Download sA -sets and decomposition of sA

South Asian Journal of Mathematics
2017 , Vol. 7 ( 1 ) : 43 ∼ 52
www.sajm-online.com ISSN 2251-1512
RESEARCH ARTICLE
sA∗I -sets and decomposition of sA∗I -continuity
K. Banumathi① ∗ , R. Premkumar②, O. Ravi②
① Department of Mathematics, St. Aloysius College of Education, Royappanpatti, Theni District, Tamil Nadu, India.
② Department of Mathematics, P. M. Thevar College, Usilampatti, Madurai District, Tamil Nadu, India.
E-mail: [email protected]
Received: Nov-19-2016; Accepted: Feb-8-2017 *Corresponding author
Abstract
The aim of this paper is to introduce and study the notions of sA∗I -sets and sCI -sets in
ideal topological spaces. Properties of sA∗I -sets and sCI -sets are investigated. Moreover, decompositions
of sA∗I -continuous functions via sA∗I -sets and sCI -sets in ideal topological spaces are established.
Key Words
sA∗I -set, sCI -set, αCI∗ -set, pre-I-regular set
MSC 2010
54A05, 54A10
1
Introduction and preliminaries
In this paper, sA∗I -sets and sCI -sets in ideal topological spaces are introduced and studied. The
relationships and properties of sA∗I -sets and sCI -sets are investigated. Furthermore, decompositions of
sA∗I -continuous functions via sA∗I -sets and sCI -sets in ideal topological spaces are provided.
Throughout this paper (X, τ ) and (Y, σ) (or simply X and Y) denote topological spaces on which
no separation axioms are assumed unless explicitly stated. For a subset A of a space X, the closure and
interior of A with respect to τ are denoted by cl(A) and int(A) respectively.
An ideal I on a topological space (X, τ ) is a nonempty collection of subsets of X which satisfies
1. A∈I and B⊆A⇒B∈I and
2. A∈I and B∈I⇒A∪B∈I [15].
If I is an ideal on X and X∈I,
/ then ̥ = {X\G : G∈I} is a filter [13]. Given a topological space (X,
τ ) with an ideal I on X and if ℘(X) is the set of all subsets of X, a set operator (.)∗ : ℘(X)→℘(X),
called a local function [15] of A with respect to τ and I, is defined as follows: for A⊆X, A∗ (I,τ )={x∈X
| U∩A∈I
/ for every U∈τ (x)} where τ (x)={U∈τ | x∈U}. A Kuratowski closure operator cl∗ (.) for a
topology τ ∗ (I,τ ), called the ⋆-topology, finer than τ is defined by cl∗ (A)=A∪A∗ (I,τ ) [13]. When there
is no chance for confusion, we will simply write A∗ for A∗ (I,τ ) and τ ∗ for τ ∗ (I,τ ). int∗ (A) will denote
the interior of A in (X, τ ∗ ). (X, τ, I) is called an ideal topological space or an ideal space.
∗
Citation: K. Banumathi, R. Premkumar, O. Ravi, sA∗
I -sets and decomposition of sAI -continuity, South Asian J Math, 2017,
7(1), 43-52.
K. Banumathi, et al: sA∗I -sets and decomposition of sA∗I -continuity
Remark 1.1. [13] The ⋆-topology is generated by τ and by the filter ̥. Also the family {H ∩ G : H ∈τ ,
G∈̥} is a basis for this topology.
Lemma 1.2. [12] Let A be a subset of an ideal topological space (X, τ , I). If N is open, then N ∩ cl∗ (A)
⊆ cl∗ (N ∩ A).
Definition 1.3. A subset A of an ideal topological space (X, τ , I) is said to be
1. pre-I-open [3] if A ⊆ int(cl∗ (A)).
2. semi-I-open [10] if A ⊆ cl∗ (int(A)).
3. α-I-open [10] if A ⊆ int(cl∗ (int(A))).
4. strongly β-I-open [11] if A ⊆ cl∗ (int(cl∗ (A))).
5. ⋆-dense [4] if cl∗ (A) = X.
6. t-I-set [10] if int(A) = int(cl∗ (A)).
7. semi∗ -I-open [7, 8] if A ⊆ cl(int∗ (A)).
The family of all semi-I-open (resp. α-I-open, pre-I-open) sets in an ideal topological space (X, τ ,
I) is denoted by SIO(X) (resp. αIO(X), PIO(X)).
Remark 1.4. [7] For several subsets defined above, we have the following implications.
pre-I-open set −→ strongly β-I-open set
↑
↑
open set −→ α-I-open set −→ semi-I-open set
The reverse implications are not true.
Lemma 1.5. [8] Every semi-I-open set is semi∗ -I-open in an ideal topological space.
Remark 1.6. The reverse implication of the above Lemma is not true in general as shown in [8].
Definition 1.7. The complement of a pre-I-open (resp. semi-I-open, α-I-open, semi∗ -I-open) set is
called pre-I-closed [3](resp. semi-I-closed [10], α-I-closed [10], semi∗ -I-closed [7, 8]).
Definition 1.8. [8] The pre-I-closure of a subset A of an ideal topological space (X, τ , I), denoted by
pI cl(A), is defined as the intersection of all pre-I-closed sets of X containing A.
Lemma 1.9. [8] For a subset A of an ideal topological space (X, τ , I), pI cl(A) = A ∪ cl(int∗ (A)).
Lemma 1.10. [6] For a subset A of an ideal topological space (X, τ , I), pI int(A) = A∩int(cl∗ (A)) and
pI int(X\A) = X\pI cl(A).
Definition 1.11. A subset A of an ideal topological space (X, τ , I) is said to be
44
South Asian J. Math. Vol. 7 No. 1
1. an ηζ-set [16] if A = L ∩ M, where L is open and M is clopen in X.
2. locally closed [2] if A = L ∩ M, where L is open and M is closed in X.
3. semi-I-regular [14] if A is a t-I-set and semi-I-open in X.
Definition 1.12. [9] An ideal space (X, τ , I) is called I-submaximal if every ⋆-dense subset of X is
open in X.
Definition 1.13. [7] An ideal space (X, τ , I) is said to be ⋆-extremally disconnected if ⋆-closure of every
open subset A of X is open.
Theorem 1.14. [9] For a subset A of an I-submaximal ideal space (X, τ , I), the following are equivalent:
1. A is semi-I-open,
2. A is strongly β-I-open.
Definition 1.15. [6] A subset A of an ideal topological space (X, τ , I) is called pre-I-regular if A is
pre-I-open and pre-I-closed in (X, τ , I).
Definition 1.16. [5, 6] A subset A of an ideal topological space (X, τ , I) is called A∗I -set if A = L ∩
M, where L is an open and M = cl(int*(M)) .
Definition 1.17. [6] Let (X, τ , I) be an ideal topological space and K ⊆ X. K is said to be
1. a C∗I -set if K = L ∩ M, where L is an open set and M is a pre-I-regular set in X.
2. a CI -set if K = L ∩ M, where L is an open set and M is a pre-I-closed set in X.
3. an ηI -set if K = L ∩ M, where L is an open set and M is an α-I-closed set in X.
Remark 1.18. [6] Let (X, τ , I) be an ideal topological space and K ⊆ X. The following diagram holds
for K. These implications are not reversible in general.
C∗I -set ⇒ CI -set
⇑
A∗I -set
⇒ ηI -set
Lemma 1.19. [1] Let (X, τ , I) be an ideal topological space. A subset A of X is α-I-open if and only
if it is semi-I-open and pre-I-open.
Proposition 1.20. [1] Let (X, τ , I) be an ideal topological space. If V ∈ PIO(X) and A ∈ αIO(X),
then V ∩ A ∈ PIO(X).
Theorem 1.21. [6] A subset A of an ideal topological space (X, τ , I) is semi∗ -I-closed if and only if A
is a t-I-set.
Definition 1.22. Let (X, τ , I) be an ideal topological space and A ⊆ X. A is said to be
45
K. Banumathi, et al: sA∗I -sets and decomposition of sA∗I -continuity
1. an αCI∗ -set [17] if A = L ∩ M, where L ∈ αIO(X) and M is a pre-I-regular set in X.
2. an ABI -set [14] if A = L ∩ M, where L ∈ τ and M is semi-I-regular set in X.
Theorem 1.23. [17] Let (X, τ , I) be an ideal topological space. Then each αCI∗ -set in X is a pre-I-open
but not conversely.
Remark 1.24. [17] In an ideal topological space, every α-I-open set and every pre-I-regular set is an
αCI∗ -set. The converses are not true in general.
sA∗I -sets and sCI -sets
2
Definition 2.1. A subset A of an ideal topological space (X, τ , I) is said to be
1. an sCI -set if A = L ∩ M, where L ∈ SIO(X) and M is a pre-I-closed set in X.
2. an sηI -set if A = L ∩ M, where L ∈ SIO(X) and M is an α-I-closed set in X.
3. an sA∗I -set if A = L ∩ M, where L ∈ SIO(X) and M = cl(int∗ (M)).
Remark 2.2. Let (X, τ , I) be an ideal topological space and A ⊆ X. The following diagram holds for A.
CI∗ -set
−→
↓
αCI∗ -set
CI -set
↓
−→
sCI -set
↑
A∗I -set
−→
sA∗I -set
−→
sηI -set
←−
ηI -set
The following Examples show that these implications are not reversible in general.
Example 2.3. Let X = {a, b, c, d}, τ = {∅, X, {a}, {b}, {a, b}, {a, b, c}} and I = {∅}. Then {a, b,
d} is αCI∗ -set but not an CI∗ -set.
Example 2.4. In Example 2.3, {a, d} is sCI -set but not an CI -set.
Example 2.5. In Example 2.3, {a, d} is sηI -set but not an ηI -set.
Example 2.6. In Example 2.3, {a, b, d} is sA∗I -set but not an A∗I -set.
Example 2.7. Let X = {a, b, c}, τ = {∅, X, {a}, {a, b}} and I = {∅}. Then {c} is sηI -set but not an
sA∗I -set.
Example 2.8. In Example 2.3, {c} is sCI -set but not an αCI∗ -set.
Example 2.9. Let X = {a, b, c}, τ = {∅, X, {a, b}} and I = {∅}. Then {a} is sCI -set but not an
sηI -set.
46
South Asian J. Math. Vol. 7 No. 1
Theorem 2.10. For a subset A of an ideal topological space (X, τ , I), the following properties are
equivalent.
1. A is an sCI -set and a semi∗ -I-open set in X.
2. A = L ∩ cl(int∗ (A)) for a semi-I-open set L.
Proof. (1) ⇒ (2): Suppose that A is an sCI -set and a semi∗ -I-open set in X. Since A is sCI -set, then we
have A = L ∩ M, where L ∈ SIO(X) and M is a pre-I-closed set in X. We have A ⊆ M, so cl(int∗ (A)) ⊆
cl(int∗ (M)). Since M is a pre-I-closed set in X, we have cl(int∗ (M)) ⊆ M. Since A is a semi∗ -I-open set in
X, We have A ⊆ cl(int∗ (A)). It follows that A = A ∩ cl(int∗ (A))=L ∩ M ∩ cl(int∗ (A)) = L ∩ cl(int∗ (A)).
(2) ⇒ (1): Let A = L ∩ cl(int∗ (A)) for a semi-I-open set L. We have A ⊆ cl(int∗ (A)). It follows
that A is a semi∗ -I-open set in X. Since cl(int∗ (A)) is a closed set, then cl(int∗ (A)) is a pre-I-closed set
in X. Hence, A is an sCI -set in X.
Theorem 2.11. Let (X, τ , I) be I-submaximal and ⋆-extremally disconnected ideal space. For a subset
A of (X, τ , I), the following properties are equivalent.
1. A is an sA∗I -set in X.
2. A is an sηI -set and a semi∗ -I-open set in X.
3. A is an sCI -set and a semi∗ -I-open set in X.
Proof. (1) ⇒ (2): Suppose that A is an sA∗I -set in X. It follows that A = L ∩ M, where L is a semi-Iopen set and M = cl(int∗ (M)). This implies A = L ∩ M = L ∩ cl(int∗ (M)) ⊆ cl∗ (int(L)) ∩ cl(int∗ (M)) ⊆
cl(int∗ (L)) ∩ cl(int∗ (M)) ⊆ cl(int∗ (L) ∩ int∗ (M)) = cl(int∗ (L ∩ M)) = cl(int∗ (A)). Thus A ⊆ cl(int∗ (A))
and hence A is a semi∗ -I-open set in X. Moreover, Remark 2.2, A is an sηI -set in X.
(2) ⇒ (3): It follows from the fact that every sηI -set is an sCI -set in X by Remark 2.2.
(3) ⇒ (1): Suppose that A is an sCI -set and a semi∗ -I-open set in X. By Theorem 2.10, A = L ∩
cl(int∗ (A)) for a semi-I-open set L. We have cl(int∗ (cl(int∗ (A)))) = cl(int∗ (A)). It follows that A is an
sA∗I -set in X.
Remark 2.12.
1. The notions of sηI -set and semi∗ -I-open set are independent of each other.
2. The notions of sCI -set and semi∗ -I-open set are independent of each other.
Example 2.13.
1. In Example 2.3, {c, d} is sCI -set as well as sηI -set but not a semi∗ -I-open set.
2. Let X = {a, b, c}, τ = {∅, X, {a}} and I = {∅, {a}}. Then {a, c} is semi∗ -I-open set but it is
neither sCI -set nor sηI -set.
Theorem 2.14. Let (X, τ , I) be an I-submaximal space and A ⊆ X. The following properties are
equivalent.
1. A is an α-I-open set.
2. A is a strongly β-I-open set and an αCI∗ -set.
47
K. Banumathi, et al: sA∗I -sets and decomposition of sA∗I -continuity
3. A is a strongly β-I-open set and a pre-I-open set.
Proof. (1) ⇒ (2): It is obvious.
(2) ⇒ (3): It follows from the fact that every αCI∗ -set is a pre-I-open set by Theorem 1.23.
(3) ⇒ (1): It follows from Theorem 1.14 and Lemma 1.19.
Definition 2.15. A subset A of an ideal topological space (X, τ , I) is said to be sgpI -open if N ⊆
pI int(A) whenever N ⊆ A and N is a semi-I-closed set in X.
Definition 2.16. A subset A of an ideal topological space (X, τ , I) is said to be semi-generalized pre-Iclosed (sgpI -closed) in X if X \ A is sgpI -open.
Theorem 2.17. For a subset A of an ideal topological space (X, τ , I), A is sgpI -closed if and only if
pI cl(A) ⊆ N whenever A ⊆ N and N is a semi-I-open set in (X, τ , I).
Proof. Let A be an sgpI -closed set in X. Suppose that A ⊆ N and N is a semi-I-open set in (X, τ , I).
Then X \ A is sgpI -open and X \ N ⊆ X \ A where X \ N is semi-I-closed. Since X \ A is sgpI -open,
then we have X \ N ⊆ pI int(X \ A), where pI int(X \ A) = (X \ A) ∩ int(cl∗ (X \ A)). By Lemma 1.10,
pI cl(A) ⊆ N. The converse is similar.
Theorem 2.18. Let (X, τ , I) be an ideal topological space and V ⊆ X. Then V is an sCI -set in X if
and only if V = G ∩ pI cl(V) for a semi-I-open set G in X.
Proof. It is similar to Theorem 32 of [6].
Theorem 2.19. Let (X, τ , I) be an ideal topological space and A ⊆ X. The following properties are
equivalent.
1. A is a pre-I-closed set in X.
2. A is an sCI -set and an sgpI -closed set in X.
Proof. (1) ⇒ (2): It follows from the fact that any pre-I-closed set in X is an sCI -set and an sgpI -closed
set in X.
(2) ⇒ (1): Suppose that A is an sCI -set and an sgpI -closed set in X. Since A is an sCI -set, then by
Theorem 2.18, A = G ∩ pI cl(A) for a semi-I-open set G in (X, τ , I). Since A ⊆ G and A is sgpI -closed
set in X, then pI cl(A) ⊆ G. It follows that pI cl(A) ⊆ G ∩ pI cl(A) = A. Thus, A = pI cl(A) and hence
A is pre-I-closed.
3
Further properties
Theorem 3.1. [9] For an ideal topological space (X, τ , I), then the following properties are equivalent.
1. X is I-submaximal.
2. Every pre-I-open set is open.
48
South Asian J. Math. Vol. 7 No. 1
3. Every pre-I-open set is semi-I-open and every α-I-open set is open.
Theorem 3.2. In an I-submaximal ideal space (X, τ , I), the following property holds.
Any αCI∗ -set is an ABI -set.
Proof. Suppose that A is an αCI∗ -set in X. It follows that A = L ∩ M, where L is an α-I-open set and M
is a pre-I-regular set in X. By Theorem 3.1, M is semi-I-open and semi-I-closed. It follows from Lemma
1.5 that M is semi-I-open and semi∗ -I-closed. By Theorem 1.21, M is semi-I-open and a t-I-set in X.
Hence M is semi-I-regular set. Thus, A is an ABI -set in X.
Theorem 3.3. [9] For an ideal topological space (X, τ , I), if X is I-submaximal and ⋆-extremally
disconnected, the following are equivalent for a subset A ⊆ X:
1. A is strongly β-I-open,
2. A is semi-I-open,
3. A is pre-I-open,
4. A is α-I-open,
5. A is open.
Theorem 3.4. For an ideal topological space (X, τ , I), if X is I-submaximal and ⋆-extremally disconnected, the following properties holds.
1. Any αCI∗ -set is an ηζ-set.
2. Any sηI -set is a locally closed set.
Proof. It follows from the above Theorem.
Definition 3.5. [8] An ideal space (X, τ , I) is said to be ⋆-hyperconnected if A is ⋆-dense for every open
subset A 6= φ of X.
Theorem 3.6. [8] The following properties are equivalent for an ideal topological space (X, τ , I) .
1. X is ⋆-hyperconnected.
2. A is ⋆-dense for every strongly β-I-open subset φ 6= A ⊆ X.
Theorem 3.7. For an ideal topological space (X, τ , I), the following properties are equivalent.
1. (X, τ , I) is ⋆-hyperconnected.
2. any αCI∗ -set in X is ⋆-dense.
Proof. (1) ⇒ (2): Let A be an αCI∗ -set in X. By Theorem 1.23, A is pre-I-open. By Remark 1.4, A is
strongly β-I-open set. Since (X, τ , I) is a ⋆-hyperconnected ideal topological space, then by Theorem
3.6, A is ⋆-dense.
(2) ⇒ (1): Suppose that any αCI∗ -set in (X, τ , I) is ⋆-dense in X. Since an open set A in X is an α-Iopen set and every α-I-open set A is an αCI∗ -set, then A is ⋆-dense. Thus, (X, τ , I) is ⋆-hyperconnected.
49
K. Banumathi, et al: sA∗I -sets and decomposition of sA∗I -continuity
Decompositions of sA∗I -continuity
4
Definition 4.1. A function f : (X, τ , I) → (Y, σ) is said to be
1. pre-I-continuous [3] if f−1 (A) is a pre-I-open set in X for every open set A in Y.
2. αCI∗ -continuous [17] if f−1 (A) is an αCI∗ -set in X for every open set A in Y.
3. PRI -continuous [6] if f−1 (A) is a pre-I-regular set in X for every open set A in Y.
Remark 4.2. For a function f : (X, τ , I) → (Y, σ), the following diagram holds. The reverses of these
implications are not true in general as shown in the following Examples.
pre-I-continuity
↑
αCI∗ -continuity
←− PRI -continuity
Example 4.3. Let X = {a, b, c}, τ = {∅, X, {a}}, I = {∅, {a}}, Y = {p, q, r}, σ = {∅, Y, {p}, {p,
q}, {p, r}} and J = {∅}. Then the function f : (X, τ , I) → (Y, σ, J ) defined by f(a) = p, f(b) = q and
f(c) = r is pre-I-continuous but not αCI∗ -continuous.
Example 4.4. Let X = {a, b, c}, τ = {∅, X, {a}}, I = {∅, {a}}, Y = {p, q, r}, σ = {∅, Y, {p}} and
J = {∅, {p}}. Then the function f : (X, τ , I) → (Y, σ, J ) defined by f(a) = p, f(b) = q and f(c) = r
is αCI∗ -continuous but not PRI -continuous.
Definition 4.5. A function f : (X, τ , I) → (Y, σ) is said to be
1. sCI -continuous if f−1 (A) is an sCI -set in X for every open set A in Y.
2. sA∗I -continuous if f−1 (A) is an sA∗I -set in X for every open set A in Y.
3. sηI -continuous if f−1 (A) is an sηI -set in X for every open set A in Y.
4. A∗I -continuous [6] if f−1 (A) is an A∗I -set in X for every open set A in Y.
Remark 4.6. For a function f : (X, τ , I) → (Y, σ), the following diagram holds. The reverses of these
implications are not true in general as shown in the following Examples.
sCI -continuity ←− αCI∗ -continuity
↑
sηI -continuity ←− sA∗I -continuity ←− A∗I -continuity
Example 4.7. Let X = {a, b, c, d}, τ = {∅, X, {a}, {b}, {a, b}, {a, b, c}}, I = {∅}, Y = {p, q, r,
s}, σ = {∅, Y, {p}, {q}, {p, q}, {p, q, s}} and J = {∅}. Then the function f : (X, τ , I) → (Y, σ, J )
defined by f(a) = p, f(b) = q, f(c) = r and f(d) = s is sA∗I -continuous but not A∗I -continuous.
50
South Asian J. Math. Vol. 7 No. 1
Example 4.8. Let X = {a, b, c}, τ = {∅, X, {a}, {a, b}}, I = {∅}, Y = {p, q, r}, σ = {∅, Y, {q},
{r}, {q, r}} and J = {∅}. Then the function f : (X, τ , I) → (Y, σ, J ) defined by f(a) = p, f(b) = q
and f(c) = r is sηI -continuous but not sA∗I -continuous.
Example 4.9. Let X = {a, b, c, d}, τ = {∅, X, {a}, {b}, {a, b}, {a, b, c}}, I = {∅}, Y = {p, q, r, s},
σ = {∅, Y, {r}, {s}, {r, s}} and J = {∅}. Then the function f : (X, τ , I) → (Y, σ, J ) defined by f(a)
= p, f(b) = q, f(c) = r and f(d) = s is sCI -continuous but not αCI∗ -continuous.
Example 4.10. Let X = {a, b, c}, τ = {∅, X, {a, b}}, I = {∅}, Y = {p, q, r}, σ = {∅, Y, {p}, {q, r}}
and J = {∅}. Then the function f : (X, τ , I) → (Y, σ, J ) defined by f(a) = p, f(b) = q and f(c) = r
is sCI -continuous but not sηI -continuous.
Definition 4.11. [6] A function f : (X, τ , I) → (Y, σ) is said to be semi∗ -I-continuous if f−1 (V) is a
semi∗ -I-open set in X for every open set V in Y.
Theorem 4.12. Let (X, τ , I) be I-submaximal and ⋆-extremally disconnected ideal space. The following
properties are equivalent for a function f : (X, τ , I) → (Y, σ):
1. f is sA∗I -continuous.
2. f is sηI -continuous and semi∗ -I-continuous.
3. f is sCI -continuous and semi∗ -I-continuous.
Proof. It follows from Theorem 2.11.
References
1
A. Acikgoz, T. Noiri and S. Yuksel, On α-I-continuous and α-I-open functions, Acta Math. Hungar., 105(1-2)(2004),
2
N. Bourbaki, Elements of Mathematics, General Topology, Part I, Hermann, Paris, Addison-Wesley Publishing Co.,
3
J. Dontchev, Idealization of Ganster-Reilly Decomposition theorems, arxiv:math.GN/9901017vl (1999).
4
J. Dontchev, M. Ganster and D. Rose, Ideal resolvability, Topology Appl., 93(1999), 1-16.
5
E. Ekici, On R-I-open sets and A∗I -sets in ideal topological spaces, Annals Univ. Craiova Math. Comp. Sci. Ser.,
27-37.
Reading, Mass-London-Don Mills, Ont., 1966.
38(2)(2011), 26-31.
6
E. Ekici, On A∗I -sets, CI -sets, CI∗ -sets and decompositions of continuity in ideal topological spaces, Analele Stiintifice
7
E. Ekici and T. Noiri, ⋆-extremally disconnected ideal topological spaces, Acta Math. Hungar., 122(1-2)(2009), 81-90.
8
E. Ekici and T. Noiri, ⋆-hyperconnected ideal topological spaces, An. Stiint. Univ.”Al. I. Cuza” Iasi. Mat. (N.S),
9
E. Ekici and T. Noiri, Properties of I-submaximal ideal topological spaces, Filomat, 24(4)(2010), 87-94.
ale Universitatii
l. I. Cuza din Iasi (S. N), LIX(2013), f.1, 173-184.
58(2012), 121-129.
10
E. Hatir and T. Noiri, On decompositions of continuity via idealization, Acta Math. Hungar., 96(4)(2002), 341-349.
11
E. Hatir, A. Keskin and T. Noiri, On a new decomposition of continuity via idealization, JP Jour. Geometry and
12
E. Hatir, A. Keskin and T. Noiri, A note on strong β-I-sets and strongly β-I-continuous functions, Acta Math.
13
D. Jankovic and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97(4)(1990), 295-310.
Topology, 3(1)(2003), 53-64.
Hungar., 108(1-2)(2005), 87-94.
51
K. Banumathi, et al: sA∗I -sets and decomposition of sA∗I -continuity
14
A. Keskin and S. Yuksel, On semi-I-regular sets, ABI sets and decompositions of continuity, RI C-continuity, AI -
15
K. Kuratowski, Topology, Vol. I, Academic Press (New York, 1966).
16
T. Noiri and O. R. Syed, On decompositions of continuity, Acta Math. Hungar., 111(2006), 1-8.
17
O. Ravi, V. Rajendran, K. Indirani and S. Vijaya, αA∗I -sets, αCI -sets, αCI∗ -sets and decompositions of α-I-continuity,
continuity, Acta Math. Hungar., 113(3)(2006), 227-241.
Italian Journal of Pure and Applied Mathematics, 35(2015), 129-142.
52