Download Some new separation axioms via $\beta$-$\mathcal{I}$

ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII “AL.I. CUZA” DIN IAŞI (S.N.)
MATEMATICĂ, Tomul LXI, 2015, f.2
SOME NEW SEPARATION AXIOMS VIA β-I-OPEN SETS
BY
N. GOWRISANKAR, A. KESKIN and N. RAJESH
Abstract. In this paper, β-I-open sets are used to define some weak separation
axioms and to study some of their basic properties. The implications of these axioms
among themselves and with the known axioms are investigated.
Mathematics Subject Classification 2010: 54D10.
Key words: ideal topological spaces, β-I-T0 space, β-I-T1 spaces β-I-T2 space.
1. Introduction
The subject of ideals in topological spaces has been introduced and studied by Kuratowski [7] and Vaidyanathasamy [12]. 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 P(X) is the set of all subsets of X, a set operator (.)∗ : P(X)
→ P(X), called the local function ([12]) of A with respect to τ and I, is
defined as follows: For A ⊂ X, A∗ (τ, I) = {x ∈ X|U ∩ A ∈
/ I for every
open neighbourhood U of x}. A Kuratowski closure operator Cl∗ (.) for a
topology τ ∗ (τ, I) called the ∗-topology, finer than τ is defined by Cl∗ (A) = A
∪ A∗ (τ, I) where there is no chance of confusion, A∗ (I) is denoted by A∗ . If
I is an ideal on X, then (X, τ, I) is called an ideal topological space. In this
paper, β-I-open sets are used to define some weak separation axioms and
to study some of their basic properties. The implications of these axioms
among themselves and with the known axioms are investigated.
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2. Preliminaries
Let A be a subset of a topological space (X, τ ). We denote the closure of
A and the interior of A by Cl(A) and Int(A), respectively. A subset A of X
is called β-open ([1]) if A ⊂ Cl(Int(Cl(A))). This notion has been studied
extensively in recent years by many topologists (see [2, 3, 11]) because βopen sets are only natural generalization of open sets. More importantly,
they also sug gest several new properties of topological spaces. A subset
S of an ideal topological space (X, τ, I) is called β-I-open ([6]) (resp. αI-open ([6])) if S ⊂ Cl(Int(Cl∗ (S))) (resp. S ⊂ Int(Cl∗ (Int(S)))). The
complement of a β-I-open set is called β-I-closed ([6]). The intersection of
all β-I-closed sets containing S is called the β-I-closure of S and is denoted
by βI Cl(S). The β-I-Interior of S is defined by the union of all β-I-open
sets contained in S and is denoted by βI Int(S). The set of all β-I-open
sets of (X, τ, I) is denoted by βIO(X). The set of all β-I-open sets of
(X, τ, I) containing a point x ∈ X is denoted by βIO(X, x). A subset S
of an ideal topological space (X, τ, I) is said to be β-I-regular ([15]) if it is
β-I-open and β-I-closed. A point x ∈ X is called the β-I-θ-cluster point
of S if βI Cl(U ) ∩ S ̸= ∅ for every β-I-open set U of (X, τ, I) containing x.
The set of all β-I-θ-cluster points of S is called the β-I-closure of S and is
denoted by βI Clθ (S). A subset S is said to be β-I-θ-closed set is said to be
β-I-θ-open. A point x ∈ X is called β-I-θ-interior point of S if there exists
a β-I-regular set U of X containing x such that x ∈ U ⊂ S. The set of all
β-I-θ-interior points of S and is denoted by βI Intθ (S). A subset A of an
ideal topological space (X, τ, I) is said to be β-I-θ-open if A = βI Intθ (A).
Equivalently, the complement of β-I-θ-closed set is β-I-θ-open. A subset
Ux of (X, τ, I) is said to be β-I-neighbourhood of a point x ∈ X if and only
if there exists a β-I-open set G such that x ∈ G ⊂ Ux .
Definition 2.1 ([6]). A function f: (X, τ, I) → (Y, σ, J ) is said to be βI-continuous (resp. β-I-irresolute) if the inverse image of every open (resp.
β-J -open) set in Y is β-I-open in X.
Definition 2.2 ([14]). An ideal topological space (X, τ, I) is said to be
β-I-regular if and only if for each closed set F of X and each point x∈X\F ,
there exist disjoint β-I-open sets U and V such that F ⊂U and x∈V .
Definition 2.3. A topological space (X, τ ) is said to be:
(i) β-T0 ([8]) (resp. semi-T0 ([9])) if to each pair of distinct points x, y of
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X there exists a β-open (resp. semiopen) set A containing x but not
y or a β-open (resp. semiopen) set B containing y but not x;
(ii) β-T1 ([8]) (resp. semi-T1 ([9])) if to each pair of distinct points x, y of
X, there exists a pair of β-open (resp. semiopen) sets, one containing
x but not y and the other containing y but not x;
(ii) β-T2 ([8]) (resp. semi-T2 ([9])) if to each pair of distinct points x, y
of X, there exists a pair of disjoint β-open (resp. semiopen) sets, one
containing x and the other containing y.
3. β-I-T0 spaces
Definition 3.1. An ideal topological space (X, τ, I) is β-I-T0 if for any
distinct pair of points in X, there is a β-I-open set containing one of the
points but not the other.
Theorem 3.2. An ideal topological space (X, τ, I) is β-I-T0 if and only
if for each pair of distinct points x, y of X, βI Cl({x}) ̸= βI Cl({y}).
Proof. Necessity. Let (X, τ, I) be an β-I-T0 space and x, y be any
two distinct points of X. There exists a β-I-open set G containing x or y,
say, x but not y. Then X\G is an β-I-closed set which does not contain x
but contains y. Since ∈ βI Cl({y}) is the smallest β-I-closed set containing
y, βI Cl({y}) ⊂ X − G, and so x ∈
/ βI Cl({y}). Consequently, βI Cl({x}) ̸=
Cl({y}).
βI
Sufficiency. Let x, y ∈ X, x ̸= y and βI Cl({x}) ̸= βI Cl({y}). Then
there exists a point z ∈ X such that z belongs to one of the two sets, say,
βI Cl({x}) but not to βI Cl({y}). If we suppose that x ∈ βI Cl({y}), then z
∈ βI Cl({x}) ⊂ βI Cl({y}), which is a contradiction. So x ∈ X\βI Cl({y}),
where X\βI Cl({y}) is β-I-open and does not contain y. This shows that
(X, τ, I) is β-I-T0 .
Definition 3.3 ([4]). Let A and X0 be subsets of an ideal topological
space (X, τ, I) such that A ⊂ X0 ⊂ X. Then (X0 , τ|X0 , I|X0 ) is an ideal
topological space with an ideal I|X0 = {I ∈ I|I ⊂ X0 } = {I ∩ X0 |I ∈ I}.
Lemma 3.4 ([14]). Let A and X0 be subsets of an ideal topological space
(X, τ, I). Then,
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(i) If A ∈ βIO(X) and X0 is α-I-open in (X, τ, I), then A ∩ X0 ∈
βIO(X0 );
(ii) If A ∈ βIO(X) and X0 is open in (X, τ, I), then A ∩ X0 ∈ βIO(X0 );
(iii) If A ∈ βIO(X0 ) and X0 ∈ βIO(X), then A ∈ βIO(X).
Theorem 3.5. Every α-I-open subspace of a β-I-T0 space is β-I-T0 .
Proof. Let Y be an α-I-open subspace of an β-I-T0 space (X, τ, I) and
x, y be two distinct points of Y . Then there exists an β-I-open set A in
X containing x or y, say, x but not y. Now by Lemma 3.4, A ∩ Y is an
β-I-open set in Y containing x but not y. Hence (Y, τ|Y , I|Y ) is β-I|Y -T0 .
Corollary 3.6. Every open subspace of a β-I-T0 space is β-I-T0 .
Definition 3.7. A function f : (X, τ, I) → (Y, σ) is said to be point β-Iclosure one-to-one if and only if x, y ∈ X such that βI Cl({x}) ̸= βI Cl({y}),
then βI Cl({f (x)}) ̸= βI Cl({f (y)}).
Theorem 3.8. If f : (X, τ, I) → (Y, σ) is point-β-I-closure one-to-one
and (X, τ, I) is β-I-T0 , then f is one-to-one.
Proof. Let x and y be any two distinct points of X. Since (X, τ, I)
is β-I-T0 , then βI Cl({x}) ̸=βI Cl({y}) by Theorem 3.2. But f is pointβ-I-closure one-to-one implies that βI Cl({f(x)}) ̸=βI Cl({f(y)}). Hence
f (x) ̸= f (y). Thus, f is one-to-one.
Theorem 3.9. Let f : (X, τ, I) → (Y, σ) be a function from β-I-T0
space (X, τ, I) into a topological space (Y, σ). Then f is point-β-I-closure
one-to-one if and only if f is one-to-one.
Proof. The proof follows from Theorem 3.8.
Theorem 3.10. Let f : (X, τ, I) → (Y, σ, I) be an injective β-I-irresolute function. If Y is β-I-T0 , then (X, τ, I) is β-I-T0 .
Proof. Let x, y ∈ X with x̸=y. Sinc f is injective and Y is β-I-T0 ,
there exists an β-I-open set Vx in Y such that f (x) ∈ Vx and f (y) ∈
/ Vx
or there exists a β-I-open set Vy in Y such that f (y) ∈ Vy and f (x) ∈
/ Vy
−1
with f (x) ̸= f (y). By β-I-irresoluteness of f , f (Vx ) is β-I-open set in
(X, τ, I) such that x ∈ f −1 (Vx ) and y ∈
/ f −1 (Vx ) or f −1 (Vy ) is β-I-open
set in (X, τ, I) such that y ∈ f −1 (Vy ) and x ∈
/ f −1 (Vy ). This shows that
(X, τ, I) is β-I-T0 .
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4. β-I-T1 spaces
Definition 4.1. An ideal topological space (X, τ, I) is β-I-T1 if to each
pair of distinct points x, y of X, there exists a pair of β-I-open sets, one
containing x but not y and the other containing y but not x.
Theorem 4.2. For an ideal topological space (X, τ, I), each of the
following statements are equivalent:
(1) (X, τ, I) is β-I-T1 ;
(2) Each one point set is β-I-closed in X;
(3) Each subset of X is the intersection of all β-I-open sets containing
it;
(4) The intersection of all β-I-open sets containing the point x ∈ X is
the set {x}.
Proof. (1)⇒(2): Let x ∈ X. Then by (1), for any y ∈ X, y ̸= x, there
exists an β-I-open set Vy containing y but not x. Hence y ∈ Vy ⊂ X\{x}.
Now varying y over X\{x} we get X\{x} = ∪ {Vy : y ∈ X\{x}}. So X\{x}
being a union of β-I-open set. Accordingly {x} is β-I-closed.
(2)⇒(1): Let x, y ∈ X and x ̸= y. Then by (2), {x} and {y} are βI-closed sets. Hence X\{x} is a β-I-open set containing y but not x and
X\{y} is an β-I-open set containing x but not y. Therefore, (X, τ, I) is
β-I-T1 .
(2)⇒(3): If A ⊂ X, then for each point y ∈
/ A, there exists a set X\{y}
such ∩
that A ⊂ X\{y} and each of these sets X\{y} is β-I-open. Hence
A=
{ X\{y}: y ∈ X\A} so that the intersection of all β-I-open sets
containing A is the set A itself.
(3)⇒(4): Obvious.
(4)⇒(1): Let x, y ∈ X and x ̸= y. Hence there exists a β-I-open set
Ux such that x ∈ Ux and y ∈
/ Ux . Similarly, there exists a β-I-open set Uy
such that y ∈ Uy and x ∈
/ Uy . Hence (X, τ, I) is β-I-T1 .
Theorem 4.3. Every α-I-open subspace of a β-I-T1 space is β-I-T1 .
Proof. Let A be an α-I-open subspace of a β-I-T1 space (X, τ, I). Let
x ∈ A. Since (X, τ, I) is β-I-T1 , X\{x} is β-I-open in (X, τ, I). Now, A
being open, A ∩ (X\{x}) = A\{x} is β-I-open in A by Lemma 3.4. Consequently, {x} is β-I-closed in A. Hence by Theorem 4.2, A is β-I-T1 .
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Corollary 4.4. Every open subspace of a β-I-T1 space is β-I-T1 .
Theorem 4.5. Let X be a T1 space and f : (X, τ ) → (Y, σ, I) be a
β-I-closed surjective function. Then (Y, σ, I) is β-I-T1 .
Proof. Suppose y ∈ Y . Since f is surjective, there exists a point x ∈ X
such that y = f (x). Since X is T1 , {x} is closed in X. Again by hypothesis,
f ({x}) = {y} is β-I-closed in Y . Hence by Theorem 4.2, Y is β-I-T1 . Definition 4.6. A point x ∈ X is said to be a β-I-limit point of A if
and only if for each V ∈ β-I(X), U ∩ (A \ {x}) ̸= ∅ and the set of all
β-I-limit points of A is called the β-I-derived set of A and is denoted by
β-Id(A).
Theorem 4.7. If (X, τ, I) is β-I-T1 and x ∈ β-Id(A) for some A ⊂
X, then every β-I-neighbourhood of x contains infinitely many points of A.
Proof. Suppose U is a β-I-neighbourhood of x such that U ∩ A is
finite. Let U ∩ A = {x1 , x2 , .... xn } = B. Clearly B is a β-I-closed
set. Hence V = (U ∩ A)\(B\{x}) is a β-I-neighbourhood of point x and
V ∩ (A\{x}) = ∅, which implies that x ∈ β-Id(A), which contradicts our
assumption. Therefore, the given statement in the theorem is true.
Theorem 4.8. In an β-I-T1 space (X, τ, I), β-Id(A) is β-I-closed for
any subset A of X.
Proof. As the proof of the theorem is easy, it is omitted.
Theorem 4.9. Let f : (X, τ, I) → (Y, σ, I) be an injective and β-Iirresolute function. If (Y, σ, I) is β-I-T1 , then (X, τ, I) is β-I-T1 .
Proof. Proof is similar to Theorem 3.10
Definition 4.10. An ideal topological space (X, τ, I) is said to be βI-R0 ([5]) if and only if for every β-I-open sets contains the β-I-closure of
each of its singletons.
Theorem 4.11. An ideal topological space (X, τ, I) is β-I-T1 if and
only if it is β-I-T0 and β-I-R0 .
Proof. Let (X, τ, I) be a β-I-T1 space. Then by definition and as
every β-I-T1 space is β-I-R0 , it is clear that (X, τ, I) is β-I-T0 and β-I-R0
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space. Conversely, suppose that (X, τ, I) is both β-I-T0 and β-I-R0 . Now,
we show that (X, τ, I) is β-I-T1 space. Let x, y ∈ X be any pair of distinct
points. Since (X, τ, I) is β-I-T0 , there exists a β-I-open set G such that x
∈ G and y ∈
/ G or there exists a β-I-open set H such that y ∈ H and x ∈
/
H. Suppose x ∈ G and y ∈
/ G. As x ∈ G implies the βI Cl({x}) ⊂ G. As y
∈
/ G, y ∈
/ βI Cl({x}). Hence y ∈ H = X\βI Cl({x}) and it is clear that x ∈
/
H. Hence, it follows that there exist β-I-open sets G and H containing x
and y respectively such that y ∈
/ G and x ∈
/ H. This implies that (X, τ, I)
is β-I-T1 .
5. β-I-T2 spaces
Definition 5.1. An ideal topological space (X, τ, I) is said to be β-IT2 ([14]) if to each pair of distinct points x, y of X, there exists a pair of
disjoint β-I-open sets, one containing x and the other containing y.
Theorem 5.2. For an ideal topological space (X, τ, I), the following
statements are equivalent:
(i) (X, τ, I) is β-I-T2 ;
(ii) Let x ∈ X. For each y ̸= x, there exists U ∈ βIO(X, x) and y ∈βI
Cl(U ).
∩
(iii) For each x ∈ X, {βI Cl(Ux ) : Ux isaβ −I-neighbourhood of x} = {x}.
(iv) The diagonal △ = {(x, x) : x ∈ X} is β-I-closed in X × X.
Proof. (i)→(ii): Let x ∈ X and y ̸= x. Then there are disjoint β-Iopen sets U and V such that x ∈ U and y ∈ V . Clearly, X\V is β-I-closed,
/ βI Cl(U ).
βI Cl(U ) ⊂ X\V and therefore y ∈
(ii)→(iii):
If y ̸= x, then there exists U ∈ βIO(X, x) and y ∈
/ βI Cl(U ).
∩
So y ∈
/ {βI Cl(U ) : U ∈ βIO(X, x)}.
(iii)→(iv): We
/ △. Then
∩ prove that X\△ is β-I-open. Let (x, y) ∈
y ̸= x and since {βI Cl(U ) : U ∈ βIO(X, x)} = {x}, there is some U ∈
βIO(X, x) and y ∈
/ βI Cl(U ). Since U ∩ X\βI Cl(U ) = ∅, U × (X\βI Cl(U ))
is β-I-open set such that (x, y) ∈ U × (X\βI Cl(U )) ⊂ X\△.
(iv)→(v): If y ̸= x, then (x, y) ∈
/ △ and thus there exist U, V ∈ βIO(X)
such that (x, y) ∈ U × V and (U × V ) ∩ △ = ∅. Clearly, for the β-I-open
sets U and V we have x ∈ U , y ∈ V and U ∩ V = ∅.
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Theorem 5.3. For an ideal topological space (X, τ, I), the following
statements are equivalent:
(i) (X, τ, I) is β-I-T2 ;
(ii) for any two distinct points x, y ∈ X, there exists a β-I-θ-open set
containing x but not y and there exists a β-I-θ-open set containing y
but not x;
(iii) for any two distinct points x, y ∈ X, there exists a β-I-θ-open set
containing x but not y or there exists a β-I-θ-open set containing y
but not x.
Proof. The proofs of implications (i)→(ii) and (ii)→(iii) are obvious.
(iii)→(i): Let x and y be any two distinct points of X. There exists a β-Iθ-open set containing y but not x. Therefore, there exists a β-I-open set V
such that x ∈ V ⊂βI Cl(V ) ⊂ U and we have y ∈ X\U ⊂ X\βI Cl(V ). By
[14], Theorem 3.1, V is both β-I-open and β-I-closed. Therefore, (X, τ, I)
is β-I-T2 .
Corollary 5.4. An ideal toological space is β-I-T2 if and only if each
singleton subsets of X is β-I-closed.
Corollary 5.5. An ideal toological space is β-I-T2 if and only if two
distinct points of X have disjoint β-I-closure.
Lemma 5.6. The product of two β-I-open sets is β-I-open.
Proof. Simillarly to the proof of Lemma 3.4 of [16].
Theorem 5.7. The product of two β-I-T2 spaces is β-I-T2 .
Proof. Let (X, τ, I) and (Y, σ, I) be β-I-T2 spaces and x, y ∈ X×Y,
such that x ̸= y. Let x = (a, b) and y = (c, d). Without loss of generality,
suppose that a ̸= c and b ̸= d. Since a and d are distinct points of X, there
exist disjoint β-I-open sets U and V of X such that a ∈ U and c ∈ V .
Similarly, let G and H are disjoint β-I-open sets in Y , such that b ∈ G
and d ∈ H. Then U×G and V×H are β-I-open sets in X×Y containing x
and y, respectively. Also (U×G) ∩ (V×H) = (U∩V) × (G∩H) = ∅. Hence
X×Y is β-I-T2 .
Theorem 5.8. Every β-I-regular T0 -space is β-I-T2 .
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Proof. Let (X, τ, I) be a β-I-regular T0 space and x, y ∈ X such that
x ̸= y. Since X is T0 there exists an open set V containing one of the
points, say, x but not y. Then y ∈ X\V , X\V is closed and x ∈
/ X\V . By
β-I-regularity of X, there exists β-I-open sets G and H such that x ∈ G,
y ∈ X\V ⊂ H and G ∩ H = ∅. Hence (X, τ, I) is β-I-T2 .
Theorem 5.9. Every α-I-open subspace of a β-I-T2 space is β-I-T2 .
Proof. Proof is similar to Theorem 4.3
Corollary 5.10. Every open subspace of a β-I-T2 space is β-I-T2 .
Theorem 5.11. If f : (X, τ, I) → (Y, σ) is injective, open and β-Icontinuous, and Y is β-I-T2 , then (X, τ, I) is β-I-T2 .
Proof. Since f is injective, f(x)̸=f(y) for each x, y ∈ X and x̸=y. Now
Y being β-I-T2 , there exists β-I-open sets G, H in Y such that f (x) ∈
G, f (y) ∈ H and G∩H=∅. Let U = f−1 (G) and V = f−1 (H). Then by
hypothesis, U and V are β-I-open in X. Also x ∈ f−1 (G) = U, y ∈ f−1 (H)
= V and U∩V = f−1 (G) ∩ f−1 (H) = ∅. Hence (X, τ, I) is β-I-T2 .
The following modification of the Theorem 5.11 where f is relaxed but
Y is restricted and is also true.
Theorem 5.12. If f : (X, τ, I) → (Y, σ, I) is injective and β-I-closed
and Y is T2 , then (X, τ, I) is β-I-T1 .
Proof. Although the proof is not identical to that of Theorem 5.11 it
is quite similar and thus omitted.
Theorem 5.13. If f : (X, τ, I) → (Y, σ, I) is β-I-irresolute and Y is
β-I-T2 . Then the set {(x1 , x2 )|f (x1 ) = f (x2 )} is β-I-closed in X × X.
Proof. Let A = {(x1 , x2 ) | f(x1 ) = f(x2 )}. If (x1 , x2 ) ∈ X×X \ A,
then f(x1 ) ̸= f(x2 ). Since Y is β-I-T2 , there exist disjoint β-I-open sets
V1 and V2 such that f(xj ) ∈ Vj for j = 1,2. Then by β-I-irresoluteness of
f , f−1 (Vj ) ∈ βIO(X, xj ) for each j. Thus, (x1 , x2 ) ∈ f−1 (V1 ) × f−1 (V2 )
∈ βIO(X1 × X2 ) by Lemma 5.6. Therefore, f−1 (V1 ) × f−1 (V2 ) ⊂ (X × X)
\ A. It follows that X × X \ A is β-I-open and hence A is β-I-closed set
in X × X.
Definition 5.14. A function f : (X, τ, I) → (Y, σ, J ) is called strongly
β-I-open if the image of every β-I-open subset of (X, τ, I) is β-J -oen in
(Y, σ, J ).
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Theorem 5.15. Let (X, τ, I) be an ideal topological space, R an equivalence relation in X and p : (X, τ, I) → X|R the identification function.
If R ⊂ (X × X) and p is a strongly β-I-open function, then X|R is β-I-T2
Proof. Let p(x) and p(y) be the distinct members of X|R. Since x and
y are not related, R ⊂ (X × X) is β-I-closed in X × X. There are β-I-open
sets U and V such that x ∈ U and y ∈ V and U × V ⊂ X\R. Thus, p(U ),
p(V ) are disjoint β-I-open sets in X|R since p is strongly β-I-open.
Definition 5.16 ([13]). Let f : (X, τ, I) → (Y, σ) be a function. The
set {(x, f(x)) | x ∈ X} of the product space X × Y is called the graph of f
and is denoted by G(f ).
Theorem 5.17. If f : (X, τ, I) → (Y, σ, I) is β-I-irresolute and (Y, σ, I)
is β-I-T2 space. Then G(f ) is β-I-closed.
Proof. Let (x, y) ∈
/ G(f) and so y ̸= f (x). Now, (Y, σ, I) being β-I-T2 ,
there exist β-I-open sets V and W such that f (x) ∈ W and y ∈ V and V
∩ W = ∅. Since f is β-I-irresolute, there exits U ∈ βIO(X, x) and f (U )
⊂ W . Therefore, we obtain (x, y) ∈ U × V ⊂ X × Y \ G(f ). Now, by
Lemma 5.6, U × V ∈ β-I(X × Y). Hence X × Y \ G(f) is β-I-open in X
× Y . Thus, G(f ) is β-I-closed in X × Y.
Definition 5.18. An ideal topological space (X, τ, I) is said to be βI-R1 if for x, y in X with βI Cl({x}) ̸= βI Cl({y}), there exists disjoint
β-I-open sets U and V such that βI Cl({x}) is a subset of U and βI Cl({y})
is a subset of V .
Theorem 5.19. The ideal topological space (X, τ, I) is β-I-T2 if and
only if it is β-I-R1 and β-I-T0 .
Proof. The proof is similar to Theorem 4.11 and thus omitted.
Remark 5.20. In the following diagram we denote by arrows the implications between the seperation axioms which we have introduced and
discussed in this paper and examples show that no other implications hold
between them.
T2 → β − I-T2 → β-T2
↓
↓
↓
T1 → β − I-T1 → β-T2
↓
↓
↓
T0 → β − I-T0 → β-T0
11
SOME NEW SEPARATION AXIOMS VIA
β -I -OPEN SETS
343
Example 5.21. Let X = {a, b, c}, τ = {∅, {a}, {b, c}, X} and I =
{∅, {a}}. Then (X, τ, I) is β-I-Ti (i = 0, 1, 2) but not Ti (i = 0, 1, 2).
Example 5.22. Let X = {a, b, c}, τ = {∅, {a}, {b}, {a, b}, {b, c}, X}
and I = {∅, {a}}. Then (X, τ, I) is β-I-T0 but not β-I-T1 .
Example 5.23. Let X = {a, b, c}, τ = {∅, {a}, {b, c}, X} and I =
{∅, {b} {c}, {b, c}}. Then (X, τ, I) is β-Ti (i = 0, 1, 2) but not β-I-Ti (i =
0, 1, 2).
Theorem 5.24. (i) An ideal topological space (X, τ, {∅}) is β-I-T0
(resp. β-I-T1 , β-I-T2 ) if and only if it is β-T0 (resp. β-T1 , β-T2 );
(ii) A topological space (X, τ, N ) is β-I-T0 (resp. β-I-T1 , β-I-T2 ) if
and only if it is β-T0 (resp. β-T1 , β-T2 ) (N is the ideal of all nowhere
dense sets of X);
(iii) A topological space (X, τ, P(X)) is β-I-T0 (resp. β-I-T1 , β-I-T2 )
if and only if it is semi-T0 (resp. semi-T1 , semi-T2 ).
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Received: 15.X.2010
Revised: 5.IX.2011
Accepted: 15.IX.2011
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