On some mappings in topological spaces

Vol. 1, No. 4, 2008, (22-29)

ISSN 1307-5543 – www.ejpam.com

On some mappings in topological spaces

Bashir Ahmad

, Sabir Hussain

, Takashi Noiri

1_{Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University,}

Multan, Pakistan.

Present Address: Department of Mathematics, King Abdul Aziz University P. O. Box 80203, Jeddah 21589, Saudi Arabia.

2_{Department of Mathematics, Islamia University Bahawalpur, Pakistan.}

3_{2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142, Japan.}

Abstract. In this paper, we continue studying the properties ofγ-semi-continuous andγ-semi-open functions introduced in _[5_], _[10_]. We also introduce _γ∗_{-irresolute functions and γ -pre-semi-open}

(closed) functions and discuss their properties.

AMS subject classifications: 54A05, 54A10, 54D10.

Key words: _γ-closed (open),_γ-closure , _γ∗_{-semi-closed (open), γ}∗_{-semi-closure,γ-regular, γ}∗

-semi-interior, _γ-semi-continuous functions, _γ-semi-open (closed) functions, _γ∗_{-irresolute functions, γ}

-pre-semi-open(closed) functions.

1. Introduction

A. Csaszar[7,8]defined generalized open sets in generalized topological spaces. In 1975, Maheshwari and Prasad[13]introduced concepts of semi- T1-spaces and semi-R0-spaces. In 1979, S. Kasahara [11] defined an operation α on topological spaces. In 1992 (1993), B. Ahmad and F.U. Rehman[1], [15] introduced the notions of γ-interior, γ-boundary and γ -exterior points in topological spaces. They also studied properties and characterizations of

(γ,β)-continuous mappings introduced by H. Ogata[14]. In 1999 (resp. 2005), B. Ahmad and S. Hussain introduced the concept of γ∗-regular spaces ( resp. γ₀-compact, γ-normal spaces) and explored their many interesting properties. They initiated and discussed the concepts ofγ∗-semi-open sets which generalizesγ-open sets introduced and discussed by H. Ogata[14], γ∗-semi-closed sets,γ∗-semi-closure,γ∗-semi-interior point in topological spaces [5], [9]. In 2006, they introduced Λγs-set and Λs

γ

-set by using γ∗-semi-open sets. More-over, they introduced theγ-semi-continuous functions andγ-semi-open (closed) functions in topological spaces and established several interesting properties.

In this paper, we continue studying the properties ofγ-semi-continuous andγ-semi-open functions introduced by B. Ahmad and S. Hussain[5],[10]. We also introduceγ∗-irresolute

∗_{Corresponding author.Email addresses: drbashir9@gmail.com(B. Ahmad)}

sabiriub@yahoo.com(S. Hussain),jt.noiri@nifty.com(T. Noiri)

functions andγ-pre-semi-open (closed) functions and discuss their properties.

Hereafter, we shall write spaces in place of topological spaces in the sequel.

2. Preliminaries

We recall some definitions and results used in this paper to make it self-contained. Definition[11]. Let (X,τ) be a space. An operationγ: τ→P(X) is a function fromτto the power set of X such that V_⊆Vγ , for each V_∈τ, whereVγ denotes the value ofγat V. The operations defined byγ(G)=G,γ(G)=cl(G) andγ(G)=intcl(G) are examples of operation γ.

Definition[14]. Let A_⊆X. A point x_∈A is said to beγ-interior point of A iff there exists an open nbd N of x such thatNγ⊆A and we denote the set of all such points byint_γ(A). Thus

int_γ (A)=_{x ∈A:x ∈N ∈τandNγ⊆A} ⊆A.

Note that A isγ-open[14]iff A=int_γ(A). A set A is calledγ- closed[14]iff X-A isγ-open. Definition[14]. A point x_∈ X is called a γ-closure point of A_⊆ X, if Uγ∩A6= φ, for each open nbd U of x. The set of allγ-closure points of A is calledγ-closure of A and is denoted by cl_γ(A). A subset A of X is calledγ-closed, ifcl_γ(A)_⊆A. Note thatcl_γ(A) is contained in every γ-closed superset of A.

Definition[14]. An operationγ onτ is said be regular, if for any open nbds U,V of x∈X, there exists an open nbd W of x such thatUγ_∩Vγ_⊇Wγ.

Definition[14]. An operation γon τis said to be open, if for every nbd U of each x ∈X, there existsγ-open set B such that x ∈BandUγ⊇B.

We defined[9]γ∗-semi-open sets usingγ-open sets in the sense of H. Ogata[14]as : Definition[9].A subset A of a space (X,τ) is said to be aγ∗-semi-open set, if there exists aγ -open set O such thatO⊆A⊆cl_γ(O). The set of allγ∗-semi-open sets is denoted bySO_γ∗(X). Definition[5].A functionf :(X,τ)_→(Y,τ)is said to beγ-semi-continuous if for anyγ-open set B of Y, f−1(B)isγ∗-semi-open in X.

Definition[5]. A function f : X →Y is said to be γ-semi-open (closed) if for each γ-open (closed) set U in X, f(U)isγ∗-semi-open (closed) in Y.

Definition[5].A set A in a space X is said to beγ∗-semi-closed if there exists aγ-closed set F such thatint_γ(F)⊆A⊆F.

Proposition[5].A subset A of a space X isγ∗-semi-closed ifX−Aisγ∗-semi-open.

Definition[10]. A subset A of a space X is said to be γ-semi-nbd of a point x ∈X if there exists aγ∗-semi-open set U such thatx ∈U ⊆A.

3.

γ

-irresolute functions

Definition 3.2[5]. An operationγis said to be semi-regular, if for any semi-open sets U and V containingx ∈X , there exists a semi-open set W containing x such thatUγ∩Vγ⊇Wγ. Theorem 3.3. A function f :X →Y isγ∗-irresolute if and only if for each x in X, the inverse of everyγ-semi-nbd of f(x)is aγ-semi-nbd of x, whereγis a semi regular operation . Proof. Let x _∈ X and A a γ-semi-nbd of f(x) . By definition of γ-semi-nbd , there exists V ∈ SO_γ∗(Y) such that f(x) ∈ V ⊆ A. This implies that x ∈ f−1(V) ⊆ f−1(A). Since f isγ∗-irresolute, so f−1(V) ∈SO_γ∗(X). Hence f−1(A) is a γ-semi-nbd of x. This proves the necessity.

Conversely, letA∈SO_γ∗(Y). PutB= f−1(A). Let x∈B, then f(x)∈A. Clearly,

A ( beingγ∗-semi-open) is aγ-semi-nbd of f(x). So by hypothesis,B= f−1(A)is aγ -semi-nbd of x. Hence by definition , there exists B_{x ∈} SO_γ∗(X) such that x ∈ B_x ⊆ B . Thus B = S_x∈BBx. Since γ is semi regular, therefore it follows that B is γ∗-semi-open in X [5].

Therefore f isγ∗-irresolute.

Remark 3.4.γ-semi-nbd of x may be replaced byγ∗-semi-open nbd of x in Theorem 3.3. Theorem 3.5. A function f : X →Y isγ∗-irresolute if and only if for each x in X and each γ-semi-nbd A of f(x) , there is a γ-semi-nbd B of x such that f(B) ⊆A, whereγ is a semi regular operation.

Proof. Let x ∈X and A aγ-semi-nbd of f(x) . Then there existsOf(x)∈SOγ∗(Y)such that f(x)∈O_f(x)⊆A. It follows that x∈ f−1(O_f(x))⊆ f−1(A). By hypothesis,

f−1(O_f(x)) _∈ SO_γ∗(X) . Let B = f−1(A). Then it follows that B is γ-semi-nbd of x and f(B) = f f−1(A)⊆A. This proves the necessity.

Conversely, let U ∈SO_γ∗(Y). TakeO= f−1(U). Let x ∈O, then f(x)∈U . Thus U is a γ-semi-nbd of f(x). So by hypothesis, there exists aγ-semi-nbdV_x of x such that

f(Vx)⊆U . Thus it follows that x ∈Vx ⊆ f−1f(Vx)⊆ f−1(U) =O . So Vx is aγ-semi-nbd

of x, which implies there exists anO_x ∈SO_γ∗(X) such that x ∈O_x ⊆O. ThusO=S_x∈oO_x . Sinceγis semi regular, then it follows that O isγ∗-semi-open in X. Thus f isγ∗-irresolute. Definition 3.6[10].Let X be a space.A⊆X andp∈X. Then p is called aγ∗-semi-limit point of A ifU∩(A− {p})6=φ, for anyγ∗-semi-open set U containing p. The set of allγ∗-semi-limit points of A is called aγ∗-semi-derived set of A and is denoted bysd_γ∗(A).

Clearly ifA⊆Bthensd_γ∗(A)⊆sd_γ∗(B)... (*)

DenoteΓ(X), the set of all monotone operators on X. Then we have:

Definition 3.7[9].Let A be a subset of space X andγ_∈Γ(X). The intersection of allγ∗ -semi-closed sets containing A is calledγ∗-semi-closure of A and is denoted byscl_γ∗(A).

Remark 3.8.From the definition 3.6, it follows that p is aγ∗-semi-limit point of A if and only if p∈scl_γ∗(A− {p}).

Theorem 3.9[10].For anyA,B⊆X , theγ∗-semi-derived sets have the following properties: (1)scl_γ∗(A) =A∪sd_γ∗(A).

(2)S_isd_γ∗(A_i) =sd_γ∗(

S

iAi).

(3)sd_γ∗(sd_γ∗(A))⊆sd_γ∗(A). (4)scl_γ∗(sd_γ∗(A)) =sd_γ∗(A).

func-tions as:

Theorem 3.10.A function f :X →Y isγ∗-irresolute if and only if f(sd_γ∗(A))⊆scl_γ∗(f(A)), for allA⊆X .

Proof. Let f :X →Y beγ∗-irresolute. LetA⊆X , and x ∈sd_γ∗(A). Assume that f(x)∈/ f(A) and let V denote aγ-semi-nbd of f(x). Since f isγ∗-irresolute, so by Theorem 3.5, there exists aγ-semi-nbd U of x such that f(U)⊆V. From x ∈sd_γ∗(A), it follows that U∩A6=φ; there exists, therefore, at least one elementa∈U∩Asuch that f(a)∈ f(A)and f(a)∈V . Since f(x)∈/ f(A), we have f(a)=6 f(x). Thus everyγ-semi-nbd of f(x)contains an element f(a) of f(A)different from f(x). Consequently, f(x)∈sd_γ∗(f(A)).This proves the necessity.

Conversely, suppose that f is not γ∗-irresolute. Then by Theorem 3.5, there exists x ∈X and aγ-semi-nbd V of f(x)such that every γ-semi-nbd U of x contains at least one element a_∈U for which f(a)_∈/V . PutA=_{a_∈X :f(a)_∈/V_}. Since f(x)_∈V , thereforex _∈/Aand hence f(x)∈/ f(A). Since f(A)∩(V−f(x)) =φ, therefore f(x)∈/sd_γ∗(f(A)). It follows that f(x)∈f(sd_γ∗(A))−(f(A)∪sd_γ∗(f(A)))6=φ, which is a contradiction to the given condition. This proves sufficiency.

Theorem 3.11.Let f :X →Y be one-to-one function. Then f isγ∗-irresolute if and only if f(sd_γ∗(A))⊆sd_γ∗(f(A)), for allA⊆X.

Proof. LetA⊆X , x ∈sd_γ∗(A)and V be aγ-semi-nbd of f(x). Since f isγ∗-irresolute, so by Theorem 3.5, there exists aγ-semi-nbd U of x such that f(U)⊆V. But x∈sd_γ∗(A)gives there exists an elementa∈U∩Asuch thata6= x. Clearly f(a)∈ f(A)and since f is one to one,

f(a)6= f(x). Thus everyγ-semi-nbd V of f(x) contains an element f(a) of f(A) different from f(x). Consequently f(x) _∈sd_γ∗(f(A)). Therefore we have f(sd_γ∗(A))⊆ sd_γ∗(f(A)). This proves the necessity.

Sufficiency follows from Theorem 3.10. This completes the proof.

H. Ogata[14]defined the notion ofγ-T₂ spaces in topological spaces, we generalize this notion and define:

Definition 3.12. A space X is said to beγ-semi-T₂ , if for each two distinct points x,y _∈X there existU,V ∈SO_γ∗(X)such thatx ∈U, y∈V andU∩V =φ.

Theorem 3.13.If f :X →Y is aγ∗-irresolute injection and Y isγ-semi-T2, then X is γ-semi-T₂.

Proof. Letx1and x2be two distinct points of X. Since f is injective, therefore

f(x1) 6= f(x2) . Since Y is γ-semi-T2 , there exist V1,V2 ∈ SOγ∗(Y) such that f(x₁) ∈ V₁, f(x₂)_∈V₂ andV_1∩V₂ =φ. Then x_{1 ∈} f−1(V₁), x_{2 ∈} f−1(V₂)and f−1(V₁)_∩f−1(V₂) =φ. Since f isγ∗-irresolute, so f−1(V1),f−1(V2)∈SOγ∗(X). This proves that X isγ-semi-T₂.

4.

γ

-pre-semi-open functions

Remark 4.2. The class of γ-pre-semi-open functions is a subclass of class of γ-semi-open functions defined in[5].

Note that if f :X →Y and g:Y →Z be any twoγ-pre-semi-open functions, then the compositiong o f :X →Zis aγ-pre-semi-open function.

The following theorem is easy to prove:

Theorem 4.3. A function f :X →Y isγ-pre-semi-open if and only if for each x ∈X and for everyA_∈SO_γ∗(X)such thatx∈A, there existsB∈SO_γ∗(Y)such that f(x)∈BandB⊆ f(A). Theorem 4.4. A function f :X →Y isγ-pre-semi-open if and only if for each x ∈X and for everyγ-semi-nbd U of x in X, there exists aγ-semi-nbd V of f(x)in Y such thatV ⊆ f(U). Proof. Let U be aγ-semi-nbd of x _∈X . Then by definition there existsW _∈SO_γ∗(X) such that x∈W⊆U . Then f(x)∈f(W)⊆f(U). Since f isγ-pre-semi-open , therefore

f(W)∈SO_γ∗(Y). HenceV = f(W)is aγ-semi-nbd of f(x)andV ⊆ f(U).

Conversely, let U _∈SO_γ∗(X)and x ∈U . Then U is aγ-semi-nbd of x. So by hypothesis, there exists aγ-semi-nbd V of f(x)such that f(x)∈V ⊆ f(U). That is, f(U)is aγ-semi-nbd of f(x). Thus f(U)is aγ-semi-nbd of each of its points. Thus f(U)isγ∗-semi-open. Hence f isγ-pre-semi-open. This completes the proof.

Definition 4.5[9].Let A be a subset of a space X andγ_∈Γ(X). The union of allγ∗-semi-open sets of X contained in A is calledγ∗-semi-interior of A and is denoted bysint_γ∗(A).

Theorem 4.6.A function f :X _→Y isγ-pre-semi-open if and only if f(sint_γ∗(A))⊆sint_γ∗(f(A)), for allA⊆X .

Proof. Let x ∈sint_γ∗(A). Then there exists U ∈SO_γ∗(X) such that x ∈ U ⊆ A. So f(x)∈ f(U)⊆ f(A). Since f isγ-pre-semi-open , therefore f(U)isγ∗-semi-open in Y. Hence f(x)∈

sint_γ∗(f(A)). Thus f(sint_γ∗(A))⊆sint_γ∗(f(A)). This proves necessity.

Conversely letU ∈SO_γ∗(X). Then by hypothesis, f(U) = f(sint_γ∗(U))⊆sint_γ∗(f(U))⊆ f(U)or f(U)⊆sint_γ∗(f(U))⊆ f(U). This implies f(U)isγ∗-semi-open in Y. So f isγ -pre-semi-open.

We use Theorem 4.6 and prove:

Theorem 4.7.A function f :X →Y isγ-pre-semi-open if and only if sint_γ∗(f−1(A))⊆ f−1(sint_γ∗(A)), for allA⊆X .

Proof.Let A be any subset of Y. Clearly,sint_γ∗(f−1(A))isγ∗-semi-open in Y. Also f(sint_γ∗(f−1(A))⊆ f(f−1(A))⊆A. Since f isγ-pre-semi-open, by Theorem 4.6 we have

f(sint_γ∗(f−1(A))⊆sint_γ∗(A). Therefore,

sint_γ∗(f−1(A))⊆ f(f−1(sint_γ∗(f−1(A))))⊆ f−1(sint_γ∗(A))or sint_γ∗(f−1(A))⊆ f−1(sint_γ∗(A)). This proves necessity.

Conversely, letB_⊆X . By hypothesis, we obtain

sint_γ∗(B)⊆sint_γ∗(f−1f(B))⊆ f−1(sint_γ∗f(B)). This implies that f(sint_γ∗(B))⊆ f(sint_γ∗(f−1f(B)))⊆ f(f−1(sint_γ∗f(B))) ⊆ sint_γ∗(f(B)). Consequently, f(sint_γ∗(B)) ⊆ sint_γ∗(f(B)) , for all

We use Theorem 4.7 and prove:

Theorem 4.8.A function f :X →Y isγ-pre-semi-open if and only if f−1(scl_γ∗(A))⊆scl_γ∗(f−1(A)), for allA⊆X .

Proof. LetA⊆Y . By Theorem 4.7 ,sint_γ∗(f−1(Y −A))⊆ f−1(sint_γ∗(Y−A)). This implies thatsint_γ∗(X−f−1(A))⊆ f−1(sint_γ∗(Y−A)). Assint_γ∗(A) =X−scl_γ∗(X−A)[5], therefore X −scl_γ∗(f−1(A))⊆ f−1(Y −scl_γ∗(A)). or X −scl_γ∗(f−1(A)) ⊆ X − f−1(scl_γ∗(A)). Hence

f−1(scl_γ∗(A))⊆scl_γ∗(f−1(A)). This proves necessity.

Conversely, let A ⊆ Y . By hypothesis, f−1(scl_γ∗(Y −A)) ⊆ scl_γ∗(f−1(Y −A)) . This implies X −scl_γ∗(f−1(Y −A)) ⊆ X − f−1(scl_γ∗(Y −A)) . Hence X −scl_γ∗(X − f−1(A)) ⊆ f−1(Y−(scl_γ∗(Y−A))). This givessint_γ∗(f−1(A))⊆ f−1(sint_γ∗(A)). Now from Theorem 4.7, it follows that f isγ-pre-semi-open. This completes the proof.

Definition 4.9[14].A function f :(X,τ)→(Y,τ)is said to be(γ,β)-continuous, if for each x _∈X and each open set V containing f(x), there exists an open set U such that x _∈U and f(Uγ)⊆Vβ , whereγandβare operations onτandδrespectively.

Definition 4.10[14]. A function f :(X,τ)→(Y,τ) is said to be(γ,β)-open (closed), if for anyγ-open (closed) set A of X, f(A)isβ-open(closed) in Y.

In[10], we proved the following Theorem:

Theorem 4.11[10]. If f : X _→ Y is a (γ,β)-open and (γ,β)-continuous function, then f−1(B)∈SO_γ∗(X), for everyB∈SO_β∗(Y), whereβis an open operation.

We use this theorem and prove:

Theorem 4.12. Let X, Y and Z be three spaces, and let f : X → Y and g :Y → Z be two functions such that g o f :X →Z aγ-pre-semi-open function. Then

(1) If f is a(γ,β)-open and(γ,β)-continuous surjection, thengisβ-pre-semi-open. (2) If gis a(β,δ)-open and(β,δ)-continuous injection, then f isγ-pre-semi-open.

Proof.(1). Let V be an arbitrary γ∗-semi-open set in Y. Since f is a (γ,β)-open and (γ,β) -continuous, then by Theorem 4.11, f−1(V)is aγ∗-semi-open set in X. Alsog o f isγ -pre-semi-open and f is surjection, we have g(V) = (g o f)(f−1(V))is a β∗-semi-open set in Z. This proves thatgisβ-pre-semi-open.

(2). Since gis injective so for every subset A of X, f(A) =g−1(g(f(A))). Let U be an arbitrary γ∗_{-semi-open set in X. Then(g o f)(U) isγ}∗_{-semi-open. Since g is a(β,δ)-open and(β,δ)} -semi-continuous, therefore by above Theorem 4.11, f(U) is γ∗-semi-open in Y. This shows that f isγ-pre-semi-open.

Theorem 4.13. Let f : X → Y and g : Y → Z be two functions such that g o f : X → Z isγ∗-irresolute. Then

(1) If gis aβ-pre-semi-open injection, then f isγ∗-irresolute. (2) If f is aγ-pre-semi-open surjection, then gisβ∗-irresolute.

Proof.(1). Let U ∈SO_β∗(Y). Since g isβ-pre-semi-open, then g(U)∈SO_δ∗(Z). Also g o f is γ∗_{-irresolute, and therefore;(g o f)}−1_(g(U))

∈SO_γ∗(X). Since g is injective(g o f)−1(g(U)) =

proves that f isγ∗-irresolute. This proves (1).

(2). Let V ∈ SO_δ∗(Z) . Since g o f is γ∗-irresolute, then (g o f)−1(V) ∈ SO_γ∗(X). Also f is γ-pre-semi-open, so f(g o f)−1(V) is β∗-semi-open in Y. Since f is surjective, we obtain f o(g o f)−1(V) = f o(f−1o g−1)(V) = (f o f−1)o g−1(V) = g−1(V). It follows that g−1(V)∈

SO_γ∗(Y). This proves that g is aβ∗-irresolute function.

5.

γ

-pre-semi-closed functions

Definition 5.1. A function f :X_→Y isγ-pre-semi-closed if and only if the image set f(A) isγ∗-semi-closed, for eachγ∗-semi-closed subset A of X.

It is obvious that the composition of twoγ-pre-semi-closed mappings is aγ-pre-semi-closed mapping.

Theorem 5.2.A function f :X →Y isγ-pre-semi-closed if and only if scl_γ∗f(B)⊆ f(scl_γ∗(B)), for every subset B of X.

Proof. Suppose f isγ-pre-semi-closed and letB⊆X. Since f isγ-pre-semi-closed , therefore f(scl_γ∗(B))isγ∗-semi-closed in Y. Since f(B)⊆ f(scl_γ∗(B)), we obtain

scl_γ∗(f(B))⊆ f(scl_γ∗(B)). This proves necessity.

Conversely, suppose A is a γ∗-semi-closed set in X. By hypothesis, we obtain f(A) ⊆

scl_γ∗(f(A)) ⊆ f(scl_γ∗(A)) = f(A). Hence f(A) = scl_γ∗(f(A)). Thus f(A) is γ∗-semi-closed set in Y. This proves that f isγ-pre-semi-closed.

Theorem 5.3.A function f :X →Y isγ-pre-semi-closed if and only if int_γ(cl_γ(f(B)))⊆ f(scl_γ∗(B)), for every subset B of X.

Proof. Suppose f is γ-pre-semi-closed and B is any subset of X. Then f(scl_γ∗(B)) is γ∗ -semi-closed in Y. This implies that there exists a γ-closed subset A of Y such that int_γ(A)⊆

f(scl_γ∗(B))⊆A. This gives thatint_γcl_γ(f(scl_γ∗(B)))⊆int_γ(A)⊆ f(scl_γ∗(B)). Thenint_γ(cl_γ(f(B))⊆ int_γ(cl_γ(f(scl_γ∗(B))))givesint_γ(cl_γ(f(B)))⊆ f(scl_γ∗(B)). This proves necessity.

Conversely, suppose A is a γ∗-semi-closed set in X. Then by hypothesis, int_γ(cl_γ(f(A))⊆

f(scl_γ∗(A)) = f(A). PutB=cl_γ(f(A)). Clearly, B isγ-closed in Y. This implies thatint_γ(B)⊆ f(A)⊆B. Hence f(A)isγ∗-semi-closed in Y. This implies f isγ-pre-semi-closed.

Theorem 5.4. A bijective function f : X → Y is aγ-pre-semi-closed if and only if for each subset A of Y and eachγ∗-semi-open set B in X containingf−1(A), there exists aγ∗-semi-open set C in Y containing A such that f−1(C)⊆B, whereγis a semi regular operation.

Proof. Let C = Y − f(X −B) . Then Cc = f(Bc) . Since f is γ-pre-semi-closed, so C is γ∗_{-semi-open. Since f}−1_{(A) ⊆ B, we haveC}c _{= f(B}c_{) ⊆ f(f}−1_(Ac_))⊆Ac _{. Hence,A⊆ C,}

and thus C is aγ-semi-open nbd of A. FurtherBc_⊆ f−1(f(Bc)) =f−1(Cc) = (f−1(C))c. This proves that f−1(C)⊆B .

Conversely, suppose F is a γ∗-semi-closed set in X. Let y ∈ Y − f(F) . Then f−1(y) ⊆

y ∈C ⊆Y−f(F). ThusY−f(F) =S_y∈Y−f(F)C . HenceY− f(F)isγ∗-semi-open set[5]. This proves that f isγ-pre-semi-closed.

We use Theorem 4.8 and give the following characterizations:

Theorem 5.5.Let f :X →Y be a bijective function. Then the following are equivalent: (1) f isγ-pre-semi-closed.

(2) f isγ-pre-semi-open. (3) f−1isγ∗-irresolute.

Proof. (1)_⇒(2)It is straightforward.

(2) _⇒ (3). Let A ⊆ X. Since f is γ-pre-semi-open, by Theorem 4.9, f−1(scl_γ∗(f(A))) ⊆ scl_γ∗(f−1f(A))impliesscl_γ∗(f(A))⊆ f(scl_γ∗(A)). Thusscl_γ∗(f−1)−1(A)is contained in

(f−1)−1(scl_γ∗(A)), for every subset A of X. Then again by Theorem 4.8 , it follows that f−1is γ∗_-irresolute.

(3)⇒(1). Let A be a γ∗-semi-closed set in X. ThenX −Aisγ∗-semi-open in X. Since f−1 is γ∗_{-irresolute,(f}−1₎−1_(X

−A)isγ∗-semi-open in Y. But(f−1)−1(X₋A) =f(X₋A) =Y₋f(A). Thus f(A)isγ∗-semi-closed in Y. This proves that f isγ-pre-semi-closed. This completes the proof.

References

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