Articulo 4.dvi


CUBO A Mathematical Journal
Vol.12, No¯ 02, (43–52). June 2010

On subsets of ideal topological spaces

V. Renukadevi

Department of Mathematics,

ANJA College,

Sivakasi-626 124, Tamil Nadu, India.

email: renu siva2003@yahoo.com

ABSTRACT

We define some new collection of sets in ideal topological spaces and characterize them

in terms of sets already defined. Also, we give a decomposition theorem for α − I−open

sets and open sets. At the end, we discuss the property of some collection of subsets in

⋆−extremally disconnected spaces.

RESUMEN

Definimos una nueva colección de conjuntos en espacios topológicos ideales y caracteri-

zamos estos en términos de conjuntos ya definidos. También damos un teorema de descom-

posición para α−I− abiertos y conjuntos abiertos. Finalmente discutimos la probabilidad

de algunas colecciones de subconjuntos en espacios disconexos ⋆− extremos.

Key words and phrases: ⋆−extremally disconnected spaces, t−I−set, α−I−open set, pre−I−open

set, semi−I−open set, semi⋆ − I−open, semipre⋆ − I−open, CI−set, BI−set, B1I−set, B2I−set,

B3I−set, δ −I−open, RI−open, I−locally closed set, weakly I−locally closed set, AIR−set, DI−set.

2000 AMS subject Classification: Primary: 54 A 05, 54 A 10



44 V. Renukadevi CUBO
12, 2 (2010)

1 Introduction

By a space, we always mean a topological space (X, τ ) with no separation properties assumed. If

A ⊂ X, cl(A) and int(A) will, respectively, denote the closure and interior of A in (X, τ ). 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. 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 f unction [14] 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}. We will make use of the basic

facts concerning the local function [11, Theorem 2.3] without mentioning it explicitly. 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 ) [16]. When there is no chance for confusion, we will simply write A⋆ for

A⋆(I, τ ) and τ ⋆ or τ ⋆(I) for τ ⋆(I, τ ). int⋆(A) will denote the interior of A in (X, τ ⋆). If I is an ideal

on X, then (X, τ, I) is called an ideal space. A subset A of an ideal space (X, τ, I) is τ ⋆ − closed or

⋆ − closed [11](resp.⋆ − perf ect[10] ) if A⋆ ⊂ A(resp.A = A⋆). A subset A of an ideal space (X, τ, I) is

said to be a t − I − set[8] if int(A) = int(cl⋆(A)). A subset A of an ideal space (X, τ, I) is said to be

δ − I − open[2](resp. α − I − open [8],pre − I − open [6],semi − I − open [8], strong β − I − open[9])

if int(cl⋆(A)) ⊂ cl⋆(int(A))(resp. A ⊂ int(cl⋆(int(A))), A ⊂ int(cl⋆(A)), A ⊂ cl⋆(int(A), A ⊂

cl⋆(int(cl⋆(A))). We will denote the family of all δ − I−open (resp. α − I−open, pre−I−open,

semi−I−open, strong β−I−open) sets by δIO(X)(resp.αIO(X), P IO(X), SIO(X), sβIO(X)). The

largest preI−open set contained in A is called the pre−I −interior of A and is denoted by pIint(A).

For any subset A of an ideal space (X, τ, I), pIint(A) = A ∩ int(cl⋆(A)) [15, Lemma 1.5].

2 Subsets of Ideal Topological Spaces

Let (X, τ, I) be an ideal space. A subset A of X is said to be a semi⋆ − I−open set [7] if

A ⊂ cl(int⋆(A)). A subset A of X is said to be a semi⋆ − I−closed set [7] if its complement is a

semi⋆ − I−open set. Clearly, A is semi⋆ − I−closed if and only if int(cl⋆(A)) ⊂ A if and only if

int(cl⋆(A)) = int(A) and so semi⋆ − I−closed sets are nothing but t − I−sets. A is said to be a

semipre⋆ − I−closed set if int(cl⋆(int(A))) ⊂ A. Clearly, A is said to be a semipre⋆ − I−closed if

and only if int(cl⋆(int(A))) = int(A) if and only if A is α⋆ − I−set [8]. Clearly, X is both semi⋆ −

I−closed and semipre⋆ − I−closed. The smallest semi⋆ − I−closed (resp. semipre⋆ − I−closed) set

containing is called the semi⋆ − I − closure (resp.semipre⋆ − I − closure) of A and is denoted by

SIcl(A)(resp.spIcl(A)). A subset A of an ideal space (X, τ, I) is said to be a BI−set[8] if A = U ∩ V

where U is open and V is a t − I−set. The easy proof of the following Theorem 2.1 is omitted

which says that the arbitrary intersection of semi⋆ − I−closed (resp. semipre⋆ − I−closed) set is a

semi⋆ − I−closed (resp. semipre⋆ − I−closed) set.

Theorem 2.1. Let (X, τ, I) be an ideal space and A ⊂ X. If {Aα | α ∈ △} is a family of semi
⋆ −

I−closed (resp. semipre⋆ − I−closed) sets, then ∩Aα is a semi
⋆ − I−closed (resp. semipre⋆ −

I−closed) set.

Theorem 2.2. Let (X, τ, I) be an ideal space and A ⊂ X. Then the following hold.

(a)SIcl(A) = A ∪ int(cl⋆(A)).



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On subsets of ideal topological spaces 45

(b)spIcl(A) = A ∪ int(cl⋆(int(A))).

Proof. The proof follows from Theorem 1.3 and Theorem 3.1 of [5].

Every semi⋆ − I−closed set is a semipre⋆ − I−closed set but not the converse as shown by

the following Example 2.3. Theorem 2.4 below shows that the reverse direction is true if the set is

semi−I−open. Theorem 2.5 gives a characterization of t − I−sets.

Example 2.3. Consider the ideal space (X, τ, I) where X = {a, b, c, d}, τ = {∅, {d}, {a, c}, {a, c, d}, X}

and I = {∅, {c}, {d}, {c, d}}. If A = {a}, then

int(cl⋆(int(A))) = int(cl⋆(∅)) = ∅ ⊂ A and so A is semipre⋆ − I−closed. Since int(cl⋆(A)) =

int(cl⋆({a})) = int({a, b, c}) = {a, c} * {a}, A is not semi⋆ − I−closed.

Theorem 2.4. Let (X, τ, I) be an ideal space and A ⊂ X be semipre⋆ − I−

closed. If A is semi−I−open, then A is semi⋆ − I−closed.

Proof. If A is semi−I−open, then A ⊂ cl⋆(int(A)) and so cl⋆(A) ⊂ cl⋆(int(A)). Now int(cl⋆(A)) ⊂

int(cl⋆(int(A))) ⊂ A and so A is semi⋆ − I−closed.

Theorem 2.5. Let (X, τ, I) be an ideal space and A ⊂ X. Then the following are equivalent.

(a) A is a t − I−set.

(b) A is semi⋆ − I−closed.

(c) A is a semipre⋆ − I−closed BI−set.

Proof. Enough to prove that (c)⇒(a). Suppose A is a semipre⋆−I−closed BI−set. Then A = U ∩V

where U is open and V is a t − I−set. Now int(cl⋆(A)) = int(cl⋆(U ∩ V )) ⊂ int(cl⋆(U ) ∩ cl⋆(V )) =

int(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(A))) ⊂ A and so int(cl⋆(A)) ⊂ int(A). But int(A) ⊂ int(cl⋆(A))

and so int(A) = int(cl⋆(A)) which implies that A is a t − I−set.

The following Example 2.6 shows that the union of two semi⋆ − Iclosed (resp. semipre⋆ −

I−closed) set is not a semi⋆ − Iclosed (resp. semipre⋆ − I−closed) set.

Example 2.6. Consider the ideal space(X, τ, I) of Example 2.3. If A = {a, c} and B = {d}, then

int(cl⋆(A)) = int(cl⋆({a, c})) = int({a, b, c}) = {a, c} = A and so A is semi⋆ − I−closed and hence

semipre⋆ − I−closed. Also, int(cl⋆(B)) = int(cl⋆({d})) = int({d}) = {d} = B. Therefore, B is

semi⋆ − I−closed and so semipre⋆ − I−closed. But int(cl⋆(int(A ∪ B))) = int(cl⋆(int({a, c, d}))) =

int(cl⋆({a, c, d})) = int(X) = X * A ∪ B and so A ∪ B is not semipre⋆ − I−closed and hence A ∪ B

is not semi⋆ − I−closed.

A subset A of an ideal space (X, τ, I) is said to be a CI−set [8] if A = U ∩ V where U is open

and V is a semipre⋆ − I−closed set. We will denote the family of all CI−set by CI(X). The following

Theorem 2.7 gives a characterization of BI−sets and CI−sets.

Theorem 2.7. Let (X, τ, I) be an ideal space and A be a subset of X. Then the following hold.

(a) A is a BI−set if and only if there exists an open set U such that A = U ∩ SIcl(A).

(b) A is a CI−set if and only if there exists an open set U such that A = U ∩ spIcl(A).

Proof. (a) Suppose A is a BI−set. Then A = U ∩ V where U is open and V is a t − I−set.

Since t − I−sets are semi⋆ − I−closed sets, SIcl(V ) = V. Now A = U ∩ A ⊂ U ∩ SIcl(A) ⊂

U ∩ SIcl(V ) = U ∩ V = A and so A = U ∩ SIcl(A). Conversely, suppose A = U ∩ SIcl(A) for some



46 V. Renukadevi CUBO
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open set U. Since SIcl(A) is semi⋆ − I−closed, int(cl⋆(SIcl(A))) ⊂ SIcl(A). Also, int(SIcl(A)) ⊂

int(cl⋆(SIcl(A))) ⊂ SIcl(A) and so int(SIcl(A)) = int(cl⋆(SIcl(A))) which implies that SIcl(A) is

a t − I−set. Therefore, A is a BI−set.

(b) The proof is similar to that of (a).

A subset A of an ideal space (X, τ, I) is said to be a A1I−set (resp. B1I−set [4](αIM1−set [1]) )

if A = U ∩ V where U is open (resp.α − I−open ) and cl⋆(int(V )) = X. We will denote the family of

all B1I−sets (resp. A1I−sets) by B1I(X) (resp.A1I(X)). Clearly, A1I(X) ⊂ B1I(X). The following

Theorem 2.8 shows that B1I−sets and A1I−sets are nothing but α − I−open sets.

Theorem 2.8. Let (X, τ, I) be an ideal space. Then B1I(X) = αIO(X) = A1I(X).

Proof. Suppose A ∈ B1I(X). Then A = U ∩ V where U is α − I−open and cl
⋆(int(V )) = X. Since

V ⊂ X = int(cl⋆(int(V ))), V ∈ αIO(X). Since αIO(X) is a topology on X, A ∈ αIO(X) and so

B1I(X) ⊂ αIO(X).

Suppose A ∈ αIO(X). Then A ⊂ int(cl⋆(int(A))) and so A = int(cl⋆(int(A)))∩(X−(int(cl⋆(int(A)))−

A)) = int(cl⋆(int(A))) ∩ ((X − int(cl⋆(int(A)))) ∪ A). Also, cl⋆(int((X − int(cl⋆(int(A)))) ∪ A)) ⊃

cl⋆(int(X − int(cl⋆(int(A)))) ∪ int(A)) = cl⋆(int(X − int(cl⋆(int(A))))) ∪ cl⋆(int(A)) ⊃ cl⋆(int(X −

cl⋆(int(A))))∪ cl⋆(int(A)) ⊃ int(X−cl⋆(int(A)))∪cl⋆(int(A)) ⊃ int((X−cl⋆(int(A)))∪cl⋆(int(A))) =

int(X) = X. Therefore, A ∈ A1I(X) which implies that αIO(X) ⊂ A1I(X).

Clearly, A1I(X) ⊂ B1I(X). This completes the proof.

A subset A of an ideal space (X, τ, I) is said to be an RI−open set [17] if A = int(cl⋆(A)). We

will denote the family of all RI−open sets by RIO(X). In [17], it is established that RIO(X) is a

base for a topology τI and τs ⊂ τI ⊂ τ where τs is the semiregularization of τ. The following Theorem

2.9 gives characterizations of pre−I−open sets in terms of RI−open sets.

Theorem 2.9. Let (X, τ, I) be an ideal space and A ⊂ X. Then the following are equivalent.

(a) A is pre−I−open.

(b) There exists an RI−open set G such that A ⊂ G and cl⋆(G) = cl⋆(A).

(c)A = G ∩ D where G is RI−open and D is τ ⋆−dense.

(d)A = G ∩ D where G is open and D is τ ⋆−dense.

Proof. (a)⇒(b). Suppose A is pre−I−open. If G = int(cl⋆(A)), then A ⊂ G and int(cl⋆(G)) =

int(cl⋆(int(cl⋆(A)))) = int(cl⋆(A)) = G which implies that G is an RI−open set containing A. Also,

cl⋆(A) ⊂ cl⋆(G) = cl⋆(int(cl⋆(A))) ⊂ cl⋆(A) which implies that cl⋆(A) = cl⋆(G). This proves (b).

(b)⇒(c). Suppose G is an RI−open set such that A ⊂ G and cl⋆(G) = cl⋆(A). If D = A ∪ (X − G),

then A = G ∩ D and D is τ ⋆−dense. This proves (c).

(c)⇒(d) is clear.

(d)⇒(a) follows from Lemma 4.3 of [3].

A subset A of an ideal space (X, τ, I) is said to be a A2I−set (resp. B2I−set [4](αIM2−set [1])

) if A = U ∩ V where U is open (resp.α − I−open ) and cl⋆(V ) = X. We will denote the family of

all A2I−sets (resp. B2I−sets) by A2I(X) (resp.B2I(X)). Clearly, A2I(X) ⊂ B2I(X). The following

Theorem 2.10 shows that A2I−sets and B2I−sets are nothing but pre−I−open sets. Also, it shows

that the converse of Proposition 2.6 of [4] is true.

Theorem 2.10. Let (X, τ, I) be an ideal space. Then A2I(X) = P IO(X) = B2I(X).



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On subsets of ideal topological spaces 47

Proof. By Theorem 2.9(d), A2I(X) = P IO(X). Since A2I(X) ⊂ B2I(X), it is enough to prove

that B2I(X) ⊂ A2I(X). Suppose A ∈ B2I(X). Then A = U ∩ V where U is α − I−open and

cl⋆(V ) = X. Now A ⊂ U ⊂ int(cl⋆(int(U ))) = int(cl⋆(int(U ∩ X))) = int(cl⋆(int(U ∩ cl⋆(V )))) ⊂

int(cl⋆(int(cl⋆(U ∩ V )))) = int(cl⋆(U ∩ V )) = int(cl⋆(A)) and so A ∈ P IO(X). This completes the

proof.

Clearly, A1I(X) ⊂ A2I(X). The following Example 2.11 shows that an A2I−set need not be

an A1I−set.

Example 2.11. Consider the ideal space (X, τ, I) where X = {a, b, c}, τ = {∅, {c}, X} and I =

{∅, {c}}. If A = {a, c}, then A is an A2I−set. But cl
⋆(int(A)) = int(A) ∪ (int(A))⋆ = {c} 6= X.

Hence A is not an A1I−set.

A subset A of an ideal space (X, τ, I) is said to be an αIN5−set [1] if A = U ∩ V where U is

α −I−open and V is ⋆−closed. We will denote the family of all αIN5−sets of an ideal space (X, τ, I)

by αIN5(X). A subset A of an ideal space (X, τ, I) is said to be an I−locally closed [6] (resp. weakly

I−locally closed [13]) set if A = U ∩ V where U is open and V is a ⋆−perfect (resp. ⋆−closed) set.

By Theorem 2.9 of [15], A is weakly I−locally closed if and only if A = U ∩ cl⋆(A) for some open

set U. The family of all weakly I−locally closed sets is denoted by W ILC(X). Clearly, every weakly

I−locally closed set is an αIN5−set but not the converse as shown by the following Example 2.12.

Theorem 2.13 below gives a characterization of αIN5−sets.

Example 2.12. [4, Example 2.2]Consider the ideal space (X, τ, I) where X = {a, b, c},

τ = {∅, {a}, {a, c}, X} and I = {∅, {b}, {c}, {b, c}}. If A = {a, b}, then int(cl⋆(int(A))) =

int(cl⋆(int({a, b}))) = int(cl⋆({a})) = int({a, b, c}) = X ⊃ A and so A is α − I−open and hence

an αIN5−set. But there is no open set U such that A = U ∩ cl
⋆(A) where cl⋆(A) = X. Hence A is

not a weakly I−locally closed set.

Theorem 2.13. Let (X, τ, I) be an ideal space and A ⊂ X. Then A is αIN5−set if and only if

A = U ∩ cl⋆(A) for some U ∈ αIO(X).

Proof. If A is an αIN5−set, then A = U ∩ V where U is α − I−open and V is ⋆−closed. Since

A ⊂ V, cl⋆(A) ⊂ cl⋆(V ) = V and so U ∩ cl⋆(A) ⊂ U ∩ V = A ⊂ U ∩ cl⋆(A) which implies that

A = U ∩ cl⋆(A). Conversely, suppose A = U ∩ cl⋆(A) for some U ∈ αIO(X). Since cl⋆(A) is ⋆−closed,

A is an αIN5−set.

A subset A of an ideal space (X, τ, I) is said to be an IR−closed set [1] if A = cl⋆(int(A)). A subset

A of an ideal space (X, τ, I) is said to be an αAI−set [4](αIN2−set [1]) (resp.AIR−set [1]) if A = U ∩V

where U is an α−I−open (resp. open) set and V is an IR−closed set. AIR−sets are called as AI−sets

in [4]. We will denote the family of all αAI −sets (resp.AIR−sets) by αAI(X)(resp.AIR(X)). Clearly,

every AIR−set is an αAI −set but the converse is not true [4, Example 2.2]. Theorem 2.14 below

shows that αAI −sets are nothing but semi−I−open sets which shows that the reverse direction of

Proposition 2.4 of [4] is true and each such set is both a strong β − I−open set and an αIN5−set.

Theorem 2.14. Let (X, τ, I) be an ideal space. Then αAI(X) = sβIO(X) ∩ αIN5(X) = SIO(X).

Proof. Suppose A ∈ αAI(X). Then A = U ∩V where U ∈ αIO(X) and V is an IR−closed set. Now

A = U ∩ V ⊂ int(cl⋆(int(U ))) ∩ cl⋆(int(V )) ⊂ cl⋆(int(cl⋆(int(U ))) ∩ int(V )) = cl⋆(int(cl⋆(int(U )) ∩

int(V ))) ⊂ cl⋆(int(cl⋆(int(U )∩int(V )))) = cl⋆(int(cl⋆(int(U∩V )))) = cl⋆(int(U∩V )) = cl⋆(int(A)) ⊂



48 V. Renukadevi CUBO
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cl⋆(int(cl⋆(A))) and so A ∈ sβIO(X). Since V is ⋆−closed, A ∈ αIN5(X) and so αAI(X) ⊂

sβIO(X) ∩ αIN5(X). Conversely, suppose A ∈ sβIO(X) ∩ αIN5(X). A ∈ sβIO(X) implies that

A ⊂ cl⋆(int(cl⋆(A))) and A ∈ αIN5(X) implies that A = U ∩ cl
⋆(A) where U ∈ αIO(X). Since

A ⊂ U, A ⊂ U ∩ cl⋆(int(cl⋆(A))) ⊂ U ∩ cl⋆(A) = A and so A = U ∩ cl⋆(int(cl⋆(A))). Since

cl⋆(int(cl⋆(A))) is IR−closed, A ∈ αAI(X) and so sβIO(X) ∩ αIN5(X) ⊂ αAI(X). Therefore,

αAI(X) = sβIO(X) ∩ αIN5(X).

If A ∈ SIO(X), then A ∈ sβIO(X) by Proposition 1(d) of [9]. Moreover, if V = A ∪ (X −

cl⋆(int(A))), then A = V ∩cl⋆(int(A)). Also, int(cl⋆(int(V ))) = int(cl⋆(int(A∪(X −cl⋆(int(A)))))) ⊃

int(cl⋆(int(A)∪int(X−cl⋆(int(A))))) = int(cl⋆(int(A)) ∪ cl⋆(int(X−cl⋆(int(A))))) ⊃ int(cl⋆(int(A))

∪ int(X − cl⋆(int(A)))) ⊃ int(int(cl⋆(int(A)) ∪ (X − cl⋆(int(A))))) = int(X) = X ⊃ V and so

V is α − I−open. Therefore, A ∈ αIN5(X) and hence SIO(X) ⊂ sβIO(X) ∩ αIN5(X). Con-

versely, suppose A ∈ sβIO(X) ∩ αIN5(X). A ∈ αIN5(X) implies that A = U ∩ V where U is

α − I−open and V is ⋆−closed. Since A ∈ sβIO(X), A ⊂ cl⋆(int(cl⋆(A))) = cl⋆(int(cl⋆(U ∩ V ))) ⊂

cl⋆(int(cl⋆(int(cl⋆(int(U ))) ∩ V ))) ⊂

cl⋆(int(cl⋆(int(cl⋆(int(U )))) ∩ V )) = cl⋆(int(cl⋆(int(U )) ∩ V )) = cl⋆(int(cl⋆(int(U ))) ∩ int(V )) ⊂

cl⋆(int(cl⋆(int(U ) ∩ int(V )))) = cl⋆(int(cl⋆(int(U ∩ V )))) = cl⋆(int(U ∩ V )) = cl⋆(int(A)). Therefore,

A ∈ SIO(X) which implies that sβIO(X) ∩ αIN5(X) ⊂ SIO(X). Hence sβIO(X) ∩ αIN5(X) =

SIO(X).

Corollary 2.15. Let (X, τ, I) be an ideal space and A ⊂ X. Then the following are equivalent.

(a) A is α − I−open.

(b) A is pre−I−open and semi−I−open [4, Proposition 1.1].

(c) A is a B2I−set and αAI−set[4, Theorem 2.3].

Proof. (a) and (b) are equivalent by Proposition 1.1 of [4]. (b) and (c) are equivalent by Theorem

2.10 and Theorem 2.14.

Corollary 2.16. Let (X, τ, I) be an ideal space and A ⊂ X. Then the following are equivalent.

(a) A is open.

(b) A is α − I−open and AIR − set.

(c) A is pre−I−open and AIR−set.

(d) A is α − I−open and weakly I−locally closed.

(e) A is α − I−open and BI−set.

(f) A is α − I−open and CI−set.

Proof. (a) and (b) are equivalent by Theorem 2.1 of [4].

That (b) implies (c) is clear.

(c) and (d) are equivalent by Proposition 2.2 of [4].

(d) implies (e) and (e) implies (f) are clear.

(f) implies (a) follows from Proposition 3.3 of [8].

A subset A of an ideal space (X, τ, I) is said to be a A3I−set (resp.B3I−set [4](αIN1−set [1]) )

if A = U ∩ V where U is open (resp.α − I−open ) and cl⋆(int(V )) ⊂ V. We will denote the family of

all A3I−sets (resp. B3I−sets) by A3I(X) (resp.B3I(X)). Clearly, A3I(X) ⊂ B3I(X). The following

Example 2.17 shows that the reverse direction is not true. Example 2.18 below shows that A2I−sets

and A3I−sets are independent concepts. Theorem 2.19 below gives a characterization of AIR−sets

in terms of A3I−sets.



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On subsets of ideal topological spaces 49

Example 2.17. Consider the ideal space (X, τ, I) of Example 2.12. If A = {a, b}, then int(cl⋆(int(A)))

= int(cl⋆(int({a, b}))) = int(cl⋆({a})) = int(X) = X ⊃ A and so A is an α − I−open set and hence

A is a B3I−set. Since cl
⋆(int(A)) * A and X is the only open set containing A, A is not an A3I−set.

Example 2.18. (a) Consider the ideal space (X, τ, I) of Example 2.12. If A = {a, b}, then A is not

an A3I−set. Since cl
⋆(A) = A ∪ A⋆ = {a, b} ∪ X = X, and so A is an A2I−set.

(b) Consider the ideal space (X, τ, I) where X = {a, b, c, d}, τ = {∅, {d}, {a, c},

{a, c, d}, X} and I = {∅, {c}, {d}, {c, d} }. If A = {a, b, c}, then cl⋆(int(A)) = int(A) ∪ (int(A))⋆ =

{a, c} ∪ {a, b, c} = {a, b, c} = A and so A is an A3I−set. Since cl
⋆(A) = A ∪ A⋆ = {a, b, c} 6= X, A is

not an A2I−set.

Theorem 2.19. Let (X, τ, I) be an ideal space . Then AIR(X) = SIO(X) ∩ A3I(X).

Proof. Suppose A ∈ AIR(X). Clearly, A ∈ A3I(X). By Theorem 3.3 of [1], A ∈ SIO(X). Therefore,

AIR(X) ⊂ SIO(X)∩A3I (X). Conversely, suppose A ∈ SIO(X)∩A3I (X). A ∈ A3I(X) implies that

A = U ∩ V where U is open and cl⋆(int(V )) ⊂ V. A ∈ SIO(X) implies that A ⊂ cl⋆(int(A)) and so

A = A ∩ cl⋆(int(A)) = (U ∩ V ) ∩ cl⋆(int(U ∩ V )) ⊂ U ∩ cl⋆(int(U ∩ V )) = U ∩ cl⋆(U ∩ int(V )) ⊂

U ∩ cl⋆(U ) ∩ cl⋆(int(V )) ⊂ U ∩ V = A and so A = U ∩ cl⋆(int(U ∩ V )) = U ∩ cl⋆(int(A)). Since

cl⋆(int(A)) is IR−closed, A ∈ AIR(X). Therefore, AIR(X) = SIO(X) ∩ A3I(X).

Corollary 2.20. Let (X, τ, I) be an ideal space and A ⊂ X. Then the following are equivalent.

(a)A ∈ AIR(X).

(b)A ∈ SIO(X) ∩ A3I(X).

(c) A ∈ αAI(X) ∩ A3I(X).

(d)A ∈ sβIO(X) ∩ αIN5(X) ∩ A3I(X).

(e)A ∈ sβIO(X) ∩ W ILC(X).

Proof. (a), (b), (c) and (d) are equivalent by Theorem 2.14 and Theorem 2.19.

(a) and (e) are equivalent by Theorem 2.10 of [15].

By Remark 3.3 of [8], every BI−set is a CI−set but the reverse direction is not true. The following

Theorem 2.22 gives characterizations of BI−sets in terms of CI−sets. A subset A of an ideal space

(X, τ, I) is said to be an αBI−set (αIN3−set [1]) if A = U ∩V where U ∈ αIO(X) and V is a t−I−set.

A subset A of an ideal space (X, τ, I) is said to be an αCI−set [4](αIN4−set [1]) if A = U ∩ V where

U ∈ αIO(X) and V is a α⋆ − I−set. Clearly every αBI−set is an αCI−set [1, Proposition 3.2(c)]

but not the converse [1, Example 3.4]. We will denote the family of all αBI−sets (resp.αCI−sets)

in (X, τ, I) by αBI(X) (resp.αCI(X)). We define DI(X) = {A ⊂ X | int(A) = pIint(A)} and if

A ∈ DI, then A is called a DI−set. The following Lemma 2.21 characterizes αBI−sets and αCI−sets,

the proof, which is similar to the proof of Theorem 2.7, is omitted. Corollary 2.23 follows from

Theorem 2.22.

Lemma 2.21. Let (X, τ, I) be an ideal space and A be a subset of X. Then the following hold.

(a) A is a αBI−set if and only if there exists an α − I−open set U such that A = U ∩ SIcl(A).

(b) A is an αCI−set if and only if there exists an α − I−open set U such that A = U ∩ spIcl(A).

Theorem 2.22. Let (X, τ, I) be an ideal space and A ⊂ X. Then the following are equivalent.

(a) A is a DI−set and a CI−set.

(b) A is a δ − I−open set and a CI−set.



50 V. Renukadevi CUBO
12, 2 (2010)

(c) A is a BI−set.

(d) A is an αBI−set and a CI−set.

Proof. (a)⇒(b). Suppose A ∈ DI(X) ∩ CI(X). If A ∈ DI(X), then int(A) = pIint(A). Now

int(cl⋆(A)) = cl⋆(A) ∩ int(cl⋆(A)) ⊂ cl⋆(A ∩ int(cl⋆(A))) = cl⋆

(pIint(A)) = cl⋆(int(A)) and so A is a δ − I−open set. This proves (b).

(b)⇒(c). Suppose A is a δ − I−open set and a CI−set. Then, by Theorem 2.4 of [12], int(cl
⋆(A)) =

int(cl⋆(int(A))) and so A∪int(cl⋆(A)) = A∪int(cl⋆(int(A))) which implies that SIcl(A) = spIcl(A).

If A is a CI−set, then Theorem 2.7, A = U ∩ spIcl(A) for some open set U and so A = U ∩ SIcl(A)

for some open set U which implies that A is a BI−set.

(c)⇒(a). Clearly, every BI−set is a CI−set. If A is a BI−set, then A = U ∩ V where U is open and

int(cl⋆(V )) = int(V ). Now pIint(A) = A ∩ int(cl⋆(A)) = A ∩ int(cl⋆(U ∩ V )) ⊂ A ∩ int(cl⋆(U ) ∩

cl⋆(V )) = A ∩ int(cl⋆(U )) ∩ int(cl⋆(V )) = (U ∩ V ) ∩ int(cl⋆(U )) ∩ int(V ) = U ∩ int(V ) = int(U ∩ V ) =

int(A). But always, int(A) ⊂ pIint(A) and so int(A) = pIint(A) which implies that A is a DI−set.

This proves (a).

(c)⇒(d) is clear.

(d)⇒(c). If A is an αBI−set, then A = U ∩ V where U is α − I−open and int(cl
⋆(V )) = int(V ).

Now A ⊂ U implies that A ⊂ int(cl⋆(int(U ))) and so int(cl⋆(A)) ⊂ int(cl⋆(int(cl⋆(int(U ))))) =

int(cl⋆(int(U ))) ⊂ int(cl⋆(U )). Again,A ⊂ V implies that int(cl⋆(A)) ⊂ int(cl⋆(V )) = int(V ). There-

fore,

int(cl⋆(A)) ⊂ int(cl⋆(U )) ∩ int(V ) ⊂ cl⋆(int(U ) ∩ int(V )) ⊂ cl⋆(int(U ∩ V )) = cl⋆(int(A)) and so

int(cl⋆(A)) = int(cl⋆(int(A))) which implies that A ∪ int(cl⋆(A))

= A ∪ int(cl⋆(int(A))). Hence SIcl(A) = spIcl(A). Since A is a CI−set, by Theorem 2.7, A =

G ∩ spIcl(A) for some open set G and so A = G ∩ SIcl(A). Therefore, A is a BI−set.

Corollary 2.23. Let (X, τ, I) be an ideal space. Then the following hold.

(a) Every BI−set is a DI−set.

(b) Every BI−set is a αBI−set.

(c) Every DI−set is a δ − I−open set (Proof follows from (a)⇒(b) of Theorem 2.22).

The following Theorem 2.24 characterizes αBI−open sets in terms of δ − I−open sets and

αCI−open sets. Example 2.25 below shows that δ − I−openness and αCI−openness are indepen-

dent concepts.

Theorem 2.24. Let (X, τ, I) be an ideal space. Then αBI(X) = δIO(X) ∩ αCI(X).

Proof. Clearly, αBI(X) ⊂ αCI(X). If A ∈ αBI(X), then A = U ∩ V where U is α − I−open

and V is a t − I−set. A ⊂ U implies that int(cl⋆(A)) ⊂ int(cl⋆(U )) ⊂ int(cl⋆(int(cl⋆(int(U ))))) ⊂

int(cl⋆(int(U ))) ⊂ cl⋆(int(U )). Also, A ⊂ V implies that int(cl⋆(A)) ⊂ int(cl⋆(V )) = int(V ) and so

int(cl⋆(A)) ⊂ cl⋆(int(U )) ∩ int(V ) ⊂ cl⋆(int(U ) ∩ int(V )) = cl⋆(int(U ∩ V )) = cl⋆(int(A). Therefore,

A ∈ δIO(X). Hence αBI(X) ⊂ δIO(X) ∩ αCI(X). Conversely, suppose A ∈ δIO(X) ∩ αCI(X).

A ∈ δIO(X) implies that int(cl⋆(A)) = int(cl⋆(int(A))) and so SIcl(A) = spIcl(A). A ∈ αCI(X)

implies that A = U ∩ spIcl(A) for some α − I−open set U by Lemma 2.21 and so A = U ∩ SIcl(A)

for some α −I−open set U which implies that A ∈ αBI(X). Therefore, δIO(X) ∩ αCI(X) ⊂ αBI(X).

This completes the proof.

Example 2.25. (a) Let X = {a, b, c, d}, τ = {∅, {d}, {a, b}, {a, b, d}, X} and I = {∅, {c} }. If

A = {a, c}, then int(cl⋆(int(A))) = int(cl⋆(int({a, c}))) = int(cl⋆(∅)) = ∅ = int(A). Therefore, A is



CUBO
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On subsets of ideal topological spaces 51

an α⋆ − I−set and hence an αCI−set. But int(cl
⋆(A)) = int({a, b, c}) = {a, b} and cl⋆(int(A)) =

cl⋆(∅) = ∅ and so A is not a δ − I−set.

(b) Let X = {a, b, c, d}, τ = {∅, {a}, {c}, {a, c}, X} and I = {∅, {a} }. If A = {a, b, c}, then A is

neither open nor an α⋆ − I−set and so A is not an αCI−set. But int(cl
⋆(A)) = int({a, b, c, d}) = X

and cl⋆(int(A)) = cl⋆({a, c}) = X and so A is a δ − I−set.

An ideal space (X, τ, I) is said to be ⋆−extremally disconnected [7] if the τ ⋆−closure (⋆−closure)

of every open set is open. Clearly, B3I(X) ⊂ αCI(X). By Example 3.6 of [1] the reverse direction

is not true. The following Theorem 2.26 shows that for ⋆−extremally disconnected spaces, the two

collection of sets coincide. Example 2.27 below shows that αCI (X) = B3I(X) does not imply that

the space is ⋆−extremally disconnected.

Theorem 2.26. Let (X, τ, I) be a ⋆−extremally disconnected ideal space. Then B3I(X) = αCI(X).

Proof. Enough to prove that αCI(X) ⊂ B3I(X). Suppose A ∈ αCI(X). Then A = U ∩ V where U is

α − I−open and int(cl⋆(int(V ))) = int(V ). Since (X, τ, I) is ⋆−extremally disconnected, cl⋆(int(V ))

is open and so int(V ) = int(cl⋆(int(V ))) = cl⋆(int(V )). Therefore, A ∈ B3I(X). This completes the

proof.

Example 2.27. Consider the ideal space (X, τ, I) where X = {a, b, c}, τ = {∅, {b}, {c}, {b, c}, X}

and I = {∅, {a}}. If A = {b}, A is open and cl⋆(A) = {b} ∪ {a, b} = {a, b}, which is not open. Hence

(X, τ, I) is not ⋆−extremally disconnected but ℘(X) = αCI(X) = B3I(X).

Received: November 2008. Revised: February 2009.

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