The purpose of this paper is to study αgs-continuous multifunctions. Some basic characterizations, preservation theorems and several properties concerning upper and lower αgs-continuous multifunctions are investigated

Pallavi Mirajakar

Rajeshwari K K.

Sarika M. Patil Patil

T.D.Rayangoudar Rayanagoudar

Mirajakar, Pallavi

K., Rajeshwari K

Patil, Sarika M. Patil

Rayanagoudar, T.D.Rayangoudar

APAV - Academy of Sciences, Letters, Arts and Technology (E-Journals)

Ratio Mathematica Volume 48, 2023Weaker forms of Contra ContinuousMultifunctionsRajeshwari K.*T.D.Rayanagoudar†Sarika M. Patil‡P. S. Mirajakar§AbstractThe purpose of this paper is to study the properties of αgs-continuousmultifunctions.Some basic characterizations, preservation theorems and several prop-erties concerning upper and lower contra αgs-continuous multifunc-tions are investigated. Further, the basic properties of upper and loweralmost contra αgs-continuous multifunctions are studied.Keywords: αgs-closed set, αgs-open sets, αgs-continuous functions,αgs-contra continuous multifunctions.2010 AMS subject classifications: 54A05, 54C08.1*Research Scholar, R and D Department, Bharathiar University, Coimbatore-641046, Tamil-nadu, India; rajeshwaribhat10@yahoo.com†Department of Mathematics, Government First Grade College, Rajnagar, Hubballi-580032,Karnataka, India; rgoudar1980@gmail.com‡Department of Mathematics, Government First Grade College, Rajnagar, Hubballi-580032,Karnataka, India; sarupatil@rediffmail.com§Department of Mathematics, P. C. Jabin Science College, Vidyanagar, Hubballi-580031, Kar-nataka, India; psmirajakar@gmail.com1Received on September 8, 2023. Accepted on December 3, 2023. Published on December 31,2023. doi: 10.23755/rm.v48i0.1374. ISSN: 1592-7415. eISSN: 2282-8214. ©Rajeshwari K et al.544Rajeshwari K., T.D.Rayanagoudar, Sarika M. Patil and P. S. Mirajakar1 IntroductionThe research in topology over last two decades has reached a high levelin many directions. Topological methods are widely used in many other branchesof modern mathematics such as differential equation, functional analysis, classicalmechanics, general theory of relativity, mathematical economics, quantum theory,biology etc. One of the important and basic topics in the theory of classical pointset topology and in several branches of mathematics, which has been investigatedby many authors is continuity of functions. This concept has been extended to thesetting of multifunctions.The concept of upper and lower multifunctions were introduced and stud-ied by Berge [2]. After this work several authors have given the several weakerand strong forms of continuity of multifunctions [1], [7], [8], [9], [10], [11], [15]).In this direction, we will continue the study of new type of multifunction calledαgs-continuous multifunctions and study some of basic properties.In this paper, (P, τ) and (Q, σ) stand for topological spaces with no sep-aration axioms assumed, unless otherwise stated. A multifunction f : (P, τ) →(Q, σ) is a point to set correspondence, and we always assume that f(p) ̸= ϕ, forevery point p ∈ P . For each A ⊂ P and each B ⊂ Q, f(A) = ∪{f(p) : p ∈ A},f+(B) = {p ∈ P : f(p) ⊂ B} and f−(B) = {p ∈ P : f(p) ∩B ̸= ϕ}.A multifunction f : P → Q is said to be upper semi continuous [13] (resp.lower semi continuous) at a point p ∈ P , and each open set V ⊆ Q with f(p) ⊂ V(f(p) ∩ V ̸= ϕ) there is an open neighborhood U of p such that f(U) ⊆ V(f(a)∩V ̸= ϕ for each a ∈ U ). Then f is semi continuous if and only if f is upperand lower semi continuous.In this paper, the first section deals with the concept of Upper and Lower con-tra αgs-continuous multifunctions in topological spaces. Several characterizationsof these functions have been introduced. The second section contains the proper-ties of Upper and Lower Almost contra αgs-continuous functions in topologicalspaces. Many properties concerning to these functions have been introduced inthis paper.2 PreliminariesDefinition 2.1. [12] A subset A of a topological space (P, τ) is called αgs-closedif αcl(A)⊆ U whenever A ⊆ U and U is gs-open in P .545Weaker forms of Contra Continuous MultifunctionsThe family of all αgs-open (resp. αgs-closed) sets of P is denoted by αgs-O(P)(resp. αgs-C(P)).If B ∈ αgs-O(P) then P −B ∈ αgs-C(P).Definition 2.2. [12] Let A ⊂ P . Then(i) αgs-interior of A is denoted by αgs-int(A) and is defined as αgs-int(A) = ∪{G :G ⊂ A, where G ∈ αgs-O(P)}.(ii) αgs-closure of A is denoted by αgs-cl(A) and is defined as αgs-cl(A) = ∩{G :A ⊂ G, where G ∈ αgs-C(P)}.Definition 2.3. [12] Let f : P → Q is a function. Then f is said to be(i) αgs-continuous, if f−1(U) ∈ αgs-C(P) for every U ∈ C(Q).(ii) αgs-irresolute, if f−1(U) ∈ αgs-C(P) for every U ∈ αgs-C(Q).Lemma 2.1. [10] For a multifunction f : (P, τ) → (Q, σ) we have the following:(i) G+f (A×B) = A ∩ f+(B)(ii) G−f (A×B) = A ∩ f−(B).For any subsets A of P and B of Q, we have Gf : P → P ×Q is defined asGf (p) = {p} × f(p) for every p ∈ P .3 Upper and Lower Contra αgs-continuous Multi-functionsDefinition 3.1. A multifunction f : (P, τ) → (Q, σ) is called a(i) Upper αgs-continuous if for each p ∈ P , V ∈ (Q, f(p)) there exists U ∈ αgs-O(P, p) such that f(U) ⊆ V .(ii) Lower αgs-continuous if for each p ∈ P , V ∈ O(Q, f(p)) with f(p)∩V ̸= ϕ,then there exists U ∈ αgs-O(P, p) such that U ∈ f−1(V ).Definition 3.2. A function f : P → Q is said to be Contra αgs-continuous iffor each p ∈ P and V ∈ O(Q, f(p)) there exists V ∈ αgs-O(Q, f(p) such thatf(U) ⊆ V .Definition 3.3. A multifunction f : (P, τ) → (Q, σ) is called a(i) Upper contra αgs-continuous if for each p ∈ P , V ∈ C(Q, f(p)) there existsU ∈ αgs-O(P, p) such that f(U) ⊆ V .(ii) Lower contra αgs-continuous if for each p ∈ P , V ∈ C(Q) with f(p)∩V ̸= ϕ,then there exists U ∈ αgs-O(P, p) such that U ∈ f(V ).Theorem 3.1. The following statements are equivalent for a multifunction in f :(P, τ) → (Q, σ)(i) f is upper contra αgs-continuous546Rajeshwari K., T.D.Rayanagoudar, Sarika M. Patil and P. S. Mirajakar(ii) f+(V ) ∈ αgp-O(P ), for every V ∈ C(Q)(iii) for each V ∈ O(Q), ⇒ f−(V ) ∈ αgs-C(Q)(iv) for each p1 ∈ P and each K ∈ C(Q, f(p1)), there exists U ∈ αgs-O(P, p1)with q1 ∈ U , then p1(q1) ⊆ K.Proof: (i) ⇔ (ii): Let V ∈ C(Q) and p1 ∈ f+(V ) as p is upper contra αgs-continuous, there exists U ∈ αgs-O(P, p1) with f(U) ⊆ V . Thus f+(V ) ∈ O(P ).We can prove the other part similarly.(ii) ⇒ (iii): We have f+(Q \ V ) = P \ f−(V ) for every V ⊆ Q.(iii) ⇒ (iV ): Obvious from the definition.Theorem 3.2. The following statements are equivalent for a multifunction in f :(P, τ) → (Q, σ)(i) f is lower almost contra αgs-continuous.(ii) f+(V ) ∈ αgs-O(P ), for each V ∈ C(Q).(iii) f−(K) ∈ αgs-C(P ), for each K ∈ O(Q)(iv) for each p ∈ Q and each K ∈ C(P ) with f(p1)∩K ̸= ϕ there exists U ∈ αgs-O(P, p1) such that q1 ∈ U then p1(q1) ⊆ K ̸= ϕ.Proof: The proof is similar to the Theorem 3.1.Definition 3.4. Let K ∈ P . Then K is said to be strongly S-closed(αgs-compact)relative to P if every cover of K by closed (resp. αgs-open) sets of P has a finitesubcover.A space P is said to be strongly S-closed(αgs-compact) if P is strongly S-closedrelative to P .Theorem 3.3. Let f : (P, τ) → (Q, σ) be an upper contra αgs-continuous sur-jective multifunction and f(p) is strongly S-closed relative to Q holds for eachp ∈ P . Then f(A) is strongly S-closed if A is αgs-compact relative to P .Proof: Let {Ai : i ∈ α} be any cover of f(A) by closed sets in P . Then for eachp ∈ A there exists a finite subset α(p) of α such that f(p) ⊆ ∪{Ai : i ∈ α(p)}.Let V (p) = ∪{Ai : i ∈ α(p)}, then f(p) ⊆ V (P ) and so there exist U(P ) ∈ αgs-O(P, p) with p(U(p)) ⊆ V (P ) as {U(P ) : p ∈ A} is cover of A by αgs-opensets in P . Then, there exist finite number of points say p1, p2, .....pm such thatA ⊆ ∪{U(pi) : i = 1, 2, ...n}. Hence f(A) ⊆ f(n⋃i=1U(pi)) ⊆n⋃i=1f(U(pn)) ⊆n⋃i=1V (pi) ⊆n⋃i=1⋃i=α(pi)Vi. Thus f(A) is strongly S-closed relative to Q.Lemma 3.1. [12] Let A,B ⊆ P . Then(i) If A ∈ αgs-O(P ) and B ∈ O(P ) then A ∩B ∈ αgs-O(P ).(ii) If A ∈ αgs-O(B) and B ∈ αgs-O(P ) then A ∈ αgs -O(P ).547Weaker forms of Contra Continuous MultifunctionsTheorem 3.4. Let f : (P, τ) → (Q, σ) be a multifunction with V ⊆ O(P ). If f isupper contra αgs-continuous (lower contra αgs-continuous) then f|v : U → Q isupper contra αgs-continuous(lower contra αgs-continuous).Proof: Let V ⊆ C(Q) with p ∈ U with p ∈ P|v−1(V ) by lower contra αgs-continuous of f . Then there exist G ∈ αgs-O(P, p) such that G ⊆ P−1(V ) andso p ∈ G∩U ∈ αgs O(A) with G∩U ⊆ F|U−1(V ) which satisfies the conditionsof lower contra αgs-continuous multifunction.Similarly, we can show for upper contra αgs-continuous multifunction.Theorem 3.5. Let {Ui : i ∈ α} be an open cover of a space P . A functionp : (P, τ) → (Q, σ) is upper contra αgs-continuous if and only if the restrictionP|Ui : Ui → Q is upper contra αgs-continuous for each i ∈ α.Proof: Let p is upper contra αgs-continuous and i ∈ α. Let P ∈ Ui and V ⊆C(N,P|Ui(pi))) as p is upper contra αgs-continuous and P (m) = P|Ui(p1). Thenthere exists G ∈ αgs-O(P, p1) with P (G) ⊆ V . Put U = G ∩ Ui, then p1 ∈U ∈ αgs-O(U, p1) and P|Ui(U) = P (U) ⊆ V . Hence P|Ui is upper contra αgs-continuous.Conversely, let m ∈ P , V ∈ αO(Q). Then there exist i < α with p1 ∈ Uiby upper contra αgs continuity of p and P (p1) = P|Ui(p1). Then there existsU ∈ αgs-O(Ui, p1) with P|Ui(p1) ⊆ V . Thus V ∈ αgs-O(P, p1) and P (U) ⊆ V .Hence p is upper contra αgs-continuous.Theorem 3.6. The set of points p of P at which a multifunction f : (P, τ) →(Q, σ) is not upper contra αgs-continuous (lower contra αgs-continuous) is iden-tical with the union of the αgs-frontiers of the upper(lower) inverse images of aclosed sets containing (meeting) f(p).Proof: Let p ∈ P at which f is not upper contra αgs-continuous. Then thereexist V ∈ C(Q) with U ∩ (P \ f+(V )) ̸= ϕ holds for each U ∈ αgs-O(P, p)and so p ∈ αgs-cl(P \ f+(V )). As p ∈ f+(V ), so p ∈ αgs-cl(f+(Q)) and sop ∈ αgs-Fr(f+(V )). On the other hand, let V ∈ C(Q, f(p)) with p ∈ αgs-Fr(f+(V )). Assume that f is upper contra αgs-continuous at point p, then thereexists U ∈ αgs-O(P, p) with f(U) ⊂ V . Thus, p ∈ U ⊂ αgs-int(f+(V )), whichcontradicts the fact that p ∈ αgs-Fr(f+(V )). Thus, f is not upper contra αgs-continuous.Similarly, we can prove for lower contra αgs-continuous.4 Upper and Lower Almost Contra αgs-ContinuousMultifunctionsDefinition 4.1. A multifunction f : (P, τ) → (Q, σ) is said to be(i) upper almost contra αgs-continuous if for each p ∈ P and V ∈ RC(P ) with548Rajeshwari K., T.D.Rayanagoudar, Sarika M. Patil and P. S. Mirajakarp ∈ f+(V ), there exists U ∈ αgs-O(P, p) with U ⊂ f+(V ).(ii) lower almost contra αgs-continuous if for each p ∈ P and V ∈ RC(P ) withp ∈ f−(V ), there exists U ∈ αgs-O(P, p) with U ⊂ f−(V ).Theorem 4.1. Every upper almost contra αgs-continuous (lower almost contraαgs-continuous) multifunction is upper (lower) weakly αgs-continuous.Proof: Let p ∈ P and V ∈ O(Q) with f(p) ⊂ V , that is cl(V ) is regular closedwith f(p) ⊆ cl(V ). By upper almost contra αgs-continuity, there exists U ∈ αgs-O(P, p) with U ⊂ f+(cl(V )) and so f is upper weakly αgs-continuous.Example 4.1. Let P = Q = {p1, p2, p3}. Let τ = {P, ϕ, {p1}, {p1, p3}} andσ = {Q, ϕ, {p1}, {p2}, {p1, p2}, {p1, p3}}. Let A = {p1, p2} and U = {p1, p2}.Then p(U) = {p1, p2} ⊂ cl(A) = P . Then f is weakly αgs-continuous but notupper almost contra αgs-continuous. For V = {p2} and U = {p1, p2}, we haveU /∈ f+(int(cl(V )).Theorem 4.2. Every upper contra αgs-continuous (lower contra αgs-continuous)multifunction is upper almost contra αgs-continuous (lower almost contraαgs-continuous).Proof: Follows from the definition 4.1.Theorem 4.3. The following are equivalent for a multifunction f : (P, τ) →(Q, σ):(i) f is upper almost contra αgs-continuous(ii) for every A ⊂ RC(Q), f+(A) ⊂ αgs-O(Q)(iii) for every U ⊂ RO(Q), f−(U) ∈ αgs-C(P)(iv) for every A ⊂ O(Q), f−(int(cl(A))) ∈ αgs-C(P)(v) for every A ⊂ O(Q), f+(cl(int(A))) ∈ αgs-O(P).(vi) for all p ∈ P and V ∈ SO(Q) with f(p) ⊂ V , there exists U ∈ αgs-O(P, p)such that f(U) ⊂ cl(V ).(vii) f+(V ) ⊂ αgs-int(f+(cl(V ))) holds for all V ∈ SO(Q).Proof: (i) → (ii): Let A ⊂ RC(Q) with p ∈ f+(A)). As f is upper almostcontra αgs-continuous, there exists U ∈ αgs-O(P, p) such that U ⊂ f+(A). Thus,f+(A) ∈ αgs-O(P).(ii) → (i): Follows from the definition.(ii) ⇔ (iii) and (iv) ⇔ (v): We have f+(Q \ A) = P \ f−(A) holds for allA ⊂ Q.(iii) ⇔ (iv): Let A ∈ O(Q). As int(cl(A)) ∈ RO(Q) and so f−(int(cl(A)) ∈αgs-C(P).Similarly, we can show (v) ⇔ (ii).(vi) → (vii): Let V ∈ SO(Q) and p ∈ f+(V ). Then, f(p) ⊂ V . By definition,there exists U ∈ αgs-O(P, p) such that f(U) ⊂ cl(V ), that is p ∈ U ⊂ f+(cl(V )).549Weaker forms of Contra Continuous MultifunctionsThus, p ∈ αgs-int(f+(cl(V ))) and f+(V ) ⊂ αgs-int(f+(cl(V ))).(vii) → (ii): Let A ∈ RC(Q). As A ∈ SO(Q) and from (vii), f+(A) ⊂ αgs-int(f+(cl(A))). Thus, f+(A) ∈ αgs-O(P).(ii) → (vi): Let p ∈ P and V ∈ SO(Q) with f(p) ⊂ V . As cl(V ) ∈ RC(Q),then there exists A ∈ αgs-O(P, p) with A ⊂ f+(cl(A)). Thus, f(A) ⊂ cl(V ).Lemma 4.1. [14] If A and B are disjoint compact subsets of an Urysohn spaceP, then there exist U ∈ O(P ) and V ∈ O(Q) with A ⊂ U , B ⊂ V with cl(U) ∩cl(V ) = ϕ.Theorem 4.4. Let f : (P, τ) → (Q, σ) be upper almost contra αgs-continuousinjective multifunction into an Urysohn space Q with f(p) is compact for eachpoint p ∈ P . Then P is αgs-T2 space.Proof: Let p, q ∈ P with f(p) ∩ f(q) = ϕ by injectiveness of f, where f(p)and f(q) are disjoint compact sets in P. Then there exist V1, V2 ∈ O(P ) withf(p) ⊂ V1 and f(q) ⊂ V2 such that cl(V1) ∩ cl(V2) = ϕ. Since cl(V1) and cl(V2)are regular closed sets, then by upper almost contra αgs-continuity of f, there existU1 ∈ αgs-O(P, p) and U2 ∈ αgs-O(P, q) with f(U1) ⊂ cl(V1), f(U2) ⊂ cl(V2).Thus, U1 ∩ U2 = ϕ. Thus P is αgs-T2 space.Discussion and ConclusionTopology developed as a field of study of geometry and set theory, throughanalysis of such concepts as space, dimensions and transformation. The topolog-ical structures are modelled suitably in the field of Computer graphics, Patternrecognition, Artificial intelligence, Data mining, Rough set theory, Informationsystems, Quantum physics etc.Continuous functions stands among the most fundamental point in the wholeof the Mathematical Science. Many different forms of stronger and weaker formsof functions have been introduced over the years. As a generalization of closedsets, Levine introduced the concept of generalized closed sets which are weakerthan closed sets in topological spaces.In this way, this paper deals with the properties of weaker forms of contin-uous functions contra and almost contra continuous multifunctions in topolog-ical spaces. We have defined the new weaker froms of continuous multifunc-tions called contra and almost contra αgs-continuous multifunctions in topologi-cal spaces. The properties and preservation theorems on upper and lower contraαgs-continuous multifunctions are introduced. Further, upper and lower almost550Rajeshwari K., T.D.Rayanagoudar, Sarika M. Patil and P. S. Mirajakarcontra αgs-continuous multifunctions are also introduced and studied basic char-acterizations of these multifunctions in topological spaces.References[1] M.E. Abd El-Monsef, A.A. Nasef, On multifunctions, Chaos, Solitons andFractals, Vol.12, (2001), 2387-2394.[2] C. Berge, Topological Spaces, Oliver and Boyed, Edinburg-London, 1st En-glish edition, (1963).[3] T. Bnzaru, Multifunctions and M-product spaces, Bull. Stin. Tech. Inst. Po-litech. Timisoara, Ser. Mat. Fiz. Mer. Teoe. Apl. 17(31), (1972), 17-23.[4] C. Carlos Carpintero,N. Rajesn, Ennis Rosas and S. Saranyasri, On upperand lower contra ω continuous multifunctions, Novi Sad Jl. Math, Vol 44,No. 1, (2014), 143-151.[5] C. Carpintero, N. Rajesh, E.Rose and S. Saranyasri, On Upper and lower al-most contra ω-continuous multifunctions, Italian Jl. of Pure and Appl. Math-ematics, Vol. 32, (2014), 445-460.[6] E. Ekiri, S. Jafari and V. 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Weaker forms of Contra Continuous Multifunctions

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