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K. V. Tamil Selvi et al., International Journal of Advanced Engineering Technology E-ISSN 0976-3945
Research Paper
α-SETS IN IDEAL TOPOLOGICAL SPACES
1K. V. Tamil Selvi, 2P. Thangaraj, 3O. Ravi
Address for Correspondence
1
Department of Mathematics, Kongu Engineering College, Perundurai, Erode District,
Tamil Nadu, India.
2
Department of Computer Science and Engineering, Bannari Amman Institute of Technology,
Sathyamangalam, Erode District, Tamil Nadu, India.
3
Department of Mathematics, P.M. Thevar College, Usilampatti, Madurai District,
Tamil Nadu, India.
ABSTRACT
The aim of this paper is to introduce α-sets in ideal topological spaces. The relationships
between the class of α-sets and the related sets are discussed.
KEYWORDS AND PHRASES. - set, α- set, weak - set, t- -set, α*- -set, α- -openset,semi-regular set, regular--closed set.
1. INTRODUCTION AND PRELIMINARIES
In 1986, Tong [32] introduced -sets and in 1989, Tong [33] introduced -sets in topological
spaces. In 1998, Dontchev [7] introduced the class of -sets which lies between the class
of -sets and the class of -sets. In 2009, Ekici and Noiri [11] introduced the class of αsets which is weaker form of the class of -sets. They also studied the relationships
between α-sets and the related sets; and some decompositions of α-continuity, αcontinuity, continuity and -continuity were provided by them.
In this paper, the class of α-sets is introduced in ideal topological spaces. Some new
relationships between the class of α-sets and the related sets are obtained. Also,
properties of α-sets are discussed.
In the present paper(X,τ) or(Y,σ) will denote topological spaces with no separation properties
assumed. For a subset V of X, let cl(V) and int(V) denote the closure and the interior of V,
respectively, with respect to the topological space(X,τ).
An ideal on a topological space (X, τ) is a non-empty collection of subsets of X which
satisfies the following conditions.
(1) A∈and B⊆A imply B∈and
(2) A ∈and B ∈imply A∪ B ∈.
Given a topological space (X, τ ) with an ideal on X if (X) is the set of all subsets
of X, a set operator (•)* : (X) →(X), called a local function [34] of A with respect to τ and
is defined as follows: for A  X, A* (, τ ) = {x ∈X | U ∩A  for every U∈ τ(x)} where
τ(x)={U ∈ τ | x ∈ U}. A Kuratowski closure operator cl* (•) for a topology τ * ( , τ ), called the
* -topology, finer than τ is defined by cl* (A) = A  A* (, τ ) [25]. We will simply write A* for
A* (, τ ) and τ * for τ * ( , τ ). If is an ideal on X, then (X, τ ,) is called an ideal topological
space. Notice that int* (A) denotes the interior of A in (X, τ* ).
Definition 1.1.A subset H of an ideal topological space (X, τ, ) is called
1. α--open [16] if H ⊆int(cl* (int(H))),
2. semi--open [16] if H ⊆cl* (int(H)),
3. pre--open [8] if H ⊆int(cl* (H)),
4. t--set[16] if int(cl* (H)) = int(H),
5. an α* --set[16]if int(H)=int(cl* (int(H))),
6. regular--closed[22]if H=(int(H))*,
7. * -closed [21] if H* ⊆ H or cl* (H) = H,
8. semi* --open[12] if H⊆cl(int * (H)),
9. semi* --closed[12] if its complement is semi*--open,
10. β--open [16] if H ⊆cl(int(cl* (H))),
02010 Mathematics Subject Classification: 54A10, 54C10, 54D10, 54D15
11. β--closed[16] if its complement is β--open
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K. V. Tamil Selvi et al., International Journal of Advanced Engineering Technology E-ISSN 0976-3945
12. semi --regular [23] if H is both a t--set and a semi -open set,
13. β--regular [35] if H is both a β-I-open set and an α*--set,
14. strong β--open [18] if H ⊆cl* (int(cl* (H))),
15. τ*-dense [20] if cl* (H) =X,
16. an α-set [28] or αI N3 -set[3] if H∈α (X) ={A∩B:A is α--open and B is
at--set},
17. an α -set [28] or α N4 -set[3] if H∈α (X)={A∩B: A is α--open and B is an
α* --set},
18. an  -set [16] if H∈(X)={A∩B:A is open and B is an α* --set},
19. an   -set [23] if H∈  (X) ={A∩B : A is open and B is semi--regular},
20. weak  -set [35] if H ∈W (X)={A∩B:A is open and B is β-- regular},
21. weakly--locally closed [24] if H∈WLC(X)={A∩B:A∈τ and B is* -closed},
22. an  -set [22] if H∈ (X)={A∩B:A∈τ and B is regular--closed},
23. a  -set [16] if H∈ (X)={A∩B:A∈τ and B is at--set}.
Proposition 1.2.
[23] Every regular--closed set is a semi--regular but not conversely.
Proposition 1.3.
[16] Every t--set is an α* --set.
Proposition 1.4.
[23] Every-se tis a -set.
Proposition 1.5.
[16] Every open set is an α--open but not conversely.
Proposition 1.6.
[2]The following are equivalent for a subset H of an ideal topological space (X, τ, ):
(1) α--open.
(2) semi--open and pre--open.
Proposition 1.7.
[18] Every semi-I-open set is a strong β--open.
Proposition 1.8. [23]
(1) Every semi--regular set is at--set.
(2) Every semi-I-regular set is a semi-I-open set.
Proposition 1.9.
[23] For a subset of an ideal topological space (X, τ, ), the following property holds:
Every AB-set is semi- -open.
Remark 1.10.
[35]The following properties hold for a subset of an ideal topological space (X, τ, ):
(1) Every semi--regular set is β--regular but not conversely.
(2) Every-set is a weak -set but not conversely.
(3) Every weak -set is a -set but not conversely.
(4) The notions of α--open sets and weak -sets are independent.
Remark 1.11.
[14] Let H be a subset of an ideal topological space (X, τ, ). Then the β--closure of
H, denoted by β--cl(H), is the smallest β--closed set containing H.
Remark 1.12.
[22] Every regular--closed set is * -closed.
Remark 1.13.
[29] Let H be a subset of an ideal topological space (X, τ, ). The smallest semi*-closed set containing H is called the semi*--closure of H and is denoted by s*-cl(H).
Theorem 1.14.
[29] Let (X, τ, ) be an ideal topological space and H ⊆ X. Then the following hold:
s *cl(H) =H ∪ int (cl* (H)).
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Theorem 1.15.
[29] Let (X, τ, ) be an ideal topological space and H be a strong β--open subset of X.
Then s* cl(H) is semi--regular.
Lemma1.16.
[28]The following are equivalent for a subset H of an ideal topological space (X, τ, ):
(1) H is an α -set.
(2) H=U∩s*cl(H) for some α--open set U
Proposition 1.17.
[2] Let H be a subset of an ideal topological space (X, τ, ).
(1) If V is semi--open and A is α--open ,then H=V∩A is semi--open.
(2) If V is pre--open and A is α--open, then H=V∩A is pre--open.
Lemma 1.18.
[4] Let (X, τ, ) be an ideal topological space and A⊆ X. Then A is
α--open if and only if A = U∩V where U is open and int(V) is τ* -dense.
Definition 1.19.
[6] A subset H of an ideal topological space (X, τ, ) is said to be-dense if H* = X.
Definition 1.20.
[6] An ideal topological space (X, τ, ) is said to be -hyper connected if every non
empty open subset of X is -dense.
Theorem 1.21.
[30] Let H be a subset of an ideal topological space (X, τ, ). Then the following holds.
H is semi--regular if and only if H is both strong β--open and semi* --closed.
Proposition 1.22.
In an ideal topological space (X, τ, ),
(1) if U ∈ τ and W is α--open set, then U∩W is α--open[26].
(2) if U ∈ τ and W is β--open set, then U∩W is β--open[17].
Definition 1.23.
[12] An ideal topological space X is called *-extremally disconnected if the *-closure of
every open subset of X is open.
Lemma 1.24.
[19] Let K be a subset of an ideal topological space (X, τ, ). If N is open, then
N∩cl * (K) ⊆ cl* (N∩K).
Definition1.25.
An ideal topological space (X, τ,  ) is called I-submaximal [1, 13]if every τ*-dense subset
of X is open.
Theorem1.26.
[13]For an ideal topological space(X, τ, ),the following properties are equivalent:
(1) X is-submaximal.
(2) Every pre--open set is open.
(3) Every pre--open set is semi--open and every α--open set is open.
2. α-sets
Definition 2.1.
A subset H of an ideal topological space (X, τ, I) is called
(1) an α-set if H ∈ α(X) ={U  V:U is α- -open and V is semi- - regular}.
(2) an  -set if cl* (int(H))=X.
Remark 2.2.
(1) Every α- -open (semi- -regular)set is an α-set but not conversely.
(2) The following diagram holds for a subset H of an ideal topological
space (X, τ, ):
-set
 -set
 -set
 -set
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K. V. Tamil Selvi et al., International Journal of Advanced Engineering Technology E-ISSN 0976-3945
α-set
α- set
α -set
None of these implications is reversible as shown in the following examples.
Example 2.3.
Let X = {a, b, c, d}, τ = {∅, {a},{b},{a, b},{b, c},{a, b, c},X} and I={∅,{b}}. Then {a, d} is
an α -set but not an α- -open.
Example 2.4.
Let X, τ and be as in Example2.3. Then {a, b, c} is an α-set but not a semi-Iregular.
Example 2.5.
Let X={a, b, c, d},τ={∅,{a},{d},{a, d},X} and I={∅,{d}}. Then {a, b} is an -set but
not an - set.
Example 2.6.
Let X={a, b, c, d},τ={∅,{d},{a, c},{a, c ,d},X}and={∅,{c},{d},{c, d}}.Then {b } is a
- set but not an -set.
Example 2.7.
In Example 2.6,{a, b, d} is a - set but not a - set.
Example 2.8.
Let X, τ and be as in Example 2.5.Then {a, b, d} is an α -set but not an  - set.
Example 2.9.
Let X={a, b, c, d},τ={∅,{a},{a, b},X} and ={∅,{c}}.Then {a, b, c} is an α- set but not a
- set.
Example 2.10.
In Example 2.6, {b} is an α- set but not an α- set.
Example 2.11.
Let X, τ and be as in Example2.5.Then {a, b, d } is an α - set but not a - set.
Example 2.12.
Let X={a, b, c},τ={∅,{a, b}, X} and ={∅,{c}}. Then {b, c} is an α- set but not an
α-set.
Proposition 2.13.
For a subset of an ideal topological space (X ,τ, ),the following property holds:
Every α- set is semi-  -open.
Proof. Let H be an α- set. Then H = U∩V where U is an α-- open set and V is a semi-regular set. By Proposition 1.8, V is a semi--open set. By Proposition 1.17(1), H is semi-open.
Theorem 2.14.ThefollowingareequivalentforasubsetHofanidealtopological space (X, τ, ):
(1) H is an α- set.
(2) H is a semi- -open and an α-set.
(3) H is a strong β- -open and an α- set.
Proof. (1)⇒(2): Every α- set is a semi- -open and an α- set.
(2) ⇒(3):Obvious.
(3) ⇒(1):Let H be a strong β- -open and an α- set. By Lemma1.16, H= U∩s*cl(H)
for some α--open set U. Since H is strong β--open, by Theorem1.15, s* cl (H) is semi -regular. Hence, H is an α- set.
Example 2.15.
(1) Let X, τ and be as in Example 2.3. Then {a, c} is an α-set but not a semi--open.
(2) Let X = {a, b, c, d, e}, τ = {∅, X, {a}, {e}, {a, e}, {c, d}, {a, c, d}, {c, d, e},{b, c, d,e},
{a, c ,d, e}} and ={∅} then {b, c ,d} is a semi--open but not an α -set.
Example 2.16.
(1) Let X, τ and be as in Example 2.3. Then {a, c} is an α- set but not a strong β-open.
(2) Let X, τ and be as in Example 2.6. Then {a} is a strong β--open but not an α- set.
Theorem 2.17.
The following are equivalent for a subset H of an ideal topological space (X, τ, ):
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(1) H is an α--open.
(2) H is a pre- -open and an α-set.
Proof.(1) ⇒(2): Since every α--open set is a pre--open and an α-set, it is obvious.
(2) ⇒(1): Let H be a pre--open and an α-set. By Proposition 2.13, H is semi-open. Since H is semi--open and pre-- open, by Proposition 1.6, it is an α--open.
Example 2.18.
(1) Let X = {a, b, c}, τ = {∅, {a}, {b}, {a, b}, X} and ={∅,{a}}.Then {b, c} is an αset but not a pre--open.
(2) In Example 2.12,{a, c} is a pre- -open but not an α- set.
Theorem 2.19.
The following are equivalent for a subset H of an ideal topological space (X, τ, ):
(1) H is an α-set.
(2) H=A∩B where A is an-set and B is an  - set.
Proof.(⇒) : Let H be an α-set. This implies H = C∩D where C is α--open and D is
semi--regular. By Lemma 1.18, we have C = E∩F where E is open and F is an  -set.
Moreover, we have H=C∩D=E∩F∩D=(E∩D)∩F such that A= E∩D is an -set and B=F is
an -set.
(⇐) : Let H=A∩B where A is an - set and B is an  -set. Since A is an -set,
there exist an open set U and a semi--regular set V such that A = U∩V. We have
H=A∩B=U∩V∩B=(U∩B)∩V where U∩B is, by Lemma1.18,an α-- open. Thus, H is an
α-set.
Example 2.20.
(1) Let X = {a, b, c, d}, τ = {∅, {c}, {a, c}, {b, c}, {a, b, c},{a, c, d},X} and
={∅,{c}}.Then{c} is anset but not an  -set.
(2 )Let X , τ and be as in Example 2.9. Then {a, b, c} is an  -set but not an
-set.
Theorem 2.21.
Let (X, τ, ) be an ideal topological space. Then the following are equivalent.
(1) X is- hyper connected.
(2) Every semi--open set is an -dense set.
(3) Every α-set is an -dense set.
(4) Every-set is an -dense set.
Proof .(1) ⇒(2): Let H be a semi--open set. Then there exists an open set G such that G  H
 cl* (G). Since G is -dense, G* = X and so H*= X which implies that H is an-dense.
(2) ⇒(3): is clear, by Proposition 2.13.
(3) ⇒(4): is clear, by Remark 2.2.
(4) ⇒(1):Let H be a non empty open set. Then H is an-set and so by (4),H is an dense set.
Theorem 2.22.
Let H be a subset of an - submaimal ideal topological space (X, τ,  )Then the following
are equivalent.
(1) H is at--set.
(2) H is a semi*--closed set.
(3) H is both an α*--set and an α-set.
Proof.
(1) ⇒(2): H is a t--set implies that int(H) = int(cl*(H)) which implies int(cl*(H))  H
and so H is a semi*--closed set. Conversely, if H is semi*--closed, then int(cl*(H))  H and so
it follows that int(cl*(H))=int(H).Hence H is a t--set.
(2) ⇒(3): Clearly int(H)  int(cl*(int(H))). But int(cl*(int(H)))  int(cl*(H))  H and
so int(cl*(int(H)))  int(H). Hence int(cl*(int(H))) = int(H) which implies that H is an α*--set.
Also, H = H∩X where H is a t--set by (1) and X is α--open. Thus H is an α-set.
(3) ⇒(1): Suppose H is both an α*--set and an α-set. Th en H = U∩V where U
is an α-- open and V is a t--set. Now int (cl*(H)) = int (cl* (U∩V))  int [cl*(U)∩cl* (V)]=int
Int J AdvEngg Tech/Vol. VII/Issue II/April-June,2016/1012-1019
K. V. Tamil Selvi et al., International Journal of Advanced Engineering Technology E-ISSN 0976-3945
(cl*(U))∩int (cl*(V)) = int (cl*(U)) ∩ int (V)=int [cl* (U)∩ int (V)]  int (cl* [U∩ int (V)]) =
int (cl* (int (U∩V))) = int (cl* (int (H))) = int (H) and so int (cl*(H))  int (H). But
int (H)  int(cl*(H)) which implies that H is a t--set.
Example 2.23.
Let X = {a, b, c, d}, τ = {  , {a, b}, {a, b, c}, {a, b, d}, X} and  = {  , {b}}. Then {a}
is an α*--set but not an α- set and {a, b} is an α-set but not an α*--set. This shows that
α*--sets and α- sets are independent of each other, in general.
Theorem 2.24.
Let H be a subset of an -sub maximal ideal topological space(X, τ,  )Then the following
are equivalent.
(1) H is semi--regular set.
(2) H is semi*--closed set and an α-set.
(3) H is an α*--set and α -set.
Proof.(1) ⇒(2): is clear. [see Theorem 1.21]
(2) ⇒(3):is clear.[see Theorem 2.22]
(3) ⇒(1): If H is an α-set,byTheorem 2.14, H is both a semi--open and an αset. Again, if H is an α*- -set, by Theorem 2.22, H is a t- -set and so H is a semi-- regular set.
Remark 2.25.
The following examples show that the concepts of semi*--closed set and α-set are
independent of each other in general.
Example 2.26.
Let X, τ and be as in Example 2.12.Then {c}is a semi*-I-closed but not an α-set.
Example 2.27.
Let (R, τ) be the real numbers with the usual topology τ with ideal={∅}. Then R\{0} is
an α-set, but it is not a semi*- -closed set. If ={∅}, then H* =cl(H) and cl* (H)=cl(H)
for every subset H of an ideal topological space. Let A = R \{0}. Then cl*(A) = R and
int(cl* (A)) = R. Since int(cl* (A))  A, A is not a semi*--closed set. On the other hand,
A=A∩R where A is open (and hence α- -open) and R is semi- -regular. This shows that A is an
α-set.
Example 2.28.
Let X, τ and be as in Example2.12.Then {a, b } is an α-set but not an α* - -set and
{c} is an α* - - set but not an α-set. This shows that α- sets and α*- -sets are independent
of each other, in general.
Theorem 2.29.
Let(X, τ, )bean* - extremely disconnected ideal space. Then the following property
holds:
α  O(X)=α(X) where α O(X)denotes the family of α--open subsets of X.
Proof.
We know that every α--open set is α-set. Hence α O(X) ⊆α(X). Suppose H
∈ α(X). Then H = U∩V where U is an α--open and V is a semi--regular. Now V is semi-regular implies that V is a t--set and also a semi--open. Hence int(V) = int(cl*(V)) and V
 cl* (int(V)) which implies that int(V) = int(cl* (V)) and cl* (V) = cl* (int(V)). Since X is * extremally disconnected, int(cl* (int(V))) = cl* (int(V)) = cl* (V). Thus, int(V) =
int(cl* (V)) = int(cl* (int(V)))= cl* (V)  V and so V is open. We have U is α--open set and V
is open set. By Proposition 1.22, H = U∩V is α--open. Therefore α(X) ⊆ αO(X). Hence
αO(X) =α(X).
3. Further Properties
Theorem 3.1.
The following are equivalent for a subset H of an ideal topological space (X, τ, ):
(1) H is open,
(2) H is α- -open and-set,
(3) H is α- -open and weak-set,
(4) H is α- -open and a -set.
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K. V. Tamil Selvi et al., International Journal of Advanced Engineering Technology E-ISSN 0976-3945
Proof.(1) ⇒(2):Obvious.
(2) ⇒(3): Obvious, by Remark1.10.
(3) ⇒(4): Obvious, by Remark1.10.
(4) ⇒(1): By the α--openness of H, H ⊆ int(cl* (int(H))) = int(cl* [int(U∩V)]) where
U ∈ τ and V is an α*--set. Hence H  U∩H  U∩int(cl* [int(U∩V)]) = U∩int (cl* [int(U)∩int
(V)]  U∩int[cl* (int(U))∩cl* (int(V))] =U∩int(cl* (int(U)))∩int(cl* (int(V))) =U∩int(V) = int (U∩V)
= int (H). This shows that H is open.
Example 3.2.
The notions of α- -open sets and-sets are independent as is shown in[23].
Example 3.3.
See Remark 1.10.
Example 3.4.
The notions of α--open sets and -sets are independent as is shown in [16].
Definition 3.5.
A subset H of an ideal topological space(X, τ,) is called a-set if H = G∩B,
where G is open and B is β--closed.
The family of all  -sets of a space(X, τ , )will be denoted by  (X).
Theorem 3.6.
Let H be a subset of an ideal topological space(X, τ, ).Then
H∈  (X) if and only if H=G∩β- -cl(H)for some open set G.
Proof.(⇐) : Assume that H = G∩β--cl(H) for some open set G. Since β--cl(H) is β- -closed,
H∈  (X),
(⇒) : Let H∈(X). We have H=G∩A where G is open and A is β-- closed. Since H
⊆ A, β--cl(H) ⊆ β--cl(A) = A. Hence, G ∩β--cl(H) ⊆ G∩A = H ⊆G∩β--cl(H) and hence
H = G ∩β--cl(H).
Definition 3.7.
A subset H of an ideal topological space (X, τ, ) is said to be
(1) gβ--closed if β--cl(H) ⊆M whenever H ⊆M and M is open set in X.
(2) gβ--open if XH is gβ--closed.
Theorem 3.8.
For a subset A of an ideal topological space (X, τ, ),the following are equivalent.
(1) A is β--closed,
(2) A is a  -set and gβ- -closed.
Proof.(1)⇒(2):Since every β--closed set is a-set and gβ--closed, it is completed.
(2) ⇒(1):Let A be a  -set. Then we have A=G∩β- -cl(A) for an open set G in X. We
have A  G. Since A is gβ--closed, then β--cl(A) ⊆G. Thus, β--cl(A)  G ∩ β-
-cl(A) = A and also, A is β--closed.
Theorem 3.9.Let H be a subset of an ideal topological space (X, τ, ). If H ∈(X),then
(1) β--cl(H)|H is β--closed.
(2) H∪(X|β--cl(H)) is β--open.
Proof.(1)Let H∈(X).By Theorem 3.6, H=V∩β--cl(H) for some open set V. Hence β-cl(H)\H = β--cl(H)\(V∩β--cl(H)) = β--cl(H)∩(X\(V∩β--cl(H))) = β--cl(H)∩ ((X\V)∪ (X\β-cl(H))) = (β--cl (H)∩(X\V))∪ (β--cl(H)∩ (X\β--cl(H))) = (β--cl (H)∩(X\V))∪∅= β--cl(H)
∩(X\V). Thus, β--cl(H)\H is β--closed.
(2)Sinceβ--cl(H)\Hisβ--closed,X\(β--cl(H)\H)isβ--open.HenceX\(β- - cl(H)\H)=X\
(β--cl(H)∩(X\H))=(X\β--cl(H))∪ H.Thus,H∪(X\β--cl(H))is β--open.
Remark 3.10. We obtain the following diagram for the subsets stated above:
α(X)
(X)
(X)W
(X)
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K. V. Tamil Selvi et al., International Journal of Advanced Engineering Technology E-ISSN 0976-3945
WLC(X)
(X)
(X)
Remark 3.11.
In the above diagram, none of the implications is true as is shown by the following
four Examples and the above Examples(2.5,2.6,2.7and2.8).
(1) InExample2.23,{a}is weak-set but not an -set.
(2) InExample2.23,{b}is-set but not weak -set.
(3) InExample2.23,{b}is weakly- -locally closed set but not-set.
(4) InExample2.20,{a}is-set but not weakly- -locally closed set.
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