Communications on Applied Nonlinear Analysis 

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spgα-Continuous Multifunctions 
 

M. M. Holliyavar1, T.D. Rayanagoudar2, Sarika M. Patil3 

1R and D Department, Bharathiar University, Coimbatore- 641046, Tamilnadu, India, 

mmholliyavar@yahoo.com 
2, 3 Department of Mathematics, Government First Grade College, Rajnagar, Hubli-580032, 

Karnataka, India 

rgoudar1980@gmail.com and sarupatil@rediffmail.com 

 

 

Article History: 

Received: 31-01-2024 

Revised: 16-04-2024 

Accepted: 02-05-2024 

Abstract: The purpose of this paper is to introduce and study spgα-

continuous multifunctions such as upper (lower) spgα-continuous 

multifunctions, basic characterizations and several properties concerning to 

upper and lower spgα-continuous multifunctions are investigated. Further, 

upper (lower) spgα-irresolute multifunctions are also studied with basic 

properties. 

   Keywords: spgα-closed set, spgα-open set, spgα-Ti-spaces, spgα- 

continuous function. 

2010 AMS subject classifications: 54A05, 54C08  

1. Introduction 

Many authors have continued their research and studied several stronger and weaker forms of 

continuous functions and multifunctions in topological spaces. It is observed that continuity 

and multifunctions are basic topics in general topology and in set valued analysis in 

mathematics. Continuous functions and continuous multifunctions stand among the most 

fundamental and most related points in the whole of mathematical sciences. 

In the literature, the upper and lower continuity for multifunctions were firstly studied and 

introduced by Berge[2]. After the work of him, many authors turned continued their research 

to investigate several weak and strong forms of continuity. In 1999, Mahmoad[12] introduced 

the concept of pre-irresolute multivalued functions. Neubrunn [13] introduced and studied the 

notion of upper(lower) α-continuous multifunctions. 

The purpose of this paper is to give a new an idea about new weaker form of continuous 

functions called upper spgα-continuous multifunctions and lower spgα-continuous 

multifunctions. Further, basic properties and preservation theorems of these upper spgα-

continuous multifunctions (lower spgα-continuous multifunctions) are studied. 

In the next section, the notion of upper spgα-irresolute multifunctions (lower spgα-

continuous multifunctions) is introduced and characterizations and some basic properties are 

investigated. 

Throughout this paper (R, ), (S, ) and (Q, ) stands for topological spaces with no separation 

axioms are assumed, for any set A of a space R, closure of A and interior of A is denoted by 

cl(A) and int(A). 

A multifunction P: R → S is a point to set correspondence and always we assume that P(p)  

 for every p  R. Let A be any subset of R and B be any subset of S. Then P(A) =  {P(p) : 



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p  A}. For a multifunction P : R → S, following [8] we will denote the upper and lower 

inverse of a set B of S by P+(B) and P-(B) respectively, 

that is P+(B) = {p  R : P(p)  B} and P-(B) = {p  R : P(p)  B  0}. So P- : S → R(p) and 

if s  S. Then P-(s) = {p  S : s  P(p)} where P(A) be the collection of the subsets of P. 

Thus, for a subset B in S, P-(B) = {P-(s) : s  B}. Then P is said to be a surjection if P(p) = S 

or equivalently, if for each s  S, there exists p  R such that s  P(p). 

 

For multifunction P : R → S, the graph multifunction GP : R → R X S is defined as  

GP (p) = {p} x P(p) for each p  R and the subset {{p} x P(p) : p  R}  R x S is called the 

multifunction of P and is denoted by G(P)[8]. 

 

2. spgα-Continuous Multifunctions 

This section deals with the characterizations of upper and lower spgα-continuous 

multifunctions and introduced basic properties related to them in topological spaces. 

 

Definition 2.1. A multifunction P : R → S is called 

1. Upper spgα-continuous (briefly u.spgα.C) at a point r  R, if for each V  O(S) with P(r) 

 V , there exist U  spgα-O(R, r) such that P(U)  V . 

2. Lower spgα-continuous (briefly l.spgα.C) at a point r  R, if for every V  O(S) with 

P(r)  V  , then there exist V  spgα-O(R, z) such that P(z)  V   holds for each z  R. 

3. Upper(lower) spgα-continuous, if it is upper(lower) spgα-continuous at every point of R. 

Example 2.1. Let R = S = {p1, p2, p3}, = {R, , {p1}} and  = {S, , {p1}}, {p2}} 

Let P be m.f and P1 be the identity m.f from R to S.  

spgα-O(R) = R, , {p1}, {p1, p2}, {p1, p3}. 

spgα-O(S) = S, , {p1}, {p2}, {p1, p2}, {p1, p3}, {p2, p3}. 

Let p1  R and V = {p1, p2} is a open set of S where P({p1}) = {p1}  V. 

Then, there exists a spgα-open set U = {p1, p2} in R containing the point p1 with P(U) = P({p1, 

p2}) = {p1, p2}  V 

Thus P(U)  V and so P1 is u.spgα.C. 

Example 2.2. Let R = S = {p1, p2, p3}  = {R, , {p1}, {p1, p2}}. 

Here spgα-open sets are: R, , {p1}, {p2}, {p1, p2} 

 = {S, , {p1}}. Here spgα-open sets are: S, , {p1}, {p1, p2}, {p1, p3} 

Let P1 : R → S be a m.f and R = {p2}  R 

spgα-open sets in S containing p2 are :S, {p1, p2} 

Let V = S. 

V = S = {p1, p2, p3} → P({p2})  V = {p2}  {p1, p2, p3} = {p2}   

Consider V = {p1, p2} 

V = {p1, p2} → P({p2})  V = {p2}  {p1, p2} = {p2}  . 

Then there exists spgα-open set U = {p1, p2} containing {p2} are: S, {p1, p2} 



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1. {p2}  {p1, p2} = {p2}  . 

 2. {p2}  N = {p2}  {p1, p2, p3} = {p2}  . 

Thus P1 is a l.spgα.C. 

 

Theorem 2.1. If R and S be any TS. Then for a m.f P: R → S the following properties are 

equivalent: 

1. R is u.spgα.C. 

2. for each r  R, for each V  O(S) such that r  P+(V ), there exists U  spgα-O(R, r) with 

U  P+(V ). 

3. for each r  R and each K  C(S) such that r  P+(S - K), there exists H  spgα-C(R) such 

that r  S - H and P-(K)   H. 

4. for each V  O(S), P+(V)  spgα-O(R). 

5. for each K  C(S), P-(V)  spgα-C(R). 

6. for each r  R and for each nbd. V of P(R), P+(V ) is spgα-nbd. of r. 

7. for each r  R and each nbd. V of P(R), there exists a spgα-nbd. U of r such that P(U)  V 

. 

Proof: (1) → (2): Obvious from the definition. 

(2) → (3): Let r  R and K  C(S) with r  P+(S - K). From (2) there exists U spgα-O(R, 

r) such that U  P+(S - K). Put H = S - K, then H  spgα-C(R) with r = R - H. Also U  P+(S 

- K) = R – P-(K), that is P-(K)  R - V = H. 

(3) → (2): Let r  R and V  O(S) with r  P+(V ). Put K = S - V , where K  C(S) with r  

P+(S - K). From (3), there exists H  spgα-C(R) such that r  R - H and P+(K)  H. Let U = 

R - H then U  spgα-O(R, r) and also P-(K)  H. Thus, R- P-(S - K)  H and so R - H  P-

(S - K). So U  P+(V). 

(2) → (4): Let V  O(S) and r  P-(V). Then from (2), there exists U  spgα-O(R, r) with U 

 P+(V ). So P+(V) = ⋃ 𝑈𝑈𝑟𝑃+(𝑉) . We know that, arbitrary union of spgα-open set is again 

spgα-open and so P+(V)  spgα-O(R). 

(4) → (2): Let r  R and V  O(S) with r  P+(V). From (4) P+(V)  spgα-O(R). Let U = 

P+(V), then U  spgα-O(R, r) and so U  P+(V). 

(4) → (5): U  C(S) and so S - U = O(S). But from (4), P+(S - U)  spgα-O(R), since P+(S 

- U) = R – P-(U) and so R – P-(U)  spgα-O(R). Thus P-(U)  spgα-O(R). 

(5) → (6): V  O(S) and so S - V  C(S). But from (6), P- (S - V)  spgα-C(R), since P-(S - 

V) = R - P+(V ) and so R - P+(U)  spgα-C(R). Thus P+(V)  spgα-O(R) 

(4) → (6): Let r  R and V be a nbd. of P(r). Then U  O(N) such that P(r)  U  V , that is r 

 P+(U)  P+(V). But from (4), P+(U)  spgα-O(R) and so P+(V)  spgα.nbd. of r. 

(6) → (7): Let r  R and V be a nbd. of P(r). From (6), P+(V) is a spgα-nbd. of the point r. 

Put U = P+(V ), so U is a spgα-nbd. of r with P(U)  V . 

(7) → (1): Let r  R and V  O(S) with P(r)  V . Then V is a nbd. of P(r). By (7), there exists 

spgα-nbd. U of r such that P(U)  V . So, there exists a G  spgα-O(R) with r  G  U and 

so P(r)  P(G)  P(U)  V. Thus R is u.spgα.C for each point r  R. 



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Theorem 2.2. The following properties are equivalent for a m.f P: R → S: 

1. P is l.spgα.C. 

2. for each r  R and V  O(R) with r  P-(V) there exists U  spgα-O(R) containing r such 

that U  P-(V). 

3. for each r  R and K  C(S) with r  P-(S - K) there exists H  spgα-C(R) such that r  

R - H and P+(K)  H. 

4. for each V  O(S), P-(V)  spgα-O(R). 

5. for each K  C(S), P+(K)  spgα-C(R). 

 

Definition 2.2: [4] A space R is said to be Submaximal if every dense subset of R is open. 

 

Theorem 2.3. If a m.f P: R → S is u.p.C and S is submaximal, then P is u.spgα.C. 

Proof. Let A  p-O(S). As S is submaximal, A  O(S). Since P is u.p.C, P+(A)  p-O(R) and 

hence P+(A)  spgα-O(R). Thus P is u.spgα.C. 

 

Theorem 2.4. A m.f P: R → S is u.spgα.C if and only if for all B  O(S), P-(B)  O(R). 

Proof. Let B  O(S) and r  P+(B). Then by u.spgα.C, there exists V  spgα-O(R) with 

P(V)  B, where P+(B)  O(R). Let P+(B) is open and r  P-(B). Then P+(B) = {r  B : P(R) 

 B}. So P is u.spgα.C. 

 

Theorem 2.5. A m.f P: R → S is l.spgα.C if and only if for all open set B in S, P-(B)  O(R). 

Proof. Let B  O(S) and r  P+(B). Then by l.spgα.C, there exists V  spgα-O(R) with 

P(V)  B  . As v  V then P-(B)  O(R). Suppose P-(B)  O(R) and r  P-(B), then P-(B) 

= {r  R : P(R)  B  }. So P is l.spgα.C. 

 

Theorem 2.6. The following holds good for a m.f P: R → S 

1. P is u.spgα.C. 

2. P(spgα-cl(B))  cl(P(B)) for every B  R. 

3. spgα-cl(P+(A))  P+(cl(A)) for every A  S. 

4. P-(int(A))  spgα-int(P-(A)) for every A  S. 

5. int(P(B))  P(spgα-int(B)) for every B  R. 

Proof. (1) → (2): Let B  R. Then P(B)  cl(P(B)), where cl(P(B))  C(S). As P is u.spgα.C, 

B  P+(cl(P(B))). From Theorem 2.1, P+(cl(P(B)))  spgα-C(R). Thus spgα-cl(B)  

P+cl(P(B))) and so P(spgα-cl(B))  cl(P(B)). 

(2) → (3): Let A  S and so P+(A)  R. From (2), P(spgα-cl(P+(A)))  cl(P(P+(A))) = cl(A). 

Thus spgα-cl(P+(A))  P+cl(A). 

(3) → (4): Let A  S. Apply (3) to S - A, then spgα-cl(P+(S - A))  P+cl(S - A)) , spgα-

cl(R–P-(A))  P+(S - int(A)) , R - spgα-int(P-(A))  R – P-(int(A)), P-(int(A))  spgα-int(P-

(A)). 



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(4) → (5): Let B  R, so P(B)  S. From (4), P-(int(P(A))  spgα-int(P-(P(A))) = spgα-

int(A). Thus int(P(A))  P(spgα-int(A)). 

(5) → (1): Let r  R and A  O(S, P(r)). So, r  P+(A) where P+(A)  R. From (5), we have 

int(P(P+(A)))  P(spgα-int(P+(A))). Then int(A)  P(spgα-int(P+(A))). As A  O(R), then 

A  P(spgα-int(P+(A))), that is P+(A)  spgα-int(P+(A)). Thus P+(A)  spgα-O(R, r) and 

P(P+(A))  A. Hence P is u.spgα.C. 

 

Theorem 2.7. Let P: R → S be u.spgα.C with Q  S. If Q is closed in S, then P: R → Q is 

u.spgα.C. 

Proof. Let Q  C(S), so Q - S  C(S). Then Q - S  spgα-C(S). By u.spgα.C, P+(Q - S)  

spgα-C(R) with P(r)  S. Thus P+(Q) = P+(Q - S)  spgα-C(R). From theorem 2.1 P: R → 

Q is u.spgα.C. 

 

Theorem 2.8. [6] Intersection of any two spgα-closed set is again spgα-closed. 

 

Theorem 2.9. Let P: R → S be u.spgα.C and A  spgα-C(R). Then P - A: A → S is 

u.spgα.C. 

Proof. Let B  C(S) as P is u.spgα.C. Then P+(B)  spgα-C(R). As intersection of two 

spgα-closed set is closed, then P+(B) - A = A1, where A1  spgα-C(R). Then (P - A)+(B) = 

A1  spgα-C(R). Hence P - A is u.spgα.C. 

 

Theorem 2.10. If P: R → S is u.spgα.C injective with S is T1, then R is spgα-T1. 

Proof. Let S be T1 space, so for each distinct points p1, p2  R, there exist A, B  O(S) such 

that P(p1)  A, P(p2)  A and P(p1)  B, P(p2)  A. Since P is u.spgα.C, there exist U, V  

spgα-O(R) with p1  U, p1  V and p2  U, p2  V, that is p1  U, p2  V and P1(U)  A, 

P1(V )  B. Hence R is u.spgα-T1. 

 

Theorem 2.11. Let P: R → S is u.spgα.C injective and S is T2 -space. Then R is spgα-T2. 

Proof. Let p1, p2  R be any two distinct points. Then, G, H  O(S) such that P(p1)  G, P(p2) 

 H. As P is u.spgα.C, there exist A, B  spgα-O(R) such that P(G)  A, P(H)  B with A 

 B =  and so G  H = . Hence R is spgα-T2. 

 

Theorem 2.12. For a m.f P: R → S is u.spgα.C, image of spgα-connected space is spgα-

connected. 

Proof. Let P: R → S is u.spgα.C. Suppose S is not connected and S = A  B with a partition 

of S, where A  O(S) and B  C(S). Since P is u.spgα.C, P+(A), P+(B)  spgα-O(R) where 

P+(A)  P+(B) =  and R = P+(A)  P+(B) is a partition of R, which contradicts that, R is spgα-

connected. 

 



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Definition 2.3 (7). For a m.f P: R → S, the graph m.f Gp: R → R x S is defined as GP (p) = {p} 

x P(p) for every p  R. 

 

Lemma 2.1 (7). For a m.f P: R → S, 

1. G+P (A x B) = A x P+(B) 

2. G-P (A x B) = A x P-(B) for A  R and B  S. 

 

Theorem 2.13. Let P: R → S be a m.f. If the graph m.f GP is u.spgα.C, then P is u.spgα.C, 

where GP : R → R x S is defined as GP (p) = {p} x P(p). 

Proof. Let p  R and V  O(S, P(p)). Then R x V  O(R x S) and GP (p)  R x V . As GP is 

u.spgα.C, there exists GP (V )  R x V . Thus U  G+P (R x V). By lemma 2.1, G+P (R x V ) = 

P+(V) and so U  P+(V ). Thus P is u.spgα.C. 

 

Theorem 2.14. Let P: R → S be a m.f. If the graph m.f GP is l.spgα.C. Then P is l.spgα.C. 

Proof. Let p  R and V  O(S) with p  P-(V). Then R x V is open in R x S. Also, we have 

GP (p)  (R x V ) = ({p}) x P(p))  (R x V ) = ({p}) x P(p)  V  . As GP l.spgα.C, there 

exists U  spgα-O(R, r) such that U  G-p (R x V) and from lemma 2.1 G-P (R x V ) = P-(V) 

and so U x P_(V). Thus P is l.spgα.C. 

 

Theorem 2.15. Let (R, ), (S, ), (Q, γ) be TS and P1: R → S and P2 : S → Q be m.f. Let P1 x 

P2 : R → S x Q be a m.f defined by (P1 x P2)(p) = P1(p) x P2(p) for each p  R if (P1 x P2) is 

u.spgα.C. Then P1 and P2 are u.spgα.C. 

Proof. Let p  R and V x S and W x Q be open sets with p  P+1 (V) and p  P+2 (W) and so 

P1(p)  V and P2(p)  W. Thus (P1 x P2)(p) = P1(p) x P2(p)   V x W and so p  (P1x P2)+(V x 

W) as P1 x  P2 is u.spgα.C, there exists spgα-open set U containing p such that U  (P1 x 

P2)+(V x W) that is U  P+1 (V) and U  P+2 (W) Hence P1 and P2 are u.spgα.C 

Similarlly we can prove the results for l.spgα.C. 

 

Theorem 2.16. Let P: R → S be compact m.f. Then GP is u.spgα.C if and only if P is 

u.spgα.C. 

Proof. Suppose GP : R → S be u.spgα.C. Let p  R and V  O(S, P(p)). Since R x V is open 

in R x S and GP (p)  R x S and so there exists U  spgα-O(R, p) such that GP (U)  R x S. 

From lemma 2.1, U  G+P (R x V ) = P+(V) and P(U) x V and hence P is u.spgα.C. 

Conversely, Let P be u.spgα.C. Let p  R and W be any open set in R x S containing GP(p). 

Then for each n  P(p), there exists U(n)  R and V (n)  S such that (p, n)  U(n) x V (n)  

W. The family {V (n) : n  P(p)} is an open cover of P(p). As P(p) is compact, there exists 

finite number of points say n1, n2, …. nk in P(p) such that P(p)  {V(ni): i = 1; 2; ….k}. Put U 

=  {U(ni): i = i=1,2….k} and V =  {V (ni): i = 1,2,…k}. Then U and V are open sets in R 

and S respectively and {p} x P(p)  U X V  W. As P is u.spgα.C there exists U  spgα-



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O(R, p) such that P(U1)  V . B lemma 2.1, U  U1  U  P+(V) = G+P (U x V )  G+P (W). 

Thus U  U1  spgα-O(R, p) and GP (U  V )  W. Hence GP is u.spgα.C. 

 

Theorem 2.17. A m.f P: R → S is if and only if GP is l.spgα.C. 

Proof. Let P be l.spgα.C. Let p  R and W  O(R x S) with p  G-P (W). As W  ({p} x 

P(p))  , there exists n  P(p) with (p, n)  W and so (p, n)  U x V  W, where U  O(R) 

and V  O(S) respectively. Since P(p)  V  , there exists G  spgα-O(R, p) with G  P-

(V). By Lemma 2.1, U  G  U  P-(V)  G-P (U x V )  G-P (W). Thus p  U x G  spgα-

O(R, p) and hence GP is l.spgα.C. 

Conversely, let GP be l.spgα.C, p  R and V  O(S) with p  P-(V). So R x V  O(R x S) 

with GP (p)  (R x V ) = ({p} x P(p))  (R x V ) = ({p} x (P(p)  V)  . Since GP is l.spgα.C, 

there exists U  spgα - O(R, p) with U   (R x V ). From Lemma 2.1, U   P-(V) and so P is 

l.spgα.C. 

 

3. Upper (Lower) spgα-Irresolute Multifunctions 

Definition 3.1. A m.f P: R → S is called 

1. Upper spgα-irresolute (briefly U.spgα.I) if for each p  R and each V  spgα - O(S, 

P(p)), there exists U  spgα - O(R, p) such that P(U)  V . 

2. Lower spgα-irresolute (briefly l.spgα.I) if for each p  R and each spgα-open set V 

with P(p)  V  , there exists U  spgα - O(R, p) such that U  P-(V). 

 

Theorem 3.1. For a m.f P: R → S the following statements are equivalent: 

1. P is u.spgα.I. 

2. for each p  R, for each spgα-nbd. V of P(p), we have P+(V) is spgα- nbd. of p. 

3. for each p  R, for each spgα-nbd. V of P(p), there exists spgα-nbd. U of p with 

P(U)  V . 

4. P+(V)  spgα- O(R) for each V  spgα - O(S). 

5. P-(V)  spgα - C(R) for each V  sgα - C(S). 

6. spgα - cl(P-(B))  P-(spgα - cl(B)) for each B  S. 

Proof. (1) → (2): Let p  R and W be spgα-nbd. of P(p). Then there exists V  spgα - O(S) 

with P(p)  V  W. As P is u.spgα.I, there exists U  spgα - O(R, p) such that P(U)  V . 

Thus p  U  P+(V )  P+(W) and so P+(W) is a spgα-nbd. of p. 

(2) → (3): Let p  R and V be a spgα-nbd. of P(p). Let U = P+(V). Then by (2), U is spgα-

nbd of p with P(U)  V . 

(3) → (4): Let V  spgα -O(S) and p  P+(V ). Then there exists a spgα - nbd G of p with 

P(G)  V . Thus for some U  spgα - O(R, p) with U  G and P(U)  V . So p  U  P+(V 

) and hence P+(V)  spgα - O(S). 

(4) → (5): Let A  spgα - C(S) and so R – P-(A) = P+(S - K)  spgα - O(R). Thus P-(A)  

spgα-C(S). 



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(5) → (6): Let B  S. Since spgα - cl(B) is spgα-closed in S, so P-(spgα - cl(B))  spgα 

-C(R) with P-(B))  P-(spgα - cl(B). Thus spgα- cl(P-(B))  P-(spgα- cl(B)). 

(6) → (1): Let p  R and V  spgα- O(R) with P(p)  V . So P(p)  (S - V ) = . Hence p  

P-(S - V). From (6), p  spgα - cl(P-(S - V)) and so there exists U  spgα-O(R, p) such that 

U  P-(S - V ) = . Thus P(U)  V and so P is u.spgα.I. 

 

Theorem 3.2. The following statements are equivalent for a m.f P: R → S: 

1. P is l.spgα.I. 

2. for each V  spgα-O(S) and each p  P-(V), there exists U  spgα - O(R, p) such 

that U  P-(V ). 

3. P-(V)  spgα- O(R) for each V  spgα-O(S). 

4. P+(K)  spgα - C(R) for each K  spgα-C(S). 

5. for each A  R, P(spgα-cl(A))  spgα-cl(P(A)). 

6. spgα-cl(P+(B))  P+(spgα-cl(B)) for each B  S. 

Proof. (1) → (2): Follows by the definition. 

(2) → (3): Let V  spgα-O(S) with p  P-(V). From (2) there exists U  spgα-O(R, p) such 

that U  P-(V). Thus, p  U  cl(int(U))  int(cl(U))  cl(int(P-(U)))  int(cl(P-(U))). So P-

(V)  spgα-O(R). 

(3) → (4): Let K  spgα-C(S). Then R - P+(K) = P-(S - K)  spgα-O(R) and so P+(K)  

spgα-C(R). 

(4) → (5) and (5) → (6) are Straight forward. 

(6) → (1): Let p  R and V  spgα-O(S) with P(p)  V  . So P(p)  (S - V) = . Then 

P(p)  S - V and p  P+(S - V). Since S - V  spgα - C(S) and by (6), p  spgα - cl(P+(S - 

V)) and so there exists U  spgα - O(R, p) with U  P-(S - V ) = U  (R – P-(V )) = . Thus 

U  R - (R – P-(V)) = P-(V), that is U  P-(V). So P is l.spgα.I. 

 

Lemma 3.1. Let P be a m.f. Then (spgα - cl(P))-(V) = P-(V) for each V  spgα - O(R). 

Proof. Let V  spgα - O(S) with p  (spgα-cl(P))-C(V) so V  (spgα - cl(P))(p)  . As 

V  spgα-O(S) and so V  P(p)  . Thus p  P-(V). 

Conversely, Let p  P-(V). Then V  P(p)  (spgα- cl(P))(p)  V   and so p  (spgα - 

cl(P))-(V). Thus (spgα-cl(P))-(V) = P-(V). 

 

Lemma 3.2. [6] Let A, B  R. Then 

1. If A  spgα-O(R) and B  R then A  B  spgα-O(B). 

2. If A  spgα-O(B) and B  spgα-O(R) then A  spgα-O(R). 

 

Theorem 3.3. Let P be a m.f. and U  O(R). If P is u.spgα.I (resp l.spgα.I) then PlU :U → S 

is u.spgα.I (resp l.spgα.I). 



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Proof. Let V  spgα-O(S) and p  U, p  P-lU(V). As P is l.spgα.I there exists G  spgα-

O(R, r) with G  P-(V ) and so p  G  U  spgα-O(U) with G  U  PlU (V). Thus PlU is 

l.spgα.I. 

Similarlly, we can prove for u.spgα.I. 

 

Definition 3.2. A subset B of a space R is said to be 

1. spgα-compact relative to R (resp. spgα-Lindelof relative to R) if every cover of B by 

spgα-open sets of R has a finite (resp. countable) subcover. 

2. spgα-compact (spgα-Lindelof) if R is spgα-compact (resp spgα-Lindelof) relative to 

R. 

 

Theorem 3.4. If P be u.spgα.I m.f and P(p) is spgα-compact relative to S for each p  R. If 

B is spgα-compact relative to R, then P(B) is spgα-compact relative to S. 

Proof. Let {Vi : i  } be a cover of P(B) by spgα-open sets in S. Then for each p  B, 

there exists a finite subset (p)   with P(p)  {Vi : i  (p)} and so P(p)  V(p)  

spgα-O(S). Thus there exists a finite number of points of B, p1, p2, p3, … pk with B  {V 

(pi): i = 1, 2, 3…. k}.Thus P(B)  P(⋃ {𝑉𝑖(𝑝𝑖)}) 𝑘
𝑖=1  ⋃ 𝑃(𝑉𝑖(𝑃𝑖))𝑘

𝑖=1   ⋃ 𝑉(𝑃𝑖)
𝑘
𝑖=1   

⋃ ⋃ 𝑉𝑖𝑖∈(𝑝𝑖)
𝑘
𝑖=1 . Thus P(B) is spgα-compact relative to S. 

 

Corollary 3.1. Let P be an u.spgα.I surjective m.f and P(B) is spgα-compact relative to S 

for each p  R. If R is spgα-compact, then S is spgα-compact. 

 

Theorem 3.5. If P is u.spgα.I and P(p) is spgα-Lindelof relative to S for each p  R and if 

B is spgα-Lindelof relative to R, then P(B) is spgα-Lindelof relative to S. 

Proof. Similar to proof of theorem 3.4. 

 

Definition 3.3. A space R is said to be spgα-normal (briefly spgα.N) if for any pair of 

distinct spgα-closed sets A and B in R there exists disjoint open sets U and V in R such that 

A  U, B  V. 

 

Theorem 3.7. The set of a point p of R at which a m.f P is not u.spgα.I (resp l.spgα.I) is 

identical with the union of the spgα-frontiers of the upper (lower) inverse image of spgα-

open sets containing (respectively meeting) P(p). 

Proof. Let p  R at which P is not u.spgα.I. Then there exists V  spgα - O(S) containing 

P(p) with U  (R - P+(V))   for each U  spgα- O(R, p). Then p  spgα- cl(R - P+(V)) as 

p  P+(V). So p  spgα - cl(P+(S)) and p  spgα-Fr(P+(B)). On the other hand V  spgα-

O(S) containing P(p) and p  spgα-Fr(P+(B)). Let P is u.spgα.I, there exists U  spgα-

O(R, p) with P(U)  V. Thus p  U  spgα-int(P+(V)) which contradicts to the fact that p  

spgα-Fr(P+(V)). Hence P is not u.spgα.I. 

Similarlly we can prove the theorem related to l.spgα.I. 



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Theorem 3.8. Let P be an u.spgα.I injective m.f and point closed from a TS R to spgα-N, 

then R is spgα-T2-space. 

Proof. Let m, n  R with m  n. Then P(m)  P(n) =  as P is injective. By spgα-normality 

of S, there exists disjoint open sets U and V containing P(m) and P(n). Thus, there exists disjoint 

spgα-open sets P+(U) and P+(V) containing m and n respectively such that G  P+(U) and H 

 P+(V) and so G  H = . Hence R is spgα-T2-space. 

 

Definition 3.4. A m.f P: R → S is said to have spgα-closed graph if for each (m, n)  G(P) 

there exists U  spgα-O(R, m) and V  spgα-O(S, n) with (U x V )  G(P) = . 

 

Theorem 3.9. Let P be a m.f from a space R into spgα-compact space S. If G(P) is spgα-

closed then P is u.spgα.C. 

Proof. Suppose P is not u.spgα.C. Then there exists a non empty closed subset B with P-(B) 

is not spgα-closed in R. Assume that P-(B)  , then there exists a point p0  spgα-cl(P-(B) 

– P-(B)). So for each point n  B, (p0, n0)  G(P), as P is spgα-closed graph. Thus there exists 

a spgα-open sets U(n) and V (n) containing p0 and n respectively with (U(n) x V (n))  G(P) 

= . Then {S – B}  {V(n) : n  B} is a spgα-open cover of SN and so it has a subcover {S 

– B}  {V(ni): ni  B: 1 < i < k}. Put U = U(ni) and V = V(ni), then B  V and (U x V)  

G(P) =  as U is spgα-nbd. of p0 U – P-(B) =  and so   (U x B)  G(P)  (U x V )  G(P) 

which is contradiction. Thus P is u.spgα.C. 

 

Discussion and Conclusion 

 Topology is a relatively new branch of Mathematics, most of the research in topology 

has been done since 1900. The topological structures are modelled suitably in the field of 

Computer graphics, Pattern recognition, Artificial intelligence, Data mining, Rough set theory, 

Information systems, Quantum physics etc. 

 The investigation on generalization of open set as well as closed set has led to 

significant contribution to the theory of generalization of continuity, separation axioms, 

covering properties and compactness with the help of open sets. Several generalized form of 

continuous functions has been introduced in the last decades which helps us to understand 

various properties of topological spaces. 

 In this way, this paper introduce some concepts of multifunctions in topological spaces. 

We first introduced the concept of upper spgα-continuous (resp. lower spgα-continuous) 

multifunctions and some properties and point out the relationship among them. Further, we 

introduce upper spgα-irresolute (resp. lower spgα-irresolute) multifunctions and several 

results of these spaces in topological spaces. 

 

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[4] E. Ekici, J.H. Park, A weak form of some types of continuous multifunctions, Filomat 20(2),(2006), 13-32. 



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