,Stud.Cerc.St.Ser.Mat.Inform, 25 (1998), 2430.

V.Popa and T.Noiri, Almost mcontinuous functions, Math.Notal 40 (1999/ 2002), 7594.

V.Popa and T.Noiri, On Mcontinuous functions, Anal.Univ.Dunarea de Jos Galati, Ser.Math.Fiz.Mec.Teor.18 (23) (2000), 3141.

V.Popa and T.Noiri, Slightly mcontinuous functions, Radovi Math, 10 (2001), 114.

V.Popa and T.Noiri, Slightly mcontinuous multifunctions, Bull Math, 1(2006).

V.Popa and J.Noiri On upper and lower weakly continuous Multifunctions, Novi sad J.Math., 32 (2002), 724.

V.Popa and T.Noiri, A unified theory of weak continuity for multifunctions, (submitted)