Some new sets and decompositions of $alpha^{*}$-$mathcal{I}$-continuity and $A_{mathcal{I}R}$-continuity via idealization

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Some new sets and decompositions of

α

∗

-

I

-continuity and

A

IR

-continuity via idealization

K. Viswanathan

a

_{, and J. Jayasudha}

b,∗

a,b_{Post-Graduate and Research Department of Mathematics, N G M College, Pollachi - 642 001, Tamil Nadu, INDIA.}

Abstract:

In this paper, we introduce and investigate the notions ofα∗-I-open sets in ideal topological spaces. Further we introduce the notions of δ-I-closed sets, weak A∗_I-sets, η_I∗-sets and C∗-I-sets to obtain decompositions of

α∗-I-continuity andAIR-continuity.

Keywords: α∗-I-open sets, δ-I-closed sets, weak A∗_I-sets, η_I∗-sets and C∗-I-sets, α∗-I-continuity, AIR

-continuity

1

Introduction

Many weaker forms of open sets were introduced over the years and using these sets numerous decompo-sitions of continuity and generalized continuities were established. In the recent years, several authors have shown remarkable interests in investigating such sets and obtaining decompositions via ideals. In 2007, Acik-goz and Yuksel introduced the notion of I-R-closed sets in [1] and using I-R-closed sets they defined the notions of AIR-sets and AIR-continuity. Further, they have obtained some decompositions of continuity and AIR-continuity. Later in 2009 Renukadevi [11] extended the study ofI-R-closed sets and AIR-sets. In [11],

characterizations of I-R-closed sets and AIR-sets are studied in terms of strong β-I-open sets and several

properties of I-R-closed sets andAIR-sets are discussed.

Ekici [2] introduced the notion ofsemi∗-I-open sets to obtain some characterizations of ∗-extremally dis-connected ideal spaces. Further in 2011, Ekici [3, 4] introduced the notions of pre∗_I-open sets, β∗_I-open sets,

A∗

I-sets,ACI-sets, BCI-sets and obtained decompositions of continuity andA∗I-continuity.

In this paper we have introduced the notion ofα∗-I-open sets, discussed some of its properties and have obtained few characterizations. Further, we have introduced the notions of η_I∗-sets and C∗-I-sets to obtain decompositions ofAIR-continuity. Also using some weaker forms of open sets we have obtained some

decom-positions ofα∗-I-continuity.

2

Preliminaries

An idealI on a topological space (X, τ) is a non-empty collection of subsets ofX satisfying the following properties:(1) A ∈ I and B ⊆ A imply B ∈ I (heredity); (2) A ∈ I and B ∈ I imply A∪B ∈ I (finite additivity). A topological space (X, τ) with an idealIonX is called an ideal topological space and is denoted by (X, τ,I).For a subsetA⊆X, A∗₍_I_{) =}_{_x_∈_X _:_U_∩_{A /}_{∈ I} _{for every}_U _∈_τ₍_x₎_}_,_{is called the local function}

[7] of A with respect toI andτ.We simply writeA∗ in case there is no chance for confusion. A Kuratowski closure operator cl∗(.) for a topologyτ∗(I) called the ∗-topology finer thanτ is defined bycl∗(A) =A∪A∗

[14]. Let (X, τ) denote a topological space on which no separation axioms are assumed unless explicitly stated.

∗_{Corresponding author.}

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In a topological space (X, τ),the closure and the interior of any subset Aof X will be denoted by cl(A) and

int(A),respectively.

Definition 2.1. A subset A of an ideal space(X, τ,I)is called

1. α-I-open [6]ifA⊆int(cl∗(int(A))).

2. semi-I-open [6]if A⊆cl∗(int(A)).

3. pre∗_I-open[3] ifA⊆int∗(cl(A)).

4. semi∗-I-open[2, 5] if A⊆cl(int∗(A)).

5. β∗_I-open[3]if A⊆cl(int∗(cl(A))).

The complements of the above mentioned sets are their respective closed sets.

Definition 2.2. An ideal space (X, τ,I) is said to be ∗-extremally disconnected [2] if the ∗-closure of every open subsetA ofX is open.

Definition 2.3. A subset A of an ideal space(X, τ,I)is said to be

1. I-R-closed[1]if A=cl∗(int(A)).

2. R-I-open [13]if A=int(cl∗(A)).

3. R-I-closed[13] if its complement isR-I-open.

Definition 2.4. A subsetAof an ideal space(X, τ,I)is calledAIR-set[1]ifA=U∩V,whereU is open and V isI-R-closed.

Definition 2.5. A subsetAof an ideal space(X, τ,I)is calledb∗-I-open[15]ifA⊂cl(int∗(A))∪int∗(cl(A)).

Definition 2.6. A functionf : (X, τ,I)→(Y, σ) is said to beAIR-continuous [1](resp. b∗I-continuous [15])

iff−1₍_V₎_{is an} _A

IR-set (resp. b∗I-open set) in(X, τ,I)for each open setV of (Y, σ).

3 On

α

∗

-

I

-open sets

Definition 3.1. A subset A of an ideal space(X, τ,I)is calledα∗-I-open ifA⊂int∗(cl(int∗(A))).

Proposition 3.2. For a subset of an ideal space(X, τ,I), the following hold.

1. Every open set isα∗-I-open.

2. Every α-open set isα∗_-_I_-open.

3. Every semi-open set is semi∗-I-open.

4. Every pre-open set is pre∗_I-open.

5. Every α∗-I-open set is semi∗-I-open.

6. Every α∗-I-open set is pre∗_I-open.

7. Every pre∗_I-open set is β_I∗-open.

Example 3.3. Let X={a, b, c, d}, τ={∅,{b},{a, d},{a, b, d}, X} andI={∅,{b}}.Then,

1. A={a, c, d} is anα∗-I-open set but not an α-open set.

2. A={b, c} is asemi∗-I-open set but not a semi-open set.

3. A={a, c} is apre∗_I-open set but not a pre-open set.

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5. A={a} is apre∗_I-open set but not an α∗-I-open set.

6. A={b, c} is aβ_I∗-open set but not a pre∗_I-open set.

Remark 3.4. For the sets defined above we have the following implications.

open⇒α-I-open⇒α∗-I-open⇒semi∗-I-open

⇓ ⇓

pre∗_I-open⇒β∗_I-open

Theorem 3.5. For an ideal space(X, τ,I), the following are equivalent.

1. X is∗-extremally disconnected.

2. int∗(A)is closed for every closed subset A ofX.

3. cl(int∗(A))⊂int∗(cl(A))for every closed subsetA of X.

4. Every semi∗-I-open set is pre∗_I-open.

5. The closure of every β_I∗-open subset of X is∗-open.

6. Every β_I∗-open set is pre∗_I-open.

7. For every subset Aof X, Ais α∗-I-open if and only if it issemi∗-I-open.

Proof. (1)⇒(2): Follows from Theorem 4 of [2].

(2)⇒(3): Letint∗(A) be closed for every closed subsetA of X,then int∗(A) =cl(int∗(A)).Since we can always have int∗(A)⊂int∗(cl(A)),it follows thatcl(int∗(A)) =int∗(A)⊂int∗(cl(A)).

(3)⇒(4): LetAbesemi∗-I-open, then by (3), we haveA⊂cl(int∗(A))⊂int∗(cl(A)).HenceAispre∗_I-open. (4)⇒(5): LetAbe aβ∗_I-open set. ThenA⊂cl(int∗(cl(A))) from which we havecl(A)⊂cl(cl(int∗(cl(A)))) =

cl(int∗(cl(A))). This shows that cl(A) is semi∗-I-open. Now by (4), cl(A) is pre∗_I-open. Thus cl(A) ⊂

int∗(cl(cl(A))) =int∗(cl(A)). This implies thatcl(A) is∗-open. (5)⇒(6): LetA be aβ∗

I-open set. By (5), cl(A) ⊂int∗(cl(A)). Hence A ⊂cl(A) ⊂int∗(cl(A)) showing

that Aispre∗_I-open.

(6)⇒(7): Let Abe a semi∗-I-open set. Since asemi∗-I-open set is β_I∗-open, by (6), it is also pre∗_I-open. Now Ais bothsemi∗-I-open and pre∗_I-open, henceAisα∗-I-open.

(7)⇒(1): Let A be an open set of X. Then cl(A) is semi∗-I-open and since cl∗(A) ⊂ cl(A), cl∗(A) is

semi∗_-_I_{-open. By (7),}_cl∗₍_A_{) is}_α∗_-_I_{-open. Therefore}_cl∗₍_A₎_⊂_int∗₍_cl₍_int∗₍_cl∗₍_A₎₎₎₎_⊂_int∗₍_cl₍_A_{)) and hence}

cl∗(A) is open andX is∗-extremally disconnected.

Definition 3.6. A subset Aof an ideal space(X, τ,I)is said to be S∗-I-set ifint∗(A) =cl(int∗(A)).

Theorem 3.7. Let(X, τ,I)be an ideal space. Then the following hold.

1. X is∗-extremally disconnected if and only if the b∗-I-open sets andpre∗_I-open sets coincide.

2. If Ais ab∗-I-open set as well as S∗-I-set, thenA is apre∗_I-open set.

Proof. (1) Suppose X is ∗-extremally disconnected. If A is a b∗_-_I_{-open subset of} _X, _{then by Proposition}

3.3(6) of [15], A is a β_I∗-open set. Since X is ∗-extremally disconnected, cl∗(A) = int(cl∗(A)) and hence

A⊂cl∗(A) =int(cl∗(A)) showing thatAis pre-I-open and hencepre∗_I-open.

Conversely, Suppose that b∗-I-open sets and pre∗_I-open sets coincide. If A is semi∗-I-open, then A is

b∗_-_I_{-open. Now by hypothesis,}_A _is_pre∗

I-open. Hence by Theorem 3.5,X is∗-extremally disconnected.

(2) If A is anS∗_-_I_{-set, then}_int∗₍_A_{) =}_cl₍_int∗₍_A₎₎_. _{Also if} _A _{is a} _b∗_-_I_{-open set, then} _A _⊂_int∗₍_cl₍_A₎₎_∪

cl(int∗(A)) =int∗(cl(A))∪int∗(A) =int∗(cl(A)).ThusA is apre∗_I-open set.

Theorem 3.8. Let (X, τ,I)be an ideal space and A, B be subsets of X. IfB is an α∗-I-open set andA is a

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Proof. By hypothesis, B ⊂ int∗(cl(int∗(B))) and A ⊂ int∗(cl(A)) ∪ cl(int∗(A)).

NowB∩A⊂int∗(cl(int∗(B)))∩[int∗(cl(A))∪cl(int∗(A))]

= [int∗(cl(int∗(B)))∩int∗(cl(A))]∪[int∗(cl(int∗(B)))∩cl(int∗(A))]

⊂int∗_[_cl₍_int∗₍_B₎₎_∩_int∗₍_cl₍_A_))]_∪_cl_[_int∗₍_cl₍_int∗₍_B₎₎₎_∩_int∗₍_A_)] ⊂int∗[cl[int∗(B)∩int∗(cl(A))]]∪cl[int∗[cl(int∗(B))∩int∗(A)]]

⊂int∗[cl[int∗(B∩cl(A)]]∪cl[int∗[cl(int∗(B)∩int∗(A))]]

⊂int∗[cl[int∗(cl(B∩A))]]∪cl[int∗[cl(int∗(B∩A))]]

⊂int∗(cl(B∩A))∪cl(int∗(B∩A)).Hence B∩Ais ab∗-I-open set.

Definition 3.9. A subset A of an ideal space(X, τ,I)is calledδ-I-closed if int∗₍_cl₍_A₎₎_⊂_cl₍_int∗₍_A₎₎_.

Definition 3.10. A subset A of an ideal space (X, τ,I) is called weak A∗_I-set if A = U ∩V, where U is

semi∗-I-open and V isR-I-closed.

Definition 3.11. A subsetA of an ideal space(X, τ,I) is calledη∗_I-set (resp. C∗-I-set) ifA=U∩V,where

U is open andV isα∗-I-closed (resp. pre∗_I-closed).

Proposition 3.12. For a subset Aof an ideal space(X, τ,I),the following conditions hold.

1. A is aC∗-I-set and a semi-I-open set inX.

2. A=U∩cl∗(int(A))for some open set U.

Proof. (1)⇒(2): Suppose thatAis aC∗-I-set and a semi-I-open set inX.SinceA is aC∗-I-set, A=U∩V

whereUis open andV ispre∗_I-closed. Also we haveA⊂V,socl∗(int(A))⊂cl∗(int(V)).SinceV ispre∗_I-closed,

cl∗(int(V))⊂V.Again, sinceA is semi-I-open, A⊂cl∗(int(A)).Hence it follows that A=A∩cl∗(int(A)) =

U∩V ∩cl∗(int(A)) =U∩cl∗(int(A)).

(2)⇒(1): LetA=U∩cl∗(int(A)) for some open setU.NowA⊂cl∗(int(A)) implies thatAis semi-I-open inX.Sincecl∗(int(A)) isτ∗-closed, it ispre∗_I-closed inX.HenceAis aC∗-I-set inX.

Proposition 3.13. For a subset of an ideal space (X, τ,I),the following conditions hold.

1. Every semi∗-I-open set is a weakA∗_I-set.

2. Every weak A∗_I-set is a δ-I-closed set.

3. Every AIR-set is anηI∗-set.

4. Every η_I∗-set is aC∗-I-set.

The converse of Proposition 3.13 need not be true as shown by the following examples.

Example 3.14. LetX ={a, b, c, d}, τ ={∅,{a},{a, b}, X}andI={∅,{a}}.Then,A={c} is a weakA∗ I-set

but not asemi∗-I-open set.

Example 3.15. LetX ={a, b, c, d}, τ ={∅,{b},{a, d},{a, b, d}, X} andI={∅,{b}}. Then,

1. A={c} is aδ-I-closed set but not a weakA∗_I-set.

2. A={c} is anη∗_I-set but not an AIR-set.

3. A={a}is aC∗-I-set but not anη∗_I-set.

Theorem 3.16. Let(X, τ,I)be an ideal space and A⊂X.Then the following are equivalent.

1. A isα∗-I-open.

2. A is bothsemi∗-I-open and pre∗_I-open.

3. A is both weakA∗_I-set andpre∗_I-open.

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Proof. The proofs of (1)⇒(2), (2)⇒(3), (3)⇒(4) follows by Propositions 3.2(5,6), 3.13(1,2).

(4)⇒(1): LetAbe bothδ-I-closed andpre∗_I-open. Thenint∗(cl(A))⊂cl(int∗(A)) and henceint∗(cl(A))⊂

int∗(cl(int∗(A))). Now by the pre∗_I-openness of A we have A ⊂ int∗(cl(A))⊂ int∗(cl(int∗(A))). This shows that Aisα∗-I-open.

Theorem 3.17. Let (X, τ,I)be an ideal space and A⊂X.Then the following are equivalent.

1. Ais an AIR-set.

2. Ais an η_I∗-set and a semi-I-open set.

3. Ais aC∗-I-set and a semi-I-open set.

Proof. (1)⇒(2): Let A be an AIR-set inX. ThenA =U ∩V,where U is open andV is I-R-closed. Hence

A = U ∩V = U ∩cl∗(int(V)) ⊂ cl∗(U ∩int(V)) = cl∗(int(U ∩V)) = cl∗(int(A)). Thus A ⊂ cl∗(int(A)),

showing that Ais semi-I-open inX.Also by Proposition 3.13(3), everyAIR-set is anηI∗-set.

(2)⇒(3): Follows from Proposition 3.13(4).

(3)⇒(1): LetAbe aC∗-I-set and a semi-I-open set inX.Then by Proposition 3.12,A=U∩cl∗(int(A)) for some open setU. Nowcl∗(int(cl∗(int(A)))) =cl∗(int(A)). Thus it follows thatAis anAIR-set.

Remark 3.18. (1)The notions ofsemi∗-I-open sets andpre∗_I-open sets are independent.

(2)The notions of weakA∗_I-sets and pre∗_I-open sets are independent.

(3)The notions of δ-I-closed sets and pre∗_I-open sets are independent.

(4)The notions of η_I∗-sets and semi-I-open sets are independent.

(5)The notions of C∗-I-sets and semi-I-open sets are independent.

Example 3.19. Let X ={a, b, c, d}, τ ={∅,{b},{a, d},{a, b, d}, X}andI ={∅,{b}}. Then,

1. A={a} is apre∗_I-open set but not asemi∗-I-open set.

2. A={b, c} is asemi∗-I-open set but not apre∗_I-open set.

3. A={d} is apre∗_I-open set but not a weakA∗_I-set.

4. A={b, c} is a weakA∗_I-set but not apre∗_I-open set.

5. A={a} is apre∗_I-open set but not aδ-I-closed set.

6. A={c} is aδ-I-closed set but not apre∗

I-open set.

Example 3.20. Let X ={a, b, c, d}, τ ={∅,{a},{a, b}, X} andI={∅,{a}}.Then,

1. A={a, b, c}is a semi-I-open set but neither anη∗_I-set nor aC∗-I-set.

2. A={b} is both anη_I∗-set and a C∗-I-set but not a semi-I-open set.

4 Decompositions of weaker forms of continuity

Definition 4.1. A function f : (X, τ,I)→(Y, σ)is said to beα∗-I-continuous (resp. pre∗_I-continuous, weak

A∗_I-continuous, η_I∗-continuous, C∗-I-continuous) if f−1₍_V₎ _{is an} _α∗_-_I_{-open set (resp.} _pre∗

I-open set, weak

A∗_I-set,η∗_I-set,C∗-I-set) in(X, τ,I)for each open setV of (Y, σ).

Definition 4.2. A functionf : (X, τ,I)→(Y, σ)is said to be contra-δ-I-continuous iff−1(V)is aδ-I-closed set in (X, τ,I)for each open setV of (Y, σ).

Proposition 4.3. For a functionf : (X, τ,I)→(Y, σ), the following hold.

1. Every α∗-I-continuous function issemi∗-I-continuous.

2. Every α∗-I-continuous function ispre∗_I-continuous.

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4. Every weak A∗_I-continuous function is contra-δ-I-continuous.

5. Every AIR-continuous function is semi-I-continuous.

6. Every AIR-continuous function isηI∗-continuous.

7. Every η_I∗-continuous function isC∗-I-continuous.

Theorem 4.4. For a function f : (X, τ,I)→(Y, σ), the following are equivalent.

1. f isα∗-I-continuous.

2. f is both semi∗-I-continuous andpre∗_I-continuous.

3. f is both weak A∗_I-continuous and pre∗_I-continuous.

4. f is both contra-δ-I-continuous andpre∗_I-continuous.

Proof. Follows from Theorem 3.16.

Theorem 4.5. For a function f : (X, τ,I)→(Y, σ), the following are equivalent.

1. f isAIR-continuous.

2. f isη∗_I-continuous and semi-I-continuous.

3. f isC∗-I-continuous and semi-I-continuous.

Proof. Follows from Theorem 3.17.

References

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Received: July 23, 2013;Accepted: September 1, 2013