Slightly m-Precontinuous Multifunctions

Slightly m-Precontinuous 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 m-precontinuous, lower slightly m-precontinuous, upper m-precontinuous, lower m- precontinuous.

AMS Subject classification : 54C10, 54C08

1. INTRODUCTION

   The notion of slightly continuous functions was introduced by R.C. Jain [2]. Nour [12] defined slightly semi-continuous 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 m-precontinuous multifunction and investigate the relationships among m- precontinuity, weak m-precontinuity and slight m- precontinuity for multifunctions.

2. 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 semi-open if ACl(Int(A)).The complement of a semi open set is called semi-closed. The union of all pre open sets of X contained in A is called the pre-interior 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-

1. 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 m-space. Each member of mX is said to be mX -open (or briefly m-open). The complement of an mX-open set is said to be mX-closed (or briefly m-closed).

   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 m-PRECONTINUOUS MULTIFUNCTIONS

   Definition:3.1

   A multifunction F:(X,)(Y,) is said to be

   1. Upper slightly precontinuous (resp. upper slightly continuous, upper slightly semi-continuous 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, semi-open, -open) set U of X containing x such that F(U)V.

   2. Lower slightly precontinuous (resp. lower slightly continuous, lower slightly semi-continuous 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, semi-open, -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 m-pre 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

1. Upper m-precontinuous( resp. upper almost m- precontinuous, upper weakly m-precontinuous), 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)).

2. Lower m-precontinuous (lower almost m- precontinuous, lower weakly m-precontinuous), 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 m-structure on

1. For a subset A of X, the mX-preclosure of A and the mX-preinterior of A are defined as follows.

   1. mX-pCl(A) = {F:AF, X-FmX}.

   2. mX-pInt(A) = {U:UA, UmX}.

Remark: 3.5

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

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

   1. mX-Cl(A) = Cl(A)(resp.sCl(A),pCl(A),

      Cl(A)).

   2. mX-Int (A) = Int(A)(resp. sInt(A), pInt(A),

Int(A)).

Definition: 3.6

A multifunction F:(X,mX) (Y,), is said to be

1. Upper slightly m-precontinuous, 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.

2. Lower slightly m-precontinuous, 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.

   1. F is upper slightly m-precontinuous

   2. F+(V) = mX-pInt(F+(V)) for each VCO(Y)

   3. F-(V) = mX-pCl( 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)

xmX-pInt(F+(V)) (2).From eqn (1) & (2) we get,F+(V) mX-pInt(F+(V)).i.e. mX-pInt (F+(V)) F+(V),

by lemma (3.1) of [20] F+(V) = mX-pInt(F+(V)) for each VCO(Y).

1. (3): Let K be any clopen set of Y. Then Y-K is clopen in Y.By (2), F+(V) = mX-pInt(F+(V)).By lemma (3.1) of [20] we have, X- F-(K) = F+(Y-K)= mX-pInt(F+(Y- K)) = X-[mX-pCl(F-(K)].Therefore F-(K) = mX-pCl(F-K)).

2. (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-(Y-B) = mX-pCl(F+(V)) = X-[mX-

pInt(F+(V)).Therefore F+(B) = mX-pInt(F+(V)).

1. (1): Let xX and V be any clopen set of Y containing F(x). Then xF+(V)= mX-pInt(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 m-precontinuous.

   Theorem: 3.8

   For a multi function F:(X, mX) (Y,) the following are equivalent.

   1. F is lower slightly m-precontinuous.

   2. F-(V) = mX-pInt((F-(V)) for each VCO(Y).

   3. F+(V) = mX-pCl(F+(V)) for each VCO(Y)

Proof:

1. (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 mX-pInt(F-(V)) xmX- pInt(F-(V)) F-(V) mX-pInt(F-(V)) and by lemma (3.1)

   of [20] we have F -(V)= mX-pInt(F-(V)).

2. (3): Let VCO(Y).Then Y-VCO(Y),by (2), we have,X- F+(V) = F-(V)=

F-(Y-V)= mX-pInt(F-(Y-V))= X-[mX-pCl(F+(V)).Hence F+(V) = mX-pCl(F+(V))

(3) (1): Let xX and VCO(Y) such that

F(x)V,then xF-(V) and

xX- F-(V) = F+(Y-V).By (3) we have,xmX-

pCl(F+(V)).By lemma (3.2) of [20], there exists preopen set UmX containing x such that UF+(Y-V) = .Thus U

F-(V).Therefore F(u)V for each uU. Hence F is lower

slightly m-precontinuous.

Theorem: 3.9

Let (Y,) be E.D. For a multifunction F:(X, mX)(Y,), the following are equivalent.

1. F is upper slightly m-precontinuous.

2. mX-pCl(F-(V)) F-(Cl(V)) for every open set V of (Y,)

3. F+(Int(C)) mX-pInt(F+(C)) for every closed set C of (Y,)

Proof:

1. (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, mX-pCl(F-(V)) mX-pCl(F-(Cl(V)))=F- (Cl(V)) mX-pCl(F-(V))

   F-(Cl(V))

2. (3): Let C be any closed set of (Y,) and V = Y-C. Then V is open in (Y,). By lemma (3.1) of [20], we have, X- [mX-pInt(F+(C))] = mX-pCl(x-F+(C))= mX-pCl(F-(Y-

   C)) F-(Y-Int(C)). Therefore X-[mX-pInt(F+(C))] = X-

   F+(Int(C)) F+(Int(C)) mX-pInt(F+(C))

3. (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 m-precontinuous.

Theorem: 3.10

Let (Y,) be E .D. For a multifunction F:(X, mX)

(Y,), the following are equivalent.

1. F is lower slightly m-precontinuous.

2. mX-pCl(F+(V)) F+(Cl(V)) for every open set V of (Y,)

3. F-(Int(C)) mX-pInt(F-(C)) for every closed set C of (Y,)

Proof:

1. (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 mX-pCl(F+(V)) mX- pCl(F+(Cl(V)))=F+(Cl(V))mX-pCl(F+(V)) F+(Cl(V))

2. (3): Let C be any closed set of (Y,) and V= Y-C,

   then V is open in (Y,)

   By lemma (3.1) of [20], we have, X- [mX-pInt (F-(C))] =

   mX-pCl(X-(F-(C)))= mX-pCl(F+(Y-C)) F+(Cl(Y-C)) (by

   given (2))=F+(Y-Int(C)).Therefore X- [mX-pInt

   (F -(C))] = X- F-(Int(C)) F-(Int(C)) mX-pInt(F-(C))

3. (1): Let xX and VCO(Y) containing F(x)V

.Let V = Y-C is open in Y. Let xF+(C) and xX- F+(C)F-(Y-C).By (3) we have, xmX-pInt (F-(Y-C)),by theorem (3.8) we have, xmX-pCl(F+(Y-C)).By lemma (3.2) of [20], there exists a preopen set UmX containing x such that U(F+(Y-C)) = .Hence U

F-(V)and F (x)V for each uU. Hence F is lower slightly m-precontinuous.

Slight m-precontinuity and other forms m-precontinuity.

Theorem: 4.1

If multifunction F:(X, mX)(Y,) is upper weakly m-precontinuous, then it is upper slightly m-precontinuous.

Proof:

Let xX and VCO(Y) containing F(x).Since F is upper weakly m-precontinuous. There exists a preopen set UmX containing x such that F(U)Cl(V) = V. Hence F is upper slightly m-precontinuous.

Theorem: 4.2

If a multi function F:(X, mX)(Y,) is lower weakly m-precontinuous then it is lower slightly m- precontinuous.

Proof:

Let xX and VCO(Y) such that F(x)V .Since F is lower weakly m-precontinuous 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 m-precontinuous.

Lemma: 4.3

A multifunction F:(X, mX)(Y,) is upper almost m-precontinuous (resp.lower almost m-precontinuous) 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 m-precontinuous 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 m-precontinuous.

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.

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