On $gamma$-Semi-Continuous Functions

On γ -Semi-Contin uous F unction s Sabir H ussain 1 , Bashir Ahmad 2 and T ak ashi N o ir i 3 1 Departmen t of Mathematics, Islamia Unive rsit y Baha w alpur, P akistan. Presen t Address: Departmen t of Mathematics, Y an bu Univ ersity , P . O. Bo x 31387, Y an bu, Saudi Ara bia. E. mail: sabiriub@y aho o.com. 2 Departmen t of Mathematics, King Abdul Aziz Univ ersit y , P . O. Bo x 8 0203, Jeddah 21589, Saudi Arabia. E. mail: drbashir9@gmail.com. 3 2949-1 Shiokita-ch a, Hinagu, Y atsushiro-shi, Kumamoto-k en, 869-51 42, Japan. E. mail: t.noiri@nifty .com. Abstract. I n this pap er, w e con tin ue studying t he prop erties of γ -semi-con tinuous and γ -semi- op en functions in tro duced in [5]. Keyw ords. γ -closed (op en), γ -closure , γ ∗ -semi-closed (op en), γ ∗ -semi-closure, γ -regular, γ ∗ - semi-in terior, γ -semi-con tin u ous function, γ -semi-op en (closed) fu nction. AMS(2000) Sub j ect Classification. Primary 54A05, 54A10, 54D10. 1 In tro duction N. Levine [11] in tro du ced the notion of semi-op en sets in to p ological spaces. A. Csaszar [ 7,8] defined generalize d op en sets in generalized top ological spaces. In 1975, Mahesh w ari and Pr asad [12] in tro duced concepts of semi- T 1 -spaces and semi- R 0 -spaces. In 1979, S. Kasahara [10] defined an op eration α on top ologic al spaces. Carpin tero, et. al [6] introdu ced the notion of α -semi-open sets as a g eneralizatio n of semi-op en sets. B. Ahmad a nd F.U. Rehman [ 1, 14] in tro duced th e notions of γ -in terior, γ -b oun dary and γ -exte rior p oin ts in top ologica l spaces. They also studied 1 prop erties and c haracterizations of ( γ , β )-con tinuous mappings introd uced b y H. Ogata [13]. In [2-4], B. Ah mad and S. Hussain in tro duced the concept of γ 0 -compact, γ ∗ -regular, γ -normal spaces and explored their man y in teresting prop erties. They initiated and discussed th e concepts of γ ∗ - semi-op en sets , γ ∗ -semi-closed sets, γ ∗ -semi-closure, γ ∗ -semi-in terior p oints in top ologica l sp aces [5,9]. In [9], they intro d uced Λ γ s -set and Λ s γ -set by using γ ∗ -semi-op en sets. Moreo v er, they also in tro duced γ -semi-con tin uous fu nction and γ -semi-op en (closed) fun ctions in top ologi cal spaces and established several interesti ng prop erties. In this pap er, w e con tinue stu dying the prop erties of γ -semi-con tinuous fun ctions and γ -semi- op en fu nction in tro duced by B. Ahmad and S. Hu s sain [5]. Hereafter, we sh all w rite space in place of top ologic al space in the s equel. 2 Preliminaries W e recall some definitions and r esults used in this pap er to m ak e it self-con tained. Definition [13]. Let (X, τ ) b e a sp ace. An op eration γ : τ → P(X) is a fun ction from τ to the p o w er set of X su ch that V ⊆ V γ , for eac h V ∈ τ , where V γ denotes the v alue of γ at V. The op erations d efined b y γ (G) = G, γ (G) = cl(G) and γ (G) = intc l(G) are examples of op eration γ . Definition [13]. Let A b e a sub set of a sp ace X. A p oin t x ∈ A is said to b e γ -int erior p oin t of A, if there exists an op en n b d N of x suc h that N γ ⊆ A. The set of all suc h p oin ts is denoted by int γ (A). Thus int γ (A) = { x ∈ A : x ∈ N ∈ τ and N γ ⊆ A } ⊆ A . Note that A is γ -op en [13] iff A = int γ (A). A set A is called γ - closed [13] iff X-A is γ -op en. Definition [10]. A p oin t x ∈ X is called a γ -closure p oin t of A ⊆ X, if U γ ∩ A 6 = φ , f or eac h op en n b d U of x. The set of all γ -closure p oin ts of A is called γ -closure of A and is denoted b y cl γ (A). A subset A of X is called γ -closed, if cl γ ( A ) ⊆ A. Note that cl γ (A) is con tained in ev ery γ -closed sup er s et of A. Definition [14]. Th e γ -exte rior of A, written ext γ ( A ) is defin ed as th e γ -in terior of ( X − A ). T hat is, int γ ( A ) = ext γ ( X − A ). Definition [14]. Th e γ -b oundary of A, written bd γ ( A ) is d efi ned as the set of p oints whic h do n ot 2 b elong to γ -in terior or the γ -exte rior of A. Definition [13]. An op eration γ on τ is said b e regular, if for an y op en nb ds U,V of x ∈ X, there exists an op en n b d W of x suc h that U γ ∩ V γ ⊇ W γ . Definition [13]. An op eration γ on τ is said to b e op en, if for ev ery op en nb d U of eac h x ∈ X , there exists γ -op en set B s u c h that x ∈ B and U γ ⊆ B . Definition [2]. L et A ⊆ X . A p oint x ∈ X is said to b e γ -limit p oint of A, if U ∩ { A − { x }} 6 = φ , where U is a γ -op en set in X. T he set of all γ -limit p oin ts of A d enoted A d γ is called γ -derive d set. Definition [9]. A sub set A of a s p ace (X, τ ) is said to b e a γ ∗ -semi-op en set, if there exists a γ -op en set O suc h that O ⊆ A ⊆ cl γ ( O ). The set of all γ ∗ -semi-op en sets is denoted by S O γ ∗ ( X ). Definition [5]. A function f : ( X , τ ) → ( Y , τ ) is said to b e γ -semi-con tinuous, if for any γ -op en B of Y, f − 1 ( B ) is γ ∗ -semi-op en in X. Definition [5]. A fu nction f : ( X , τ ) → ( Y , τ ) is said to b e γ -semi-op en (closed), if for eac h γ -op en (closed) set U in X, f ( U ) is γ ∗ -semi-op en (closed) in Y. Definition [5]. A set A in a sp ace X is s aid to b e γ ∗ -semi-closed, if there exists a γ -closed set F suc h that int γ ( F ) ⊆ A ⊆ F . Prop osition [5]. A subset A of X is γ ∗ -semi-closed if and only if X − A is γ ∗ -semi-op en. Definition 2.1. A subset A of X is said to b e γ -semi-n b d of a p oin t x ∈ X , if t here exists a γ ∗ -semi-op en set U suc h that x ∈ U ⊆ A . Definition [9]. L et A b e a su b set of space X . The int ersection of all γ ∗ -semi-closed sets con taining A is called γ ∗ -semi-closure of A and is denoted by scl γ ∗ ( A ). Note that A is γ ∗ -semi-closed if and only if s cl γ ∗ ( A ) = A . Definition [5]. Let A b e a subset of a space X . The u nion of all γ ∗ -semi-op en sets of X contai ned in A is called γ ∗ -semi-in terior of A and is denoted b y s int γ ∗ ( A ) . Lemma 2.2. Let A b e a subs et of a sp ace X. Then x ∈ scl γ ∗ ( A ) if and only if for any γ -semi-n b d N x of x in X, A ∩ N x 6 = φ . Pro of. Let x ∈ s cl γ ∗ ( A ) .Supp ose on the con trary , there exists a γ -semi-n b d N x of x in X suc h that A ∩ N x 6 = φ . Then there exists U ∈ S O γ ∗ ( A ) suc h that x ∈ U ⊆ N x . Therefore, U ∩ A = φ , so that A ⊆ X − U . C learly X − U is γ ∗ -semi-closed in X and hence scl γ ∗ ( A ) ⊆ X − U . Since x / ∈ X − U , we obtain x / ∈ s cl γ ∗ ( A ). This is con tradiction to the hyp othesis. This pro v es the necessit y . 3 Con v ers ely , su p p ose that ev ery γ -semi-n b d of x in X meets A. If x / ∈ s cl γ ∗ ( A ), then by definition there exists a γ ∗ -semi-closed F of X suc h that A ⊆ F and x / ∈ F . Th erefore w e h a ve x ∈ X − F ∈ S O γ ∗ ( X ). Hence X − F i s γ -semi-n b d of x in X. But ( X − F ) ∩ A = φ . This is con tradiction to the hyp othesis. Thus x ∈ scl γ ∗ ( A ). 3 γ -Semi-Op en F unctions Theorem 3.1. Let f : X → Y b e a fu nction from a sp ace X in to a sp ace Y and γ is an op en, monotone and r egular op eration. T h en th e follo w ing statemen ts are equiv alen t: (1) f is γ -semi-op en. (2) f ( int γ ( A )) ⊆ sin γ ∗ ( f (( A )) f or eac h su bset A of X. (3) F or eac h x ∈ X and eac h γ -op en-nb d U of x, there exists a γ -semi-nbd V of f ( x ) such that V ⊆ f ( U ). Pro of. (1) ⇒ (2). S upp ose that f is γ -semi-op en , and let A b e an arbitrary subset of X. Since f ( int γ ( A )) is γ ∗ -semi-op en and f ( int γ ( A )) ⊆ f ( A ), then f ( int γ ( A )) ⊆ sin γ ∗ ( f (( A )). (2) ⇒ (3). Le t U b e an arbitrary γ -op en -n b d of x ∈ X . Then there exists γ -op en set O suc h that x ∈ O ⊆ U . By hyp othesis, w e ha v e f ( O ) = f ( int γ ( O )) ⊆ s in γ ∗ ( f (( O )) and hence f ( O ) ⊆ sin γ ∗ ( f (( O )). Th erefore it follo w s that f ( O ) is γ -semi-op en-nb d in Y su c h that f ( x ) ∈ f ( O ) ⊆ f ( U ). This prov es (3). (3) ⇒ (1). Let U b e an arbitrary γ -open set in X. F or eac h y ∈ f ( U ), by hypothesis th er e exists a γ -semi-n b d V y of y in Y suc h th at V y ⊆ f ( U ). Sin ce V y is a γ -semi-nbd of y , there exists a γ ∗ -semi-op en set A y in Y su c h that y ∈ A y ⊆ V y . Ther efore f ( U ) = S { A y : y ∈ f ( U ) } is a γ ∗ -semi-op en in Y, since is γ regular [9]. Th is sh o ws that f is a γ -semi-op en function. Theorem 3.2. A bijectiv e function f : X → Y is γ -semi-o p en if and only if f − 1 ( scl γ ∗ ( B )) ⊆ cl γ ( f − 1 ( B )) for eve ry s ubset B of Y, where γ is an op en op eration. Pro of. Let B b e an arbitrary subs et of Y. Put U = X − cl γ ( f − 1 ( B )) ...... ( I) Clearly U is a γ -op en set in X. Th en b y h yp othesis, f ( U ) is a γ ∗ -semi-op en set in Y, or Y − f ( U ) is γ ∗ -semi-closed se t in Y. Since f is on to, f rom (I), it follo ws B ⊆ Y − f ( U ). Thus w e h a ve 4 scl γ ∗ ( B ) ⊆ Y − f ( U ). Since f is one-one, we hav e f − 1 ( scl γ ∗ ( B )) ⊆ f − 1 ( Y ) − f − 1 f ( U ) = X − f − 1 f ( U ) ⊆ X − U = cl γ ( f − 1 ( B )). T h is p ro v es the necessit y . Con v ers ely , let U b e an arbitrary -op en set in X. Put B = Y − f ( U ). S ince f is bijectiv e, ther efore b y hyp othesis, f ( U ) ∩ s cl γ ∗ ( B ) = f ( U ∩ f − 1 ( scl γ ∗ ( B ))) ⊆ f ( U ∩ cl γ ( f − 1 ( B ))). Since U is γ -op en, therefore b y Lemma 2(3) [14], we ha ve U ∩ cl γ ( f − 1 ( B )) ⊆ cl γ ( U ∩ f − 1 ( B )). Moreo ver, it is obvious that U ∩ f − 1 ( B ) = φ . Thus we ha v e f ( U ) ∩ scl γ ∗ ( B ) = φ and hen ce scl γ ∗ ( B ) ⊆ Y − f ( U ) = B . Therefore B is a γ ∗ -semi-closed in Y and hence f ( U ) is a γ ∗ -semi-op en set in Y. Th is p ro ves that f is a γ -semi-op en mapp in g. Definition 3.3 [13]. A function f : ( X , τ , γ ) → ( Y , δ, β ) is said to b e ( γ , β )-con tinuous, if for eac h x ∈ X and eac h op en set V con taining f ( x ), th ere exists an op en set U suc h that x ∈ U and f ( U γ ) ⊆ V β , wher e γ and β are op erations on τ and δ resp ectiv ely . Definition 3.4 [13]. A function f : ( X, τ , γ ) → ( Y , δ, β ) is said to b e ( γ , β )-op en (closed), if for an y γ -op en (closed) set A of X, f ( A ) is γ -op en (closed) in Y. Theorem 3 .5 [1]. Let f : ( X, τ , γ ) → ( Y , δ, β ) b e a fun ction and β b e an o p en op eration on Y. Then f is ( γ , β )-con tinuous if and only if for eac h β -op en set V in Y, f − 1 ( V ) is γ -op en in X. Theorem 3 .6 [1]. Let f : ( X, τ , γ ) → ( Y , δ, β ) b e a fun ction and β b e an o p en op eration on Y. Then the follo wing are equiv alen t: (1) f is ( γ , β )-op en . (2) f − 1 ( cl β ( B )) ⊆ cl γ ( f − 1 ( B )). (3) f − 1 ( bd β ( B )) ⊆ bd γ ( f − 1 ( B )) for any s ubset B of Y. Theorem 3.7. If a function f : ( X, τ , γ ) → ( Y , δ , β ) is a ( γ , β )-op en and a ( γ , β )-con tinuous, then the inv erse image f − 1 ( B ) of eac h β ∗ -semi-op en set B in Y is γ ∗ -semi-op en in X, where β is an op en op eration on Y. Pro of. Let B b e an arb itrary β ∗ -semi-op en set in Y. Then there exists β -op en set V in Y such that V ⊆ B ⊆ cl β ( V ). Since f is ( γ , β )-op en, us ing T h eorem 3.6, we ha ve f − 1 ( V ) ⊆ f − 1 ( B ) ⊆ 5 f − 1 ( cl β ( V )) ⊆ cl γ ( f − 1 ( V )). Since is ( γ , β )-con tinuous and V is β -op en in Y, b y Theorem 3.5 , f − 1 ( V ) is γ -op en in X. Th is sho ws that f − 1 ( B ) is γ ∗ -semi-op en set in X. Theorem 3.8. Let X, Y an d Z b e three s paces and let f : X → Y b e a fu n ction, g : Y → Z b e an injectiv e fun ction and g of : X → Z is a γ -semi-op en fun ction. Then we h a ve: (1) If f is ( γ , β )-con tinuous and surjectiv e, then g is γ -semi-op en. (2) If g is ( β , α )-op en , ( β , α )-cont in uous and in jectiv e, then f is γ -semi-op en, w h ere β is op en op- eration on Y. Pro of. (1) Let V b e a β -op en set in Y. Th en f − 1 ( V ) is γ -op en in X, b ecause f is ( γ , β )-con tin u ous. Since g of is γ -semi-op en and f is surjectiv e, therefore g ( V ) = ( g of )( f − 1 ( V )) is α ∗ -semi-op en in Z. This shows that g is a γ -semi-op en function. (2) Since g is injectiv e, therefore for A ⊆ X , f ( A ) = g − 1 ( g ( f ( A )) ). Let U b e a γ -op en set in X, then gof ( U ) is α ∗ -semi-op en. Thus b y T heorem 3.7, g − 1 ( g ( f ( U ))) = f ( U ) is β ∗ -semi-op en in Y. This shows that f is a γ -semi-op en function. Let B ⊆ X , γ : τ → P ( X ) b e an op eration. W e define γ B : τ B → P ( X ) as γ B ( U ∩ B ) = γ ( U ) ∩ B . F r om here γ B is an op eration and satisfies that cl γ B ( U ∩ B ) ⊆ cl γ ( U ∩ B ) ⊆ cl γ ( U ) ∩ cl γ ( B ). Using this fact w e p ro v e the follo wing: Theorem 3.9. Let X b e a space and B a γ ∗ -semi-op en set in X contai ning a subset A of X. If A is γ ∗ -semi-op en in the subspace B, then A is γ ∗ -semi-op en in X, w here γ is a r egular op eration. Pro of. Let A b e γ ∗ B -semi-op en in the sub space B. Then there exists a γ B -op en set U B in B suc h that U B ⊆ A ⊆ cl γ B ( U B ). Since U B is γ B -op en in B, there exists a γ -op en set U in X suc h that U B = U ∩ B [4]. Thus we hav e U ∩ B ⊆ A ⊆ cl γ B ( U ∩ B ) ⊆ cl γ ( U ∩ B ) = cl γ ( A ) ∩ cl γ ( B ). Since B is γ ∗ -semi-op en set in X and U is γ -op en in X, therefore U ∩ B is γ -op en in X. Consequent ly , A is a γ ∗ -semi-op en set in X. Theorem 3.10. Let X and Y b e spaces. If a bijectiv e fun ction f : X → Y is a γ -semi-op en, then for eac h γ -op en set V ( 6 = φ ) in Y f | f − 1 ( V ) : f − 1 ( V ) → V is γ -semi-op en, w here γ is a r egular op eration. 6 Pro of. Let U V b e an arbitrary γ f − 1 ( V ) -op en set in f − 1 ( V ). T hen ther e exists a γ -op en s et U in X suc h that U V = U ∩ f − 1 ( V ). No w w e h a ve [ f | f − 1 ( V ) ]( U V ) = f ( U ∩ f − 1 ( V )) = f ( U ) ∩ V . S ince f ( U ) is γ ∗ -semi-op en and V is γ -op en, f ( U ) ∩ V is γ ∗ -semi-op en. Hence [ f | f − 1 ( V ) ]( U V ) is also γ ∗ V -semi-op en in V. T his shows that f | f − 1 ( V ) : f − 1 ( V ) → V is a γ -semi-op en mapping. Theorem 3.11. A bijectiv e f u nction f : X → Y is γ -semi-op en if and only if for a n y su bset V of Y and for any γ -closed set F of X cont aining f − 1 ( V ), there exists a γ ∗ -semi-closed set G of Y con taining V such that f − 1 ( G ) ⊆ F . Pro of. Let V ⊆ Y and F b e a γ -closed set of X con taining f − 1 ( V ). Pu t G = Y − f ( X − F ). S ince f is γ -semi-op en, so G is γ ∗ -semi-closed sets in Y . As f is b ijectiv e, it follo ws from f − 1 ( V ) ⊆ F that V ⊆ G . C alculations giv e f − 1 ( G ) ⊆ F . Con v ers ely , su p p ose U is γ -op en set. Put V = Y − f ( U ). Then X − U is γ -closed set in X con taining f − 1 ( V ). By hypothesis, there exists a γ ∗ -semi-closed set G of Y suc h that V ⊆ G an d f − 1 ( G ) ⊆ ( X − U ). O n the other h and, it follo ws from V ⊆ G th at f ( U ) = ( Y − V ) ⊆ ( Y − G ). Therefore, we obtain f ( U ) = ( Y − G ) ∈ S O γ ∗ ( Y ). T h is sh o w s that f is γ -semi-op en. Lemma 3.12 [5]. The follo wing prop erties of a sub set A of X are equiv alen t: (1) A is γ ∗ -semi-closed. (2) int γ ( cl γ ( A )) ⊆ A . (3) X − A is γ ∗ -semi-op en. Theorem 3.13. If f : X → Y is ( γ , β )-op en and ( γ , β )-con tinuous mapp ing, then the inv erse image f − 1 ( B ) of eac h β ∗ -semi-closed B in Y is γ ∗ -semi-closed in X, wh er e β is an op en op eration on Y. Pro of. Th is follo ws from Th eorem 3.7 and Lemma 3.12. Theorem 3.14. Let f : X → Y b e surjectiv e and g : Y → Z b e an injectiv e fu nction a nd let g of : X → Z b e a γ -semi-closed function. T hen (1) If f is ( γ , β )-con tinuous and surjectiv e, then g is β -semi-closed. (2) If g is ( β , α )-op en, ( β , α )-con tin u ous and injectiv e, th en f is γ -semi-c losed, w here β is an op en 7 op eration on Y. Pro of. (1) Sup p ose H is an arbitrary β -closed set in Y. Then f − 1 ( H ) is γ -cl osed in X b ecause f is ( γ , β )-con tinuous. S ince g of is γ -semi-closed and f is s u rjectiv e, g of ( f − 1 ( H )) ⊆ g ( f ( f − 1 ( H ))) = g ( H ), is α ∗ -semi-closed in Z . This implies that g is β -semi-op en fun ction. T his prov es (1). (2) Since g is in jectiv e so for every subs et A of X, f ( A ) = g − 1 ( g ( f ( A )) ). Let F b e an arbitrary γ -closed set in X. Then g of ( F ) is γ ∗ -semi-closed. It follo ws immediately from Theorem 3.13 th at f ( F ) is γ ∗ -semi-closed set in Y. This implies that f is γ -semi-closed. 4 γ -Semi-Closed F unctions Theorem 4.1. L et γ b e an op en and mon otone op eration. A function f : X → Y is γ -semi-c losed if and only if f ( cl γ ( A )) ⊇ int γ ( cl γ ( f ( A ))) for ev ery su bset A of X. Pro of. S upp ose f is a γ -semi-c losed mapp ing and A is an arbitrary sub set of X. Then f ( cl γ ( A )) is γ ∗ -semi-closed in Y. Th en by Lemma 3.12, w e obtain f ( cl γ ( A )) ⊇ int γ ( cl γ ( f ( cl γ ( A )))) ⊇ int γ ( cl γ ( f ( A ))). This implies that f ( cl γ ( A )) ⊇ int γ ( cl γ ( f ( A ))). Con v ers ely , sup p ose that F is an arbitrary γ -closed set in X. Then by hyp othesis, we hav e int γ ( cl γ ( f ( F ))) ⊆ f ( cl γ ( F )) = f ( F ). By Lemma 3.12, f ( F ) is γ ∗ -semi-closed in Y. This implies that f is γ -semi-closed. Recall [9] th at the intersecti on of all γ ∗ -semi-closed sets con taining A is called γ -semi-closure of A and is denoted by scl γ ∗ ( A ). Clearly A is γ ∗ -semi-closed if and only if s cl γ ∗ ( A ) = A . Theorem 4.2. Let γ b e an op en and mon otone op eration. A fu n ction f : X → Y is γ -semi- closed if and only if scl γ ∗ ( A ) ⊆ f ( cl γ ( A )) for every subset A of X. Pro of. S upp ose f is a γ -semi-c losed mapp ing and A is an arbitrary sub set of X. Then f ( cl γ ( A )) is γ ∗ -semi-closed. Since f ( A ) ⊆ f ( cl γ ( A )), we obtain scl γ ∗ ( f ( A )) ⊆ f ( cl γ ( A )). This imp lies scl γ ∗ ( f ( A )) ⊆ f ( cl γ ( A )). Sufficiency follo w s from Theorem 4.1. Theorem 4.3. A surjectiv e fu nction f : X → Y is γ -semi-closed if and o nly if for eac h sub- set B in Y and eac h γ -open set U in X cont aining f − 1 ( B ), th er e exists a γ ∗ -semi-op en set V in Y 8 con taining B s u c h that f − 1 ( V ) ⊆ U , where γ is a monotone and r egular op eration. Pro of. S upp ose B is an arbitrary subs et in Y and U is an arbitrary γ -op en set in X con taining f − 1 ( B ). W e pu t V = Y − f ( X − U ) ..... (*) Then V is γ ∗ -semi-op en set in Y. Since f − 1 ( B ) ⊆ U , calculations give B ⊆ V . Moreo ver, by (*), w e ha v e f − 1 ( V ) = f − 1 ( Y ) − f − 1 ( f ( X − U )) = X − f − 1 ( f ( X − U )) ⊆ X − ( X − U ) = U . Con v ers ely , supp ose that F is an arb itrary γ -closed set in X. L et y b e an arbitrary p oin t in Y − f ( F ), then f − 1 ( y ) ⊆ X − f − 1 ( f ( F )) ⊆ X − F , and X − F is γ -op en in X. Hence by the hy- p othesis, there exists a γ ∗ -semi-op en set V y con taining y such that f − 1 ( V y ) ⊆ X − F . Th is implies that v ∈ V y ⊆ Y − f ( F ). W e obtain that Y − f ( F ) = S { V y : y ∈ Y − f ( F ) } is γ ∗ -semi-op en in Y, since union of an y collectio n of γ ∗ -semi-op en sets is γ ∗ -semi-op en. T herefore f ( F ) is γ ∗ -semi-closed. 5 γ -Semi-Con tin uous F unctions Theorem 5.1. Let f : X → Y be a function and γ is an op en op er ation. Then the follo wing are equiv alen t: (1) f is γ -semi-con tinuous. (2) int γ ( cl γ ( f − 1 ( B ))) ⊆ f − 1 ( cl γ ( B )) for eac h su bset B of Y. (3) f ( int γ ( cl γ ( A ))) ⊆ cl γ ( f ( A )) for eac h su bset A of X. Pro of. (1) ⇒ (2). Let B b e an arbitrary su b set of Y. Then by (1), f − 1 ( cl γ ( B )) is a γ ∗ -semi- closed set of X. S ince B ⊆ cl γ ( B ), by Lemma 3.12, we get f − 1 ( cl γ ( B )) ⊇ int γ ( cl γ ( f − 1 ( cl γ ( B )))) ⊇ int γ ( cl γ ( f − 1 ( B )))implies that int γ ( cl γ ( f − 1 ( B ))) ⊆ f − 1 ( cl γ ( B )). (2) ⇒ (3). L et A b e an arb itrary sub set of X. Put B = f ( A ). Th en A ⊆ f − 1 ( B ). T herefore b y hypothesis, we h a ve int γ ( cl γ ( A )) ⊆ in t γ ( cl γ ( f − 1 ( B ))) ⊆ f − 1 ( cl γ ( B )). C onsequent ly , w e hav e f ( int γ ( cl γ ( A ))) ⊆ f f − 1 ( cl γ ( B )) ⊆ cl γ ( B ) = cl γ ( f ( A )). This give s (3). (3) ⇒ (1). Let F b e an arbitrary γ -closed set of Y. Put A = f − 1 ( F ), then f ( A ) ⊆ F . Ther efore b y h yp othesis, we h a ve f ( int γ ( cl γ ( A ))) ⊆ cl γ ( f ( A )) ⊆ cl γ ( F ) = F ..... (**) 9 By (**), we ha ve int γ ( cl γ ( A )) ⊆ f − 1 f ( int γ ( cl γ ( A ))) ⊆ f − 1 ( cl γ ( f ( A ))) ⊆ f − 1 ( cl γ ( F )) = f − 1 ( F ), or int γ ( cl γ ( A )) ⊆ f − 1 ( F ),. By Lemma 3.12, f − 1 ( F ) is a γ ∗ -semi-closed set in X. T his implies that f is γ -semi-con tin u ous. Definition 5.2. Let X be a space A ⊆ X and p ∈ X . Then p is a γ ∗ -semi-limit p oint of A, for all γ ∗ -semi-op en set U con taining p su c h that U ∩ ( A − { p } ) 6 = φ . Th e set of all γ ∗ -semi-limit p oint of A is said to b e γ ∗ -semi-deriv ed set of A and is d enoted b y sd γ ∗ ( A ). Clearly if A ⊆ B then sd γ ∗ ( A ) ⊆ sd γ ∗ ( B )..... (I) Remark 5.3. F rom th e definition, it follo ws th at p is a γ ∗ -semi-limit p oint of A if and only if p ∈ scl γ ∗ ( A − { p } ). Theorem 5.4. The γ ∗ -semi-deriv ed set, sd γ ∗ , has the f ollo wing prop erties: (1) scl γ ∗ ( A ) = A ∪ sd γ ∗ ( A ). (2) sd γ ∗ ( A ∪ B ) = s d γ ∗ ( A ) ∪ sd γ ∗ ( B ). In general (3) S i sd γ ∗ ( A i ) = sd γ ∗ ( S i ( A i )). (4) sd γ ∗ ( sd γ ∗ ( A )) ⊆ sd γ ∗ ( A ). (5) scl γ ∗ ( sd γ ∗ ( A )) = sd γ ∗ ( A ). Pro of. (1) Let x ∈ scl γ ∗ ( A ). Th en x ∈ C , for ev ery γ ∗ -semi-closed sup erset C of A. Now (i) If x ∈ A , then x ∈ A ∪ sd γ ∗ ( A ). (ii) If x / ∈ A , th en w e pr o ve that x ∈ scl γ ∗ ( A ). T o prov e (ii), supp ose U is γ ∗ -semi-op en set cont aining x. Then U ∩ A 6 = φ , for otherwise, A ⊆ X − U = C , where C is a γ ∗ -semi-closed sup erset of A n ot con taining x. Th is co n tra- dicts the fact that x b elongs to ev ery γ ∗ -semi-closed sup erset C of A. Therefore x ∈ sd γ ∗ ( A ) give s x ∈ A ∪ sd γ ∗ ( A ). Con v ers ely , supp ose that x ∈ A ∪ sd γ ∗ ( A ), we show that x ∈ scl γ ∗ ( A ). I f x ∈ A then x ∈ scl γ ∗ ( A ). If x ∈ sd γ ∗ ( A ), then w e sho w that x is in ev ery γ ∗ -semi-closed sup erset of A. W e supp ose otherwise that there is γ ∗ -semi-closed su p erset C of A not con taining x. T hen x ∈ X − C = U (say), whic h is γ ∗ -semi-op en and U ∩ A = φ . This implies that x / ∈ sd γ ∗ ( A ). This con tradiction pro v es that x ∈ scl γ ∗ ( A ). Consequently s cl γ ∗ ( A ) = A ∪ sd γ ∗ ( A ). This p ro v es (1). (2) sd γ ∗ ( A ∪ B ) ⊆ sd γ ∗ ( A ) ∪ sd γ ∗ ( B ). 10 Let x ∈ sd γ ∗ ( A ∪ B ). Then x ∈ s cl γ ∗ (( A ∪ B ) − { x } ) or x ∈ scl γ ∗ (( A − { x } ) ∪ ( B − { x } ) implies x ∈ scl γ ∗ ( A − { x } ) or x ∈ scl γ ∗ ( B − { x } ). This giv es x ∈ sd γ ∗ ( A ) or x ∈ sd γ ∗ ( B ) . Therefore x ∈ sd γ ∗ ( A ) ∪ sd γ ∗ ( B ) . Th is pr o ves sd γ ∗ ( A ∪ B ) ⊆ s d γ ∗ ( A ) ∪ sd γ ∗ ( B ) Con v ers e follo ws d irectly by using the prop ert y(I). (3) Th e pr o of is immediate by pr op erty (I). (4) Supp ose that x / ∈ sd γ ∗ ( A ). Then x / ∈ scl γ ∗ ( A − { x } ). This imp lies that there is γ ∗ -semi- op en set U such th at x ∈ U and U ∩ ( A − { x } ) = φ . W e pro ve that x / ∈ sd γ ∗ ( sd γ ∗ ( A )). S upp ose on the contrary that x ∈ sd γ ∗ ( sd γ ∗ ( A )). Then x ∈ s cl γ ∗ ( sd γ ∗ ( A ) − { x } ). Since x ∈ U , we ha v e U ∩ ( sd γ ∗ ( A ) − { x } ) 6 = φ . Therefore there is a q 6 = x suc h that q ∈ U ∩ ( s d γ ∗ ( A )). It follo ws that q ∈ ( U − { x } ) ∩ ( sd γ ∗ ( A ) − { x } ). Hence ( U − { x } ) ∩ ( sd γ ∗ ( A ) − { x } ) 6 = φ , a con tradiction to the fact that ( U ∩ ( sd γ ∗ ( A ) − { x } ) = φ . This imp lies that x / ∈ sd γ ∗ ( sd γ ∗ ( A )) and so s d γ ∗ ( sd γ ∗ ( A )) ⊆ sd γ ∗ ( A ). This pr o ves (4). (5) Th is is a consequence of (1), (2) and (4). Theorem 5.5 [5]. Let f : X → Y b e a fun ction. Then the follo wing are equiv alen t: (1) f : X → Y is γ -semi-c on tin u ou s . (2) scl γ ∗ ( f − 1 ( A )) ⊆ f − 1 ( cl γ ( A )) for eac h s ubset A of Y. Theorem 5.6. Let f : X → Y b e a fu nction and γ is an op en op eration. Then the f ollo w- ing are equiv alen t: (1) f : X → Y is γ -semi-c on tin u ou s . (2) f ( sd γ ∗ ( A )) ⊆ cl γ ( f ( A )) for an y sub set A of X. Pro of. (1) ⇒ (2). Supp ose that f is γ -semi-con tin uous. Let A b e an y set in X. S ince cl γ ( f ( A )) is γ -closed in Y. f − 1 ( cl γ ( A )) is γ ∗ -semi-closed in X. A ⊆ f − 1 ( f ( A )) ⊆ f − 1 ( cl γ ( f ( A ))) giv es scl γ ∗ ( A ) ⊆ scl γ ∗ ( f − 1 ( cl γ ( f ( A )))) = f − 1 ( cl γ ( f ( A ))). Therefore f ( sd γ ∗ ( A )) ⊆ f ( scl γ ∗ ( A )) ⊆ f f − 1 ( cl γ ( f ( A ))) ⊆ cl γ ( f ( A )). Consequen tly , f ( s d γ ∗ ( A )) ⊆ cl γ ( f ( A )). (2) ⇒ (1). Su pp ose that f ( sd γ ∗ ( A )) ⊆ cl γ ( f ( A )), for A ⊆ X . Let B b e any γ -cl osed subset of Y. W e sh o w that f − 1 ( B ) is γ ∗ -semi-closed in X. By hyp othesis, f ( sd γ ∗ ( f − 1 ( B ))) ⊆ cl γ ( f ( f − 1 ( B ))) ⊆ cl γ ( B ) = B or f ( sd γ ∗ ( f − 1 ( B ))) ⊆ B giv es s d γ ∗ ( f − 1 ( B )) ⊆ f − 1 f ( sd γ ∗ ( f − 1 ( B ))) ⊆ f − 1 ( B ) o r sd γ ∗ ( f − 1 ( B )) ⊆ f − 1 ( B ) implies f − 1 ( B ) is γ ∗ -semi-closed in X. Thus f is γ -semi-con tin uous. 11 Theorem 5.7 [5]. Let f : X → Y b e a fun ction and x ∈ X . Then f is γ -semi-con tin u ous if and only if for eac h γ -op en s et B cont aining f(x), there exists A ∈ S O γ ∗ ( X ) suc h that x ∈ A and f ( A ) ⊆ B , where γ is a r egular op eration. W e use Th eorem 5.7 and pr o ve th e follo w ing: Theorem 5.8. Let f : X → Y b e an injectiv e function. If f is γ -semi-con tin u ous th en f ( sd γ ∗ ( A )) ⊆ ( f ( A )) d γ for ev ery A ⊆ X , where γ is a regular op eration. Pro of. Supp ose that f is γ -semi-con tinuous. Let A ⊆ X , a ∈ sd γ ∗ ( X ) and V b e a γ -op en-n b d of f ( a ). Since f is γ -semi-con tinuous then by Theorem 5.7, there exists a γ -semi-op en-nb d U of a such that f ( U ) ⊆ V . But a ∈ sd γ ∗ ( A ), therefore there exists an elemen t a 1 ∈ U ∩ A suc h that a 6 = a 1 ; t hen f ( a 1 ) ∈ f ( A ) and since f is an injection f ( a ) 6 = f ( a 1 ). T h us ev ery γ -op en-nb d V of f ( a ) cont ains an element f ( a 1 ) of f ( A ) d ifferen t from f ( a ). Consequent ly f ( a ) ∈ ( f ( A )) d γ . W e ha ve therefore, f ( sd γ ∗ ( A )) ⊆ ( f ( A )) d γ . The follo w ing theorem follo w s from T heorem 5.6: Theorem 5.9. Let f : X → Y b e a function. If f or ev ery A ⊆ X , f ( sd γ ∗ ( A )) ⊆ ( f ( A )) d γ , then f is γ -semi-con tinuous, where γ is an op en op eration. Theorem 5.10. A fu nction f : X → Y is γ -semi-con tinuous if and only if f − 1 ( int γ ( B )) ⊆ sint γ ∗ ( f − 1 ( B )))), for eac h B ⊆ Y , where γ is a regular op eration. Pro of. F or an y B ⊆ Y , int γ ( B ) = Y − cl γ ( Y − B ) [14]. This implies f − 1 ( int γ ( B )) = f − 1 ( Y − cl γ ( Y − B )) = X − f − 1 ( cl γ ( Y − B )). Since f is γ -semi-con tin u ous, b y Theorem 5.5 we ha v e scl γ ∗ ( f − 1 ( Y − B )) ⊆ f − 1 ( cl γ ( Y − B )). Hence f − 1 ( int γ ( B )) ⊆ X − scl γ ∗ ( f − 1 ( Y − B )). Thus f − 1 ( int γ ( B )) ⊆ X − s cl γ ∗ ( X − f − 1 ( B )). Hence f − 1 ( int γ ( B )) ⊆ X − s cl γ ∗ ( X − f − 1 ( B )) = sint γ ∗ ( f − 1 ( B )))). Con v ers ely , let B b e an arbitrary γ -op en set in Y. 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