Ratio Mathematica Volume 45, 2023 Strong forms of b-continuous multifunctions Suresh R* Pasunkilipandian S† Hari Siva Annam G‡ Selva Banu Priya T § Abstract In this paper we have introduced strong forms of b-continuous multifunctions namely b#-multicontinuity and *b-multicontinuity and studied their properties and characterizations. Also investigate the relationship with other type of functions with suitable examples. Keywords: b-open, multi-function, u.b#-c, u.*b-c 2010 AMS subject classification: 54C05, 54C08, 54C60**. *Research Scholar (19112102091007), Manonmaniam Sundaranar University, Tirunelveli-12, India. rsuresh211089@gmail.com. †Dept. of Mathematics, Aditanar College of Arts and Science, Tiruchendur, affiliated to Manonmaniam Sundaranar University, Tirunelveli-12, India. pasunkilipandian@gmail.com. ‡Dept. of Mathematics, Kamaraj College, Tuticorin, affiliated to Manonmaniam Sundaranar University, Tirunelveli-12, Tamilnadu, India. hsannam@yahoo.com. §Department of Artificial Intelligence and Data Science, Panimalar Engineering College, Chennai - 600123, India. Priya8517@gmail.com. ** Received on July 10, 2022. Accepted on October 15, 2022. Published on January 30, 2023. doi: 10.23755/rm.v45i0.1027. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY license agreement. 280 mailto:Priya8517@gmail.com Suresh, Pasunkilipandian, Hari Siva Annam, Selva Banu Priya 1. Introduction Recently topologists concentrate their research in several types of continuous multi functions. A weak form of b-continuous multifunctions was studied in [4]. The variations of multi continuity were discussed in [5]. The weak and strong forms of continuity of multi functions were introduced in [6]. Certain properties of topological spaces preserved under multivalued continuous mappings were investigated in [7]. Certain strong forms of mixed continuous multi functions were characterized in [8] and the upper and lower -continuous multi functions were studied in [11]. The notions of b#-continuity and *b-continuity were respectively discussed and studied in [9] and [3]. In this paper we have introduced strong forms of b-continuous multifunctions namely b#-multicontinuity and *b-multicontinuity and also studied their properties and characterizations with suitable examples. 2. Preliminaries Throughout this paper it is assumed that X and Y are non-empty sets and  and  are topologies on X and Y respectively and  and  denote the collections of closed sets in X and Y respectively. The notation P: X⇉Y is used for a multivalued function. For the notations in multifunction theory, the reader may consult (Thangavelu, Premakumari, 2015). We use the following abbreviations and notations. “continuous” =”c”, “upper continuous” = “u.c” and “lower continuous” = “l.c”. Further V (, x, P(x),)  V , xX and P(x)V. U[, x, P, V,]  U , xU and P(U)V. V (, x, P(x), )  V , xX and P(x)V. U [, x, P, V, ]  U, xU and P(u)V uU. {b#, *b}. Definition 2.1. The set A is called (resp. b, *b)-open[1] (resp.[2], resp.[3]) if A Cl(Int(Cl(A))) (resp. Cl(Int(A))Int(Cl(A)), Cl(Int(A))Int(Cl(A)) and b#-open [9,10] if A=Cl(Int(A))Int(Cl(A)). The complements of (resp. b,*b, b#)-open sets are (resp. b,*b, b#)-closed sets. Lemma 2.2. The set B is (i) -open  b-open (ii) open  b-open (iii) b-open-open Definition 2.3. The multifunction P is u.c [5,6,7] if  V(, x, P(x), ),  U[, x, P,V, ] and is l.c if  V(, x, P(x), ),  U[, x, P, V, ]. 281 Strong forms of b-continuous multi functions Analogously u.b-c [4] and u.-c [11] may be defined by replacing “” in [, x, P, V,] respectively by “bO(X,)” and “O(X,)”. Also l.b-c [4] and l.-c [11] may be defined by replacing “” in [, x, P, V, ] by “bO(X,)” and “O(X,)” respectively. Definition 2.4. The multifunction P is c if P is u.c and l.c. The notions b-c and -c can be similarly defined. 3.-multi continuity where {*b, b#} Definition 3.1. The multivalued function P is u.b#-c (resp. u.*b-c ) if P+(V) is b#-open (resp. *b-open)  V. Proposition 3.2. Consider the following statements. (i)P is u.-c. (ii)P −(B) is -closed  B. (iii)P −(Cl (B)) is -closed  BY. (iv)P+ (Int (B)) is -open  BY. The implications (i)  (ii)  (iii)  (iv) always hold. Proof: Suppose (i) holds. Let B that implies, P+(Y\B) is -open so that X\ P−(B) = P+(Y\B) is -open that further shows that P−(B) is -closed. This proves (i) (ii). Now we assume (ii). Let V that implies by (ii), P−(Y\V) is -closed so that X\ P+(V) is -closed that further shows that P+(V) is -open. This proves (ii) (i). Other implications follow easily. Proposition 3.3. If P is u.-c then  V(, x, P(x), ), U[ O(X,), x, P, V, ]. Proof: Let P be u.-c and V(, x, P(x), ). Since P(x)V, xP+(V). Since P+(V) is -open  a -open set U with xU P+(V). Clearly U[ O(X,), x, P, V, ]. Proposition 3.4. P is u.-c  it is u.b-c and u.-c. Definition 3.5. The multifunction P is l.b#-c(resp. l.*b-c) if P−(V) is b#-(resp.*b)-open  V. Proposition 3.6. Consider the following statements. (i)P is l.-c. (ii) P+ (B) is -closed  B. (iii) P+ (Cl (B)) is -closed  B Y. (iv) P− (Int (B)) is -open  B Y. The implications (i)  (ii)  (iii)  (iv) always hold. 282 Suresh, Pasunkilipandian, Hari Siva Annam, Selva Banu Priya Proof: Suppose (i) holds. Let B  that implies P−(Y\B) is -open so that X\ P+(B) is -open that further shows that P+(B) is -closed. This proves (i) (ii). Now we assume (ii). Let V that implies by (ii)), P+(Y\V) is -closed so that X\ P−(V) is -closed that further shows that P−(V) is -open. This proves (ii) (i). The rest follows easily. Proposition 3.7. If P is l.-c then V (, x, F(x), ),  U[O(X,), x, P, V, ]. Proof: Analogous to Proposition 3.3. Proposition 3.8. P is l.-c it is l.b-c and l.-c Definition 3.9. P is b#-c (resp.*b-c) if it is u.b#-c (resp.u.*b-c) and l.b#-c(resp. l.*b-c). The next proposition follows from previous definition, Proposition 3.2 and Proposition 3.6. Proposition 3.10. Consider the following statements. (i) P is -continuous. (ii) P+ (V) and P− (V) are -open  V. (iii) P+ (B) and P− (B) are -closed  B. (iv) P+ (Int (B)) and P− (Int (B)) are -open B  Y. (v) P+ (Cl (B)) and P− (Cl (B)) are -closed B  Y. The implications (i)  (ii)  (iii)  (iv)  (v) always hold. The following diagrams always hold. Diagram 3.11. Let t=u or l. (i) t.b#-c  t.b-c  t.*b-c. (ii) t.c  t.b-c  t.-c. Examples 3.12. In this section some examples are given to illustrate certain results in the third section. Let X = {p, q r, s}, Y = {1, 2, 3},  = {, {1}, Y},  = {, {r}, {q}, {q, r}, {p, q}, {p, q, r}, {q, r, s}, X}. (i) F1(p) = {1, 2}, F1(q)={1, 3} F1(r) = {1} and F1(s) = {1} then F1 +({1})={r, s} is b#- open so that F1 is u.b #-c. (ii) If F2(p) = {1, 2}, F2(q)={1}, F2(r) = {1, 3} and F2(s) = {3} then F2 +({1})={r} is *b-open that implies F2 is u.*b-c. 283 Strong forms of b-continuous multi functions (iii) If F3(p) = {1}, F3(q)={1}, F3(r) ={1,2} and F3(s) = {1} then F3 +({1})={p, q, s} is b- open and -open and hence F3 is u.b-c and u.-c. However F3 is not u.-c as F3 +({1})= {p, q, s} is not -open. (iv) If F4(p) ={2}, F4(q)={3}, F4(r) ={1, 2} and F4(s) = {1, 3} then F4 −({1})= {r, s} is b#-open that implies F4 is l.b #-c. (v) If F5(p) = {1, 2}, F5(q)={1, 3}, F5(r) = {2} and F5(s) = {3} then F5 −({1})={p, q} is *b-open and hence F5 is l.*b-c. (vi) If F6(p) ={2}, F6(q)={1, 2}, F6(r) = {3} and F6(s) = {1, 3} then F6 −({1})={q, s}is b-open and -open so that F6 is l.b-c and l.-c. However F6 is not l.-c as F6 −({1})= {q, s} is not -open. (vii) If F7(p) = Y, F7(q)={1, 3}, F7(r) = {1} and F7(s)={1} then F7 +({1})={r, s} and F7 −({1})= X are b#-open we see that F7 is u.b #-c and l.b#-c and hence b#-c. (viii) If G1(p) = {2}, G1(q)={1}, G1(r) = {1, 2} and G1(s) = {3} then G1 +({1})= {q} and G1 −({1})= {q, r} are *b-open so that G1 is u.*b-c and l.*b-c and hence *b-c. (ix) If G2(p) = {1, 3}, G2(q)={1}, G2(r) = {2} and G2(s) = {1} then G2 +({1})= {q, s} and G2 −({1})= {p, q, s} are b-open and -open we see that G2 is b-c and -c. However G2 is not -c as G2 +({1})={q, s} and G2 −({1})={p, q, s} are not -open . (x) If G3(p) = {2, 3}, G3(q)={1}, G3(r) ={1} and G3(s) = {1} then G3({1})= {q, r, s}is open that implies G3 is u.c. However G3 is not u.b #-c as G3 +({1})= {q, r, s} is not b#- open. If G4(p)={2, 3}, G4(q)={1, 2}, G4(r)={1} and G4(s) ={1} then G4 +({1})={r, s} is b#-open so that G4 is u.b #.c. However G4 is not u.c as G4 +({1})={r, s} is not open. (xi) If G5(p) = {2, 3}, G5(q)={1, 2}, G5(r) = {1, 3} and G5(s) = {1} then G5 −({1})= {q, r, s} is open we see that G5 is l.c . However G5 is not l.b #-c as G5 + ({1}) = {q, r, s} is not b#-open. If G6 (p) = {3}, G6 (q) = {2}, G6(r) = Y and G6(s) = {1, 3} then G6 −({1})={r, s} is b#-open we see that G6 is l.b #-c. However G6 is not l.c as G6 − ({1}) ={r, s} is not open. (xii) If F(p) = {2, 3}, F(q)={1}, F(r) = {1, 3} and F(s) = {1, 2} then F+({1})={q} and F−({1})={q, r, s} are open we see that F is u.c and l.c so that it is c. However F is neither u.b#-c nor l.b#-c as F+ ({1}) = {q} and F− ({1}) = {q, r, s} are not b#-open. If G(p) = {3}, G(q)={2, 3}, G(r) ={1} and G(s) = {1}then G+({1})={r, s}= G−({1}) is b#- open we see that G is u.b#-c and l.b#-c so that it is b#-c. However G is not c as G+ ({1}) = {r, s} = G− ({1}) is not open. 284 Suresh, Pasunkilipandian, Hari Siva Annam, Selva Banu Priya 4. Conclusions The concepts of strong forms of b-continuous multifunctions namely b#- multicontinuous and *b-multicontinuous functions are suitable for future extension research. 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