Decomposition of continuity via
θ-local function in ideal topological
spaces
C. Janakia,∗ and M. Anandhib
a,bDepartment of Mathematics, L.R.G. Govt. Arts College for Women,Tirupur-641004, Tamil Nadu, India.
Abstract
In this paper, we introduce new classes of sets called ∗θ -pre-I-open sets, ∗θ -semi-I-open sets,
∗θ-α -I-open sets and ∗θ-β -I-open sets in ideal topological spaces and study some of their
characteristics. Also, by using these sets, we obtain new decompositions of continuity in ideal
topological spaces.
Keywords: ∗θ-open set, ∗θ-pre-I-open set, ∗θ-semi-I-open set, ∗θ -α -I-open set, ∗θ - β -I-open set,
decomposition of continuity.
2010 MSC:54D10. c2012 MJM. All rights reserved.
1
IntroductionIn 1968, Velicko[18] introduced the notions ofθ-open subsets,θ-closed subsets andθ-closure, for
the sake of studying the important class of H-closed spaces in terms of arbitrary filterbases. In 1990,
Jankovic and Hamlett[10,11] defined the concept of I-open set via local function which was given by
Vaidyanathaswamy[17]. The later concept was also established utilizing the concept of ideal whose
topic in general topological spaces was treated in the classical text by Kuratowski[12]. Recently,
Hatir and Noiri [6,7,8] have introducedα-I-open sets, semi-I-open sets andβ-I-open sets to obtain
a decomposition of continuity. In this paper, we define new classes of sets called∗θ-pre-open sets,
∗θ-semi-open sets,∗θ-α -open sets and∗θ-β -open sets in ideal topological spaces. We investigate
their properties and the relationships of these sets. Moreover, by using these sets, we obtain new
decompositions of continuity in ideal topological spaces.
∗
Corresponding author.
2
Preliminaries
Let (X,τ) be a topological space with no separation properties assumed. For a subset A of a
space (X,τ), cl(A) and int(A) denote the closure of A and the interior of A respectively. (X,τ) and
(Y,σ) will be replaced by X and Y if there is no chance of confusion. Let (X,τ) be a topological space.
A subset A of X is said to be semi open[14] (resp. pre open[15], β-open[8] and α-open[2] if A ⊂
cl(int(A)) (resp. A ⊂ int(cl(A)), cl(int(cl(A))) and A ⊂ int(cl(int(A)))). A point x ∈ X is called a
θ-adherent point of A [3], if A∩cl(A)6=ϕfor every open set V containing x.
An ideal I on a topological space (X,τ) is a nonempty collection of subsets of X which satisfies
(i) A∈I and B⊆A implies B∈I and
(ii) A∈I and B∈I implies (A∪B)∈I.
A topological space (X,τ ) with an ideal I on X is called an ideal topological space and is denoted by
(X,τ, I). For a subset A⊆ X, A∗(I) = {x∈X :U∩A /∈I for every U ∈τ(x)} is called the local
function of A with respect to I andτ[4]. We simply writeA∗in case there is no chance for confusion.
A Kuratowski[12] closure operator cl∗(.)for a topologyτ∗(I)called theτ∗-topology finer than τ is
definedcl∗(A) =A∪A∗. A subset A of an ideal space (X,τ,I ) isτ∗-closed [16] ifA∗ ⊂A. A subset
A of X is said to be semi-I -open[6] (resp. pre-I-open[4] α-I-open[6] and β-I-open[6] if A⊆ cl∗
(int(A)) (resp. A ⊆ int(cl∗(A)), A ⊆ int(cl∗(int(A)))andA ⊆ cl(int(cl∗(A)))). A subset A of X is
called (1) a t-I-set[6] ifint(cl∗(A)) =int(A). (2) a B-I-set ([6]) if there exist U∈τ and a t-I-set V in X
such that A = U ∩V. A function f : (X,τ, I)→ (Y,σ ) is said to be pre-continuous[15] iff−1(V)is
pre-open for each open set V inσ.
Quite recently, Janaki and Anandhi [9] introducedθ-local function in ideal topological spaces
in the following manner. Let (X,τ,I) be an ideal topological space and A be a subset of X. Then
A∗θ(I, τ) = {x∈X :Ux∩A /∈I for everyUx∈θO(X, x)} is called the θ-local function of I on X
with respect to I andτ. A subset A of (X,τ,I) is said to be∗θ-closed[9] ifA∗θ ⊂A.
Lemma 2.1. ([9]). Let (X,τ,I)) be an ideal topological space and let A,B be subsets of X. Then for θ-local
functions the following properties hold:
(i) A∗⊂A∗θ.
(ii) A∗θ ⊂clθ(A).
(iv) for aθ- open set U,U ∩A∗θ =U∩(U ∩A)∗θ ⊂(U∩A)∗θ.
(v) cl∗(A)⊂cl∗θ(A)⊂clθ(A).
Lemma 2.2 (3). Let (X,τ,I)) be an ideal topological space and let A be subset of X. Then, the following
properties hold:
1. If A is open, thencl(A) =clθ(A).
2. If A is closed, thenint(A) =intθ(A).
3
∗
θ
pre-open,
∗
θ
semi-open,
∗
θ
-
α
open set and
∗
θ
-
β
open sets
Definition 3.1. Let (X,τ ,I) be an ideal topological space. A subset A of X is said to be
1. ∗θ-pre-open ifA⊆int(cl∗θ(A));
2. ∗θ-semi-open ifA⊆cl(int∗θ(A));
3. ∗θ-α-open ifA⊆int(cl(int∗θ(A)));
4. ∗θ-β-open ifA⊆cl(int(cl∗θ(A)));
Remark 3.1. 1. Every∗θ-α-I-open is∗θ-pre-I-open ;
2. Every∗θ-α-I-open is∗θ-semi-I-open ;
3. Every∗θ-semi-I-open is∗θ-β-I-open and
4. Every∗θ-pre-I-open is∗θ-β -I-open.
The converse need not be true as seen in the following examples.
Example 3.1. Let (X,τ ,I) be an ideal topological space andX ={a, b, c, d}with
τ ={ϕ,{a},{c},{a, c},{c, d},{a, c, d},{b, c, d}, X}andI ={ϕ,{b},{c},{b, c}}.
1. LetA={c}. Then, A is∗θ-pre-I-open but not∗θ-α-I-open.
2. LetA={d}. Then, A is∗θ-semi-I-open but not∗θ-semi-I-open.
3. LetA={b, c}. Then, A is∗θ-β-I-open but not∗θ-pre-I-open.
4. LetA={a, c}. Then, A is∗θ-β-I-open but not∗θ-semi-I-open.
Remark 3.2. 1. Every pre-I-open set is∗θ-pre-I-open.
2. Everyβ-I-open set is∗θ-β-I-open.
Example 3.2. In Example 3.1
1. The set{a, d}is a∗θ-pre-I-open set but not a pre-I-open set.
2. The set{a, b, d}is a∗θ-β-I-open set but not aβ-I-open set.
Remark 3.3. semi-I-open sets(resp. α -I-open sets) and∗θ-semi-I-open sets(resp. ∗θ-α -I-open sets) are
independent as seen in the following examples.
Example 3.3. Let (X,τ ,I) be an ideal topological space andX={a, b, c, d, e}with
τ =
{ϕ,{a},{c},{e},{a, b},{a, c},{a, e},{c, e},{a, b, c},{a, b, e},{a, c, e},{c, d, e},{a, b, c, e},{a, c, d, e}, X}
and
I ={ϕ,{c},{d},{c, d}}.
(i) {c}is a semi-I-open but not a∗θ-semi-I-open and{b, d, e}is a∗θ-semi-I-open but not a semi-I-open.
(ii) {a}is aα-I-open but not a∗θ-α-I-open and{b, e}is a∗θ-α-I-open but not aα-I-open.
Remark 3.4. (i) Every preopen set is∗θ-pre-I-open.
(ii) Everyβ-open set is∗θ-β -I-open.
The converse need not be true as seen in the following example.
Example 3.4. In example 3.3
(i) {b}is∗θ-pre-I-open but not a preopen set.
(ii) {b, c}is∗θ-β-I-open but not aβ-open set.
Remark 3.5. semi-open sets(resp. α-open sets) and ∗θ-semi-I-open sets(resp. ∗θ- α-I-open sets) are
independent as seen in the following example.
Example 3.5. In example 3.4
(i) {a}is a semi-open but not a∗θ-semi-I-open and{b, d, e}is a∗θ-semi-I-open but not a semi-open.
(ii) {a, c, d, e}isα-open but not a∗θ-α-I-open and{b, e}is a∗θ-α-I- open but not aα-open.
Theorem 3.1. A subset A of an ideal topological space (X,τ ,I) is∗θ-semi-I-open if and only if
cl(A) =cl(int∗θ(A)).
Theorem 3.2. A subset A of an ideal topological space (X,τ ,I) is ∗θ-semi-I-open if and only if for some
∗θ-open setU,U ⊆A⊆cl(U).
Theorem 3.3. Let (X,τ ,I) be an ideal topological space. If A ⊂ B ⊂ cl∗θ(A)and B is ∗θ- β -I-open and
θ-closed, then A is∗θ-β-I-open.
Theorem 3.4. Let (X,τ ,I) be an ideal topological space. IfA ⊂B ⊂cl(A)and A is∗θ-β -I-open, then B is
∗θ-β-I-open.
Theorem 3.5. Let (X,τ ,I) be an ideal topological space and{Aα:α∈∆}a family of subsets of X, where∆
is an arbitrary index set. Then,
(1) .If{Aα :α∈∆} ⊆ ∗θβIO(X, τ), then∪ {Aα :α∈∆} ∈ ∗θβIO(X, τ).
(2) IfA∈ ∗θβIO(X, τ)andU ∈τθ, thenA∩U ∈ ∗θβIO(X, τ).
Remark 3.7. The intersection of two∗θ-α -I-open sets (resp. ∗θ-pre-I-open sets,∗θ-β -I-open sets and∗θ
-semi-I-open sets) need not be∗θ-α-I-open set (resp. ∗θ-pre-I-open set,∗θ-β -I-open set and∗θ- semi-I-open
Example 3.6. Let (X,τ ,I) be an ideal topological space andX={a, b, c, d, e}
withτ ={φ,{a},{c, d},{a, c, d},{b, c, d, e}, X}
andI ={φ,{c},{d},{c, d}}.
(i) A={b, c, e}andB ={b, d, e}are∗θ-α- I-open sets. ButA∩B ={b, e}is not a∗θ-α-I-open set.
(ii) A={c, d}andB ={c, e}are∗θ-pre-I-open sets. ButA∩B ={c}is not a∗θ-pre-I-open set.
(iii) A={b, d}andB ={d, e}are∗θ-β-I-open sets. ButA∩B ={d}is not a∗θ-β-I-open set.
Example 3.7. Let (X,τ ,I) be an ideal topological space andX={a, b, c, d, e}with
τ =
{φ,{a},{c},{e},{a, b},{a, c},{a, e},{c, e},{a, b, c},{a, b, e},{a, c, e},{c, d, e},{a, b, c, e},{a, c, d, e}, X}
and I = {φ,{b},{d},{b, d}}. A = {a, b, c, e} and B = {a, b, d, e} are ∗θ-semi-I-open sets.
ButA∩B ={a, b, e}is not a∗θ-semi-I-open set.
Definition 3.2. A subset A of an ideal topological space (X,τ,I) is called
(1) a∗θ-pre-t-I-set ifint(cl∗θ(A)) =int(A);
(2) a∗θ-β-t-I-set ifcl(int(cl∗θ(A))) =int(A);
Theorem 3.6. Let (X,τ ,I) be an ideal topological space,I = {φ}, and A ⊆ X. Then the following are
equivalent:
(i) A is a∗θ-pre-t-I-set;
(ii) int(A) =int(clθ(A)).
Theorem 3.7. Let A and B be subsets of an ideal topological space (X,τ ,I). If A and B are∗θ-pre-t-I-sets, then
A∩Bis a∗θ-pre-t-I-set.
Remark 3.8. The union of two∗θ-pre-t-I- sets need not be a∗θ-pre-t-I-set as given in the following example.
Example 3.8. Let (X,τ ,I) be an ideal topological space andX={a, b, c, d}with
τ = {φ,{a},{c},{a, c},{c, d},{a, c, d},{b, c, d}, X} and I = {φ,{c},{d},{c, d}}. A = {a, c} and
B ={a, d}are two∗θ-pre-t-I-sets. ButA∪B ={a, c, d}which is not a∗θ-pre-t-I-set.
Proposition 3.1. Let A be a subset of an ideal topological space (X,τ ,I). The following properties hold:
(1) If A is∗θ-closed, then it is a∗θ-pre-t-I-set;
(2) If A is a∗θ-pre-t-I-set, then it is a t-I-set.
Theorem 3.8. Let (X,τ ,I) be an ideal topological space. If A is∗θ-semi-I-open and B is∗θ-pre-I-open, then
Theorem 3.9. Let (X,τ ,I)be an ideal topological space. If A is∗θ-β-I-open and B is∗θ-α-I-open, thenA∩B
is a∗θ-β-I-open set.
Theorem 3.10. Let (X,τ ,I) be an ideal topological space. If A is∗θ-pre-I-open (resp.∗θ-semi- I-open) and B
is∗θ-α-I-open, thenA∩B is∗θ-pre-I-open (resp.∗θ-semi-I-open).
Theorem 3.11. Let ( X,τ ,I) be an ideal topological space. The following are equivalent;
(1) The∗θ-closure of every∗θ-open subset of X is∗θ-open;
(2) cl(int∗θ(A))⊆int(cl∗θ(A))for every subset A of X;
(3) ∗θSIO(X)⊆ ∗θP IO(X);
(4) The∗θ-closure of every∗θ-β-I-open subset of X is∗θ-open;
(5) ∗θ-βIO(X)⊆ ∗θP IO(X).
4
Decomposition of continuityDefinition 4.1. Let ( X,τ ,I) be an ideal topological space. A subset A of X is called
(1) a∗θ-pre-B-I-set if there existU ∈τ and a∗θ-pre-t-I-set V in X such thatA=U∩V;
(2) a∗θ-β-B-I-set if there existU ∈τ and a∗θ-β-t-I-set V in X such thatA=U∩V;
Proposition 4.1. For a subset A of an ideal topological space (X,τ ,I), the following properties hold:
(1) If A is a∗θ-pre-B-I-set, then it is a∗θ-pre-t-I-set;
(2) If A is a∗θ-pre-B-I-set, then it is a B-I-set.
Definition 4.2. Let (X,τ ,I) be an ideal topological space. A subset A of X is called a∗θ-semi- I-closed set if
int(cl∗θ(A))⊆A.
Definition 4.3. A subset A in an ideal topological space ( X,τ ,I) is called aG∗θ-I-set ifA=U∩V, whereU
is open andV is∗θ-semi-I-closed andint(cl∗θ(V)) =cl(int∗θ(V)).
Theorem 4.1. For a subset A of an ideal topological space (X,τ ,I), the following properties are equivalent:
(1) A is open;
(2) A is preopen and aG∗θ-I-set;
(3) A is∗θ-pre-I-open and aG∗θ-I-set;
(5) A is∗θ-pre-I-open and a∗θ-pre-B-I-set;
(6) A is∗θ-β-I-open and a∗θ-β -B-I-set.
Definition 4.4. A function f : (X, τ, I) → (Y, σ) is said to be ∗θ-pre-I-continuous (resp. ∗θ-β
-I-continuous, ∗θ-semi-I-continuous, ∗θ-α -I-continuous, G∗θ-I-continuous, ∗θ-pre-B-I-continuous and ∗θ-β
-B-I-continuous ) if f−1(V) is ∗θ-pre-I- open (resp. ∗θ-β -I-open, ∗θ-semi-I-open and ∗θ-α -I-open, G∗θ
-I-open,∗θ-pre-B-I-set and∗θ-β-B-I-open ) for each open set V inσ.
Theorem 4.2. For a functionf : (X, τ, I)→(Y, σ), the following properties are equivalent:
(1) f is continuous;
(2) f is pre-continuous andG∗θ-I-continuous;
(3) f is∗θ-pre-I-continuous andG∗θ-I-continuous;
(4) f is∗θ-β-I-continuous andG∗θ-I-continuous;
(5) f is∗θ-pre-I-continuous and∗θ-pre-B-I-continuous;
(6) f is∗θ-β-I-continuous and∗θ-β-B-I-continuous.
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Received: April 15, 2015;Accepted: June 13, 2015
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