ON β - I*P -OPEN SETS VIA PRE-LOCAL FUNCTIONS

ABSTRACT. The aim of this paper is to introduce the new concepts namely, pre local

functions, β - I*P - open set, β - I* - open set, weakly- β - I*P - open sets, β – IP- open set, β – I*P-

open set and their corresponding continuous functions are defined. Some properties and several

characterizations are investigated. Also, we obtained the relationships between these classes of

open sets.

*M.GANSTER, DEPARTMENT OF MATHEMATICS, GRAZ

UNIVERSITY, AUSTRIA.

* I.AROCKIA RANI, DEPARTMENT OF MATHEMATICS, NIRMALA

COLLEGE, COIMBATORE.

**A.A.NITHYA, DEPARTMENT OF MATHEMATICS, NIRMALA

COLLEGE, COIMBATORE.

KEY WORDS: pre local functions, β - I*P - open set, β - I* - open set, weakly- β - I*P - open sets,

β – IP- open set, β – I*P- open set, β -I*P-continuous functions.

1. INTRODUCTION

In 1983, M.E.Abd El monsef,S.N.EL-Deeb and R.A.Mahmoud[1],

introduced β-open sets and β-continuous mapping in general topology. In 1992, Jankovic

and Hamlett [5] introduced the notion of I - open sets in topological spaces. El - Monsef

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[2], investigated I -open sets and I - continuous functions. In 1996, Dontchev [3]

introduced the notion of pre - I - open sets and obtained a decomposition of I -continuity.

In 2002 E.Hatir and T.Noiri introduced the concepts of β - I – open and β - I-

continuous function.

In this paper, the notions of pre local functions, β - I*P - open set, β - I* -

open set, weakly- β - I*P - open sets, β – IP- open set, β – I*P- open set, and β - I*P -continuous

functions are introduced . The fundamental properties of such functions are studied.

Throughout this paper, cl(A) and int(A) denote the closure of A and the interior of A

respectively. Let(X,) be a topological space and an ideal of subsets of X. An ideal

topological space denoted by (X, ,) is a topological space (X, ) with an ideal on X.

For a subset A of X, A*()={xX : U ∩ A for each neighbourhood U of x } is called

the local function of A with respect to and . For every ideal topological space (X,,

), there exists a topology * (I),finer than , generated by the base B(,)= {U\G: U

and G}.Additionally, cl*(A) =AA* defines a Kuratowski closure operator for *(I).

Definition1.1 A subset S of a space (X, , ) is said to be

I-open [2] if S int (S*).

pre-I-open [3] if S int(cl*(S)).

semi-I-open [6] if S cl*(int(S)).

α-I-open [6] if S int (cl*(int(S))).

b -I-open [9] if S cl*(int(S)) int(cl*(S)).

β -I-open [6] if S cl(int(cl*(S))).

weakly-semi-I-open [7] if S cl*(int(cl(S))).

Definition 1.2 A subset S of a space (X,) is said to be

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pre-open [4] if S int(cl(S)).

semi-open [6] if S cl(int(S)).

α-open [8] if S int (cl(int(S))).

b-open [9] if S cl(int(S)) int(cl(S)).

β-open [1] if S cl(int(cl(S))).

2. β - I*P - OPEN SETS

Definition 2.1. Given a space (X,, ),a set operator ()*P:P(X)→P(X) called the pre local

function of with respect to τ is defined as follows; for A X, A*P(, ) = {xX:Ux ∩

A,for every Ux PN(x)},where PN(x) = {UPO(X); x U). When there is no

ambiguity, we will simply write A*P() or A*P for(A)*P (,).

Definition 2.2. A subset S of a space (X,, ) is said to be β - I* - open if

S cl*int(cl*(S))).

Definition 2.3. A subset S of a space (X,, ) is said to be β - I*P - open if

S cl*P int(cl*P (S))).

Definition 2.4. A subset S of a space (X,, ) is said to be weakly – β-I*P - open if

S cl*P int(cl(S))).

Definition 2.5. A subset S of a space (X,, ) is said to be β – I*P - open if

S cl*int(cl*P (S))).

Definition 2.6. A subset S of a space (X,, ) is said to be β – IP open if

S cl*P int(cl*(S))).

Definition 2.7. A subset S of a space(X,,) is said to be pre-I*P-open if

S int(cl*P(S)).

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Definition 2.8. A subset S of a space(X,,) is said to be semi -I*P-open if

S cl*P(int(S)).

Theorem 2.9. Let (X,,) be a space and A,B X, then the following statements hold:

1) *P() = .

2) (A B)*P() = A*P() B*P().

3) (A ∩ B)*P() A*P() ∩ B*P().

Corollary 2.10. Let (X,,)be a space and A X, then the following hold:

1) (A*P )*P A*P = p-cl(A*P ) p-cl(A).

2) (A*P )*P p-cl(A*P) cl(A).

3) A*P A* cl(A).

Remark 2.11. 1) The collection of all pre open subsets of a space(X,) fails to form an

ideal on X.

2) Each pre local function is local function .But the converse is not true, in general.

Example 2.12. Let X = {a,b,c}, = { ,X,{a}} and = {,{b}},A={b,c} then.

{b,c}* = {b,c}, but{b,c}*P = {c}.

If (X, , ) is a space ,we denote by *P() the topology on X generated by the

subbasis {U\E:U PO(X) and E }.

The closure operator in *P() denoted by cl*P, can be described as follows; for A X,

cl*P(A) = A A*P().

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Remark 2.13.1) Every semi- I*P –open set is β - I*P- open set.

2) Every pre - I*P -open set is β - I*P- open set.

3) Every β - I*P- open set is β -I-open set.

4) Every α - I*P- open set is β -I*P- open set.

However, the converses are not true.

Example 2.14. Let X = {a, b, c, d}, = {X,,{b},{c,d},{ b,c,d}}, = {, {d}} and A =

{a,b,c}.Then A is β- I*P- open but not α - I*P -open.

From the above definitions and remarks we have the following implications

pre I*P-open pre-I-open pre-open

open α-I*P –open β-I*P –open β-I –open β-I*P –open

semi -I*P –open semi-I –open semi–open

Proposition 2.15. For a subset of an ideal topological space, the following condition

hold.1) Every β - I*P- open set is weakly-β-I*P-open set.

2) Every β - I*P- open set is β –IP-open set.

3) Every β - IP- open set is weakly-β -I*P -open set.

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4) Every weakly-β-I*P-open set is weakly-semi-I -open set.

5) Every β – I*- open set is β - I- open set.

6) Every β – I*P- open set is β-I*-open set.

7) Every β – IP- open set is β-I*-open set.

8) Every weakly semi-I-open set is β-open set.

9) Every β-I-open set is β-open set.

Proof. The proof is obvious.

From the above proposition we have the following implications:

β- I*P –open weakly β-I*P –open weakly semi-I-open

β-IP-open β-open

β- I*P-open β- I* -open β- I –open

Proposition 2.16. Let S be a b-I-open set such that int S = . Then S is β-I- open set.

Proof. Since S cl*(int(S)) int(cl*(S)) = cl*() int(cl*(S)) = int(cl*(S))

cl(int(cl*(S)).

By β I*PO(X,) we denote the family of all β - I*P -open sets of space (X,,).

Lemma 2.17. Let A and B be subsets of space (X, , ) then,

1) If A B, then A*P B*P.

2) If U PO(X), then U∩A*P (U∩A)*P.

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3) A*P is pre closed in (X,).

Proof.1) and 2) are obvious.

3) Clearly A*P p-cl(A)*P . Let x p-cl (A*P)

(i.e) UX ∩ A*P for UX PN(x). Let y UX ∩ A*P, then UX PN(y) and UX ∩ A

which implies, x A*P . Therefore p-cl(A) A*P . Therefore, A*P = p-cl(A*P).

Theorem 2.18. Let (X,,) be an ideal topological space and A,B are subsets of X.

1) If Uα β I*PO(X,) for each α then {Uα: α } β I*PO(X,)

2) If A β I*PO(X,) and B PO(X) then A B β I*PO(X,)

Proof.1) Let Uα β I*PO(X,), we have Uα cl*P(int(cl*P(Uα))) for every α Δ.

αΔUα αΔ cl*Pint(cl*P(Uα)))

αΔ{(int(cl*P(Uα))) (int(cl*P(Uα)))*P}

{(αΔ (int(cl*P(Uα)))) (αΔ(int(cl*P(Uα))))*P}

{(αΔint(Uα (Uα)*P )) (αΔint(Uα (Uα)*P))*P }

{(int(αΔ Uα ) (αΔ (Uα)*P )) (int((αΔ Uα ) (αΔ (Uα)*P))*P }

{(int((αΔ Uα ) (αΔ Uα)*P )) (int((αΔ Uα ) (αΔ Uα)*P))*P }

= {cl*P(int((αΔ Uα ) (αΔ Uα)*P ))}

= cl*P(int(cl*P (αΔUα))) which implies, αΔ Uα β I*PO(X,).

2) Let A β I*PO(X,), B PO(X), then A cl*P(int(cl*P(A))) and by

Lemma 2.17, we obtain, A ∩ B (cl*P(int(cl*P(A)))) ∩ B

= {(int(cl*P(A)) (int(cl*P (A)))*P} ∩ B

={(int(A A*P)) ∩ B) ((int(AA*P))*P∩B)}

{(int(A A*P ∩ B)) ((int(AA*P))*P∩B)}

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{(int((A ∩ B) (A*P ∩ B))) (int((A ∩ B) (A*P ∩ B)))*P)}

{(int((A ∩ B) (A ∩ B) *P)) (int((A ∩ B) (A ∩ B) *P))*P) }

= {cl*P (int((A ∩ B) (A ∩ B) *P))} = cl*P(int(cl*P(A∩B))).

Therefore, A ∩ B β I*PO(X,).

Definition 2.19. A subset S is said to be β -I*P -closed if its complement is β -I*P open.

Theorem 2.20. A subset A of a space (X,,) is β -I*P -closed if and only if

int*P(cl(int*P(A))) A.

Proof. Let A be β - I*P-closed set of (X, ,).Then X-A is β - I-*P open and hence X-A

cl*P(int(cl*P(X-A))) = X – int*P(cl(int*P(A))). Therefore, int*P(cl(int*P(A))) A.

Conversely, let int*P(cl(int*P(A))) A then X - A X – int*P(cl(int*P(A)))

= cl*P(int(cl*P(X-A))).(i.e)X - A is β - I*P-open. Therefore, A is β - I*P-closed.

Remark 2.21. For a subset A of a space (X,,) we have

X – int(cl*P(int(A))) cl*P(int(cl*P(X-A))) is shown by the following.

Example2.22. Let X = {a,b,c}, = {,X,{a}}, = { ,{a}}and A= {b,c}.

Then, X – int(cl*P(int(A))) = X but cl*P(int(cl*P(X-A))) = {a}.

Hence X – int(cl*P(int(A))) cl*P(int(cl*P(X-A))).

Theorem 2.23. If a subset A of a space(X,,) is β- I*P-closed then int(cl*P(int(A))) A.

Proof. We have cl*P(A) cl*(A) cl(A) and int(A) int*(A) int*P(A)

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Let A be a β-I*P-closed set of (X, ,),we have int(cl*P(int(A))) int*P(cl(int*P(A))) A.

Therefore, int(cl*P(int(A))) A.

Corollary 2.24. Let A be a subset of a space (X,,) such that

X – int(cl*P(int(A))) = cl*P(int(cl*P(X-A))). Then A is β - I*P -closed if and only if int(cl*P

(int(A))) A.

Proof. This is an immediate consequence of Theorem 2.23.

Definition 2.25. A subset ‘A’ of a space (X,, ) is called

a) Strong βP - I*P- set if cl*P(int(cl*P(A)) = int(A).

b) βP - I*P- set if cl*P(int(A)) = int(A).

Definition 2.26. A subset of an ideal topological space (X,,) is called

a) Strong βPI*P set if A = U ∩ V, where U and V is Strong βP-I*P-set

b) βPI*P set if A = U ∩ V, where U and V is βP-I*P-set.

Remark 2.27. a) Every Strong βP-I*P-set is βP-I*P-set.

b) Every Strong βPI*P -set is βPI*P – set.

c) Every open set is Strong βPI*P set.

Proof. a) Let A be a Strong βP-I*P -set then cl*P(int(cl*P(A))) = int(A)

cl*P(int(A)) cl*P(int(cl*P(A))) int (A), (i.e) cl*P(int(A)) int (A)

But cl*P(int(A)) = int(A) (int(A))*P and so cl*P(int(A)) (int(A).

Therefore, cl*P(int(A)) = int(A).(i.e) A is a βP -I*P-set.

b) Every Strong βP I*P -set is βP I*P - set

Let A be a Strong βP I*P-set, then A = U ∩ V, where U and V is a Strong βP -I*P-set

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A = U∩V, where U and V is βP -I*P-set (by using (a) above)

A is a βP I*P set.

c) Every open set is Strong βP I*P set.

Let A be an open set

Then A = A∩X, where A and X is a Strong βP -I*P-set

Therefore, A is a Strong βP I*P – set.

Proposition.2.28. For a subset (X,,) the following conditions are equivalent

a) A is open.

b) A is β- I*P - open and Strong βPI*P – open.

c) A is semi - I*P - open and βPI*P – open.

Proof. a) b) Let A be open set, then by Remark 2.27(c), A is Strong βPI*P -set.

Given A = int A which implies, A = int A int(cl*P(A)) cl*P(int(cl*P(A)))

Therefore, A is β- I*P –open.

b) a) Let A be β- I*P -open and Strong βP I*P -set.

Then A = U∩V, where U, V is Strong βP -I *P-set, and A cl*P(int(cl*P(A)))

Therefore, A cl*P(int(cl*P(U∩V))) . Since A U∩A, we get

A U ∩ [cl*P(int(cl*P(U))) ∩ cl*P(int(cl*P(V)))]

But U = intU int(cl*P(U)) cl*P(int(cl*P(U))) and cl*P(int(cl*P(V))) = int V

we get A U intV = intU ∩ intV = int(U∩V) = int A

A is open.

a) c) Let A be open, then A = int(A) cl*P(int(A)). Therefore A is semi- I*P-open.

Then by Remark 2.27 c), every open set is βP I*P -set. Therefore, A is both semi- I*P-open

and βP I*P -set.

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c)a) Let A be semi- I*P-open and βP I*P -set

Then A cl*P(int(A)) and A = U ∩ V, where U and V is βP-I*P -set

A U ∩ [cl*P(int U) ∩ cl*P(intV)] U ∩ cl*P(int(cl*P(V)))

= U ∩ intV = intA. Therefore, A is open.

3. β - I*P –continuous functions

Definition 3.1. A function f : (X, ,) ® (Y,s) is said to be β - I*P -continuous (resp.

pre- I*P -continuous, Strong βP I*P –continuous , b-I- continuous) if f -1(V) is β - I*P-open

(resp. pre -I*P-open, Strong βP I*P -open, b-I-open) in (X,,) for every open set V of

(Y,s).

Definition 3.2. A function f : (X, ,) ® (Y,s) is said to be weakly β - I*P -continuous if

f -1(V) is weakly β - I*P-open in (X,,) for every open set V of (Y,s).

Definition 3.3. A function f : (X, ,) ® (Y,s) is said to be β – I*P - continuous if f -

1(V) is β - I*P-open in (X,,) for every open set V of (Y,s).

Definition 3.4[7]. A function f : (X, ,) ® (Y,s) is said to be weakly -semi–I-

continuous if f -1(V) is weakly semi-I-open in (X,,) for every open set V of (Y,s).

Definition 3.5. A function f : (X, ,) ® (Y,s) is said to be β – IP - continuous if f -1

(V) is β - IP-open in (X,,) for every open set V of (Y,s).

Definition 3.6. A function f : (X, ,) ® (Y,s) is said to be β – I* - continuous if f -1

(V) is β – I*-open in (X,,) for every open set V of (Y,s).

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Remark 3.7. It is obvious that pre-I*P-continuity implies β - I*P -continuity, β -I*P-

continuity implies β-I-continuity and semi-I*P-continuity implies β - I*P -continuity.

Theorem 3.8. For a subset of an ideal topological space, the following condition hold.

1) Every β - I*P- continuous is weakly-β-I*P- continuous.

2) Every β - I*P- continuous is β –IP- continuous.

3) Every β - IP- continuous is weakly-β -I*P - continuous.

4) Every weakly-β-I*P- continuous is weakly-semi-I - continuous.

5) Every β – I*- continuous is β - I- continuous.

6) Every β – I*P- continuous is β-I*- continuous.

7) Every β – IP- continuous is β-I*- continuous.

8) Every weakly semi-I- continuous is β- continuous.

9) Every β-I- continuous is β- continuous.

Proof. The proof is obvious.

Theorem 3.9 (Decomposition of Continuity)

For function f: (X, ,) ® (Y,s) the following conditions are equivalent.

a) f is continuous.

b) A is β- I*P - continuous and Strong βPI*P – continuous.

c) A is semi - I*P - continuous and βPI*P – continuous.

Proof: - It is obvious from Proposition 2.28.

Theorem 3.10. For a function f: (X,,) ® (Y,s) the following are equivalent.

1) f is β-I*P- continuous

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2) For each x X and each V s containing f (x), there exists U β I*PO(X, )

containing x such that f (U) V.

3) The inverse image of each closed set in Y is β-I*P - closed.

Proof. Straight forward.

Definition 3.11. A function f: (X ,,) ® (Y,s, J) is called β - I*P -open (resp. β - I*P

closed), if the image of every open set (resp. closed) in (X, ) is β -I*P -open (resp. β- I*P

closed) in (Y,s, J).

Definition 3.12. A function f: (X ,,) ® (Y,s, J) is said to be β - I*P - irresolute, if

f -1(V) is β- I*P-open in (X,,) for every β- I*P-open set V of (Y,s,J).

Theorem 3.13. The following hold for functions f :(X, , ) ® (Y,s,J) and g : (Y,s,J)

®(Z,)

1) gof is β- I*P - continuous if f is β- I*P - continuous and g is continuous.

2) gof is β- I*P- continuous if f is β - I*P- irresolute and g is β- I*P- continuous.

Proof . Straight forward.

If (X,,) is an ideal topological space and A is subset of X, we denote by /A

the relative topology on A and /A = { A∩/} is obviously an ideal on A.

Lemma 3.14. Let (X,,) be an ideal topological space and A, B are subsets of X such

that B A then B*P( /A, /A) = B*P(, ) ∩ A.

Proof . B*P ( /A, /A) = {x A / Ux ∩ B /A for each pre - open set Ux in A} = {x

A / Ux ∩ B ∩ A , for each pre - open set Ux in A} = A ∩{x X/ Ux ∩ B , for each

pre - open set Ux in X} = A ∩ B*P(, ).

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Lemma 3.15. Let (X,,) be an ideal topological space A X and U then

cl*P(A) ∩ U = cl*P(A ∩ U).

Proof . cl*P(A) ∩ U = (A*P A) ∩ U = (A*P ∩ U) (A ∩ U) (A ∩ U) *P (A∩U)

= cl*P(A ∩ U).

Theorem 3.16. Let (X,,) be an ideal topological space. If U PO(X) and A

β I*PO(X,) then U ∩ A β I*PO(U,/U,/U).

Proof . Since UPO(X) and A β I*PO(X,) by Theorem.2.18.(2) we have U ∩ A

β I*PO(X,), which implies A ∩ U cl*P(int(cl*P(A ∩ U))).And hence by Lemma 3.15

A ∩ U U ∩ cl*P(int(cl*P(A∩U))) clU*P(U ∩ int(cl*P(A ∩ U))) clU

*P(intU(U ∩ cl*P (A

∩ U))) = clU*P(intU(clU

*P(U∩A ∩ U))) = clU*P(intU(cl*P

U(A ∩ U))).Therefore, A ∩ U β

I*PO(U, /U,/U).

Theorem 3.17. Let f : (X,, ) ® (Y, s) be β I*P continuous function and U PO(X)

then the restriction f/U : (U,/U,/U) ® (Y, s) is β – I*P continuous.

Proof .Let V be any open set of (Y, s). Since f is β - I*P – continuous, we have f -1(V)

β I*PO(X, ). Since U PO(X) by Theorem 3.16, we have U f-1(V) β I*PO(U,

/U,/U). On the other hand,(f/U) -1(V) = U f-1(V) and (f/U)-1(V) β I*PO

(U,/U,/U). This shows that f/U : (U, /U,/U) ® (Y, s) is β- I*P- continuous.

4. β-I*P- open and β- I*P- closed functions

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Definition 4.1. A function f: (X , ,)(Y ,σ ) is called pre-I*P-open (resp. pre- I*P - closed) if

the image of each open (resp. closed) set in X is pre-I*P-open (resp. pre- I*P- closed) in Y.

Definition 4.2. A function f: (X , ,)(Y ,σ ) is called β-I*P-open (resp. β- I*P - closed) if the

image of each open (resp. closed) set in X is β-I*P-open (resp. β- I*P- closed) in Y.

Remark 4.3. Clearly every β -I*P open (resp. β -I*P -closed) function is β-open (resp. β-

closed) but the converse is not true in general. Observe that every pre-I*P-open (resp. pre-

I*P-closed) is β -I*P-open (resp. β -I*P-closed) function but the converse is not true in

general.

Theorem 4.4. A function f : (X, ,) ® (Y, σ, J ) is β - I*P-open if and only if for each

point x of X and each neighbourhood U of x, there exists a β- I*P- open set V in Y

containing f(x) such that V f(U).

Proof. Necessity If f is β -I*P-open function and let U be an open set in(X,) containing

x, then f(U) is β -I*P-open set containing f(x). Let V = f(U),then this V satisfies the

conditions required.

Sufficiency Suppose that for each x X and each neighbourhood U of x, there exists

V β I*PO(Y,σ) such that f(x) V f(U). f(U) is β -I*P-open. (i.e.) f is β -I*P-open

function. Hence proved.

Theorem 4.5. For any bijective function f: (X,) ® (Y,σ,J) the following conditions

hold.

1) The inverse function is β - I*P-continuous.

2) f is β- I*P-open.

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3) f is β - I*P-closed.

Proof . Straight forward.

Theorem 4.6. Let f: (X,) ® (Y,σ,J) and g: (Y,σ,J) ® (Z,V,K) be two functions where

I,J and K are ideals on X,Y and Z respectively. Then the following statements hold.

1) gof is β -I*P-open, if f is open and g is β - I*P-open.

2) f is β -I*P-open if gof is open and g is β - I*P-continuous.

Proof. Straightforward .

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