Download b − I-OPEN SETS AND DECOMPOSITION OF CONTINUITY VIA

Proceedings of IMM of NAS of Azerbaijan
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Aysegul CAKSU GULER and Gulhan ASLIM
b − I-OPEN SETS AND DECOMPOSITION OF
CONTINUITY VIA IDEALIZATION
Abstract
We introduce the notion of b − I-open sets and strong BI -sets to obtain a
decomposition of continuity via idealization. Additionally, we investigate properties of b − I-open sets and strong BI -sets.
Introduction and preliminaries
In 1990, Jankovic and Hamlett [7] introduced the notion of I-open sets in ideal
topological spaces. Abd El-Monsef et al [2] further investigated I-open sets and Icontinuous functions. In 1999, Dontchev [4] has introduced the notion of pre-I-open
sets which is weaker than that of I-open sets. At last Hatir at all [5] have introduced
the notions of BI -sets, CI -sets, α − I-sets, semi-I-sets and β − I-sets. By using this
sets, they provided decompositions of continuity. In this paper, we introduced the
notions b − I-open and strong BI -sets to obtain decompositions of continuity.
Throughout this paper for a subset A of a space (X, τ ), the closure of A and
interior of A are denoted Cl(A) and Int(A) respectively. An ideal topological space
is a topological space (X, τ ) with an ideal I on X, and is denoted (X, τ , I). The
following collections form important ideals on a topological space (X, τ ): the ideal of
all finite sets F, the ideal of all closed and discrete sets CD, the ideal of all nowhere
dense sets N . A∗ (I) = {x ∈ X | U ∩ A ∈
/ If or each neighborhood U of X} is called
the local function of A with respect to I and τ [7]. When there is no chance for
confusion A∗ (I) is denoted by A∗ . Note that often X ∗ is a proper subset of X. The
hypothesis X = X ∗ was used by Hayashi in [5], while the hypothesis τ ∩ I = ∅ was
used by Samuels in [9]. In fact, those two conditions are equivalent [7, Theorem 6.1]
and so the ideal topological spaces satisfying this hypothesis are called as HayashiSamuels spaces. For every ideal topological space (X, τ , I), there exits a topology
τ ∗ (I), finer than τ , generated by the base β(I, τ ) = {U \ I | U ∈ τ and I ∈ I}. In
general β(I, τ ) is not always a topology [7]. Observe additionally that Cl∗ (A) =
A∗ ∪ A defines a Kuratowski closure operator for τ ∗ (I).
Now we recall some definitions and results, which are used in this paper.
Definition 1. A subset A of a topological space X is called
(a) α-open set [8] if A ⊂ Int(Cl(Int(A))),
(b) t-set [9] if Int(A) = Int(Cl(A)),
(c) b-open set [1] if A ⊂ Int(Cl(A)) ∪ Cl(Int(A)),
(d) strong B-set [3] if A = U ∩ V where U is an open set and V is a t-set and
Int(Cl(V )) = Cl(Int(V )).
Definition 2. A subset A of an ideal topological space (X, τ , I) is said to be
(a) I-open [7] A ⊂ Int(A∗ ),
(b) α − I-open [5] if A ⊂ Int(Cl∗ (Int(A))),
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[A.C.Guler and G.Aslim]
(c) semi-I-open [5] if A ⊂ Cl∗ (Int(A)),
(d) pre-I-open [4] if A ⊂ Int(Cl∗ (A)),
(e) t − I-set [5] if Int(Cl∗ (A)) = Int(A),
(f ) BI -set [5] if A = U ∩ V , where U ∈ τ and V is a t − I-set.
b − I-open set
Definition 3. A subset A of an ideal topological space (X, τ , I) is said to be
b − I-open if A ⊂ Int(Cl∗ (A)) ∪ Cl∗ (Int(A)).
Proposition 1. Let A be a b − I-open set such that Int(A) = ∅, then A is
pre-I-open.
For a subset of an ideal topological space the following hold:
(a) Every open set is b − I-open.
(b) Every semi-I-open set is b − I-open.
(c) Every pre-I-open set is b − I-open.
(d) Every b − I-open set is β − I-open.
(e) Every b − I-open is b-open.
Proof. (a) The proof is obvious.
(b) The proof is obvious.
(c) The proof is obvious.
(d) Let A be a b − I-open set. Then we have A ⊂ Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) ⊂
Cl(Int(Cl∗ (A)))∪((Int(A))∗ ∪Int(A)) ⊂ Cl(Int(Cl∗ (A)))∪(Cl(Int(A))∪Int(A)) ⊂
Cl(Int(Cl∗ (A))) ∪ Cl(Int(A)) ⊂ Cl(Int(Cl∗ (A)).
This shows that A is an β − I-open set.
(e) Let A be a b − I-open set. Then we have A ⊂ Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) =
Int(A∗ ∪ A) ∪ ((Int(A))∗ ∪ Int(A)) ⊂ Int(Cl(A) ∪ A) ∪ (Cl(Int(A)) ∪ Int(A)) =
Int(Cl(A)) ∪ Cl(Int(A)).
This shows that A is a b-open set.
Remark 1. From above the following implication and none of these implications
is reversible as shown by examples given below and well -known facts
open
-
α-I-open
-
semi-I-open
?
pre-I-open
b − I-open
-
@
@
R
@
β − I-open
Example 1. Let X = {a, b, c, d}, τ = {X, ∅, {b}, {a, d}, {a, b, d}} and I =
{∅, {b}}. Then A = {b, d} is b − I-open, but it is not semi-I-open. Because
Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) = Int({b, d}∗ ∪ {b, d}) ∪ Cl∗ (Int({b, d})) = Int(X) ∪
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[b-I-open sets and decom.of continuity...]
({b}∗ ∪ {b}) = X ∪ {b} = X ⊃ A and hence A is b − I-open. Since Cl∗ (Int(A)) =
Cl∗ (Int({b, d})) = ({b}∗ ∪ {b}) = {b} 6⊃ A. So A is not semi-I-open.
Example 2. Let X = {a, b, c}, τ = {X, ∅, {a}, {b}, {a, b}} and I = {∅, {b}}.
Then A = {a, c} is b−I-open, but it is not pre-I-open. For Int(Cl∗ (A))∪Cl∗ (Int(A)) =
Int({a, c}∗ ∪ {a, c}) ∪ Cl∗ ({a}) = Int({a, c}) ∪ ({a}∗ ∪ {a}) = {a, c} ⊃ A and hence
A is b − I-open. Since Int(Cl∗ (A)) = Int({a, c}∗ ∪ {a, c}) = Int({a, c}) = {a} 6⊃ A.
Hence A is not pre-I-open.
Example 3. Let X = {a, b, c}, τ = {X, ∅, {a}, {b}, {a, b}} and I = {∅, {b}}.
Then A = {b, c} is β − I-open, but it is not b − I-open. Since Cl(Int(Cl∗ (A))) =
Cl(Int({b, c}∗ ∪ {b, c})) = Cl(Int({b, c})) = Cl({b}) = {b, c} ⊃ A. Hence A is not
β − I-open. For Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) = Int({b, c}∗ ∪ {b, c}) ∪ Cl∗ ({b}) =
Int({b, c}) ∪ ({b}∗ ∪ {b}) = {b} 6⊃ A and hence A is not b − I-open.
Proposition 2. For an ideal topological space (X, τ , I) and A ⊆ X we have:
(a) If I = ∅, then A is b − I-open if and only if A is b-open.
(b) If I = P(X), then A is b − I-open if and only open A ∈ τ .
(c) If I = N , then A is b − I-open if and only if A is b-open.
Proof (a) Necessity is shown in Proposition 2.2. For sufficiency note that in
case of the minimal ideal A∗ = Cl(A).
(b) Necessity: If A is a b − I-open set, then A ⊂ Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) =
Int(A∗ ∪ A) ∪ ((Int(A)∗ ) ∪ Int(A)) = Int(∅ ∪ A) ∪ (∅ ∪ Int(A)) = Int(A) ∪ Int(A) =
Int(A). Hence A is open.
Sufficiency: It is shown in Proposition 2.2
(c) Necessity is given in Proposition 3.2.
Sufficiency: Note that the local function of A with respect to N and τ can
be given explicitly [10]. We have
A∗ (N ) = Cl(Int(Cl(A))).
Hence A is b − I-open if and only if A ⊂ Int(Cl(Int(Cl(A))) ∪ A)∪
Cl(Int(Cl(Int(A))) ∪ A. Suppose that A is b-open. Since always Int(Cl(A)) ∪
Cl(Int(A)) ⊆ A∪ Cl(Int(Cl(A))) ∪ Cl(Int(A)), then A ⊆ Int(A∪Cl(Int(Cl(A))))∪
Cl(Int(A)) = Int(A ∪ A∗ (N )) ∪ CI(Int(A)). Hence A is a b-I-open set.
The intersection of even two b − I-open sets need not to be b − I-open as shown
in the following examples.
Example 4. Let X = {a, b, c, d}, τ = {X, ∅, {a}, {b, d}, {a, b, d}} and I =
{∅, {c}, {d}, {c, d}}. Then A = {a, c} and B = {b, c} are b − I-open but A ∩ B = {c}
is not b−I-open. Since Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) = {a, c} ⊃ A and Int(Cl∗ (B)) ∪
Cl∗ (Int(B)) = {b, c, d} ⊃ B. But Int(Cl∗ (A ∩ B))∪ Cl∗ (Int(A ∩ B)) = ∅ 6⊃ A ∩ B.
Lemma 1 [7, Theorem 2.3 (g)] Let (X, τ , I) be ideal topological space and let
A ⊆ X. Then U ∈ τ ⇒ U ∩ A∗ = U ∩ (U ∩ A)∗ ⊆ (U ∩ A)∗ .
Proposition 3. Let (X, τ , I) be ideal topological space and let A, U ⊆ X. If A
is a b − I-open set and U ∈ τ . Then A ∩ U is a b − I-open set.
Proof. By assumption A ⊆ Int(Cl∗ (A)) ∪ Cl∗ (Int(A)) and U ⊆ Int(U ). Thus
applying Lemma 2.1,
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A ∩ U ⊆ (Int(Cl∗ (A)) ∪ Cl∗ (Int(A))) ∩ Int(U ) ⊆ (Int(Cl∗ (A)) ∩ Int(U )) ∪
(Cl∗ (Int(A)) ∩ Int(U )) = Int((A∗ ∩ U ) ∪ (A ∩ U )) ∪ ((Int(A))∗ ∩ Int(U )) ∪ (Int(A) ∩
Int(U )) ⊆ Int((A ∩ U )∗ ∪ (A ∩ U )) ∪ ((Int(A ∩ U ))∗ ∪ (Int(A ∩ U ))) = Int(Cl∗ (A ∩
U )) ∪ Cl∗ (Int(A ∩ U )).
Thus A ∩ U is a b − I-open set.
Strong BI -set
Definition 4. Let (X, τ , I) be an ideal topological space and be A ⊆ X is called
a strong BI -set if A = U ∩ V , where U ∈ τ and V is a t − I-set and Int(Cl∗ (V )) =
Cl∗ (Int(V)).
Proposition 4. Let (X, τ , I) be an ideal topological space and be A ⊆ X. The
following hold:
(a) If A is a strong BI -set, then a BI -set.
(b) If A is a strong B-set, then A is a strong BI -set.
Proof. The proofs are obvious by Proposition 2.3 of [5]
Remark 2. The converses of proposition (a) and (b) need not to be true as the
following examples show.
Example 5. Let X = {a, b, c, d}, τ = {X, ∅, {a}, {b, c}, {a, b, c}} and I =
{∅, {c}, {a, c}}. Then A = {a, c} is a BI -set, but it is not a strong BI -set. For
Int(Cl∗ (A)) = Int({a, c}∗ ∪ {a, c}) = Int({a, c}) = {a} = Int(A) and hence A is a
t − I-set. It is obvious that A is a BI -set. But Int(Cl∗ (A)) = Int({a, c}∗ ∪ {a, c}) =
Int({a, c}) = {a} and Cl∗ (Int(A)) = Cl∗ (Int({a, c})) = {a}∗ ∪ {a} = {a, d} i.e
Int(Cl∗ (A)) 6= Cl∗ (Int(A)). So A is not a strong BI -set.
Example 6. Let X = {a, b, c, d}, τ = {X, ∅, {b}, {c, d}, {b, c, d}} and I =
{∅, {b}, {c}, {b, c}}. Then A = {b, c} is a strong BI -set but not a strong B-set.
Int(Cl∗ (A)) = Int({b, c}∗ ∪ {b, c}) = Int({b, c}) = {b} = Int(A). Then A is a
t − I-set. Besides Int(Cl∗ (A)) = {b} = {b}∗ ∪ {b} = Cl∗ (Int(A)). So A is a strong
BI -set. But, Int(Cl(A)) = Int({X}) = X 6= Int(A). Therefore A is not a strong
B-set.
Proposition 5. For a subset A ⊆ (X, τ , I) the following conditions are equivalent:
(a) A is open.
(b) A is b − I-open and a strong BI -set.
Proof. (a) ⇒ (b) By Proposition 3.3, every open set is b − I-open. On the other
hand every open set is strong BI -set, because X is t − I-set and Int(Cl∗ (X)) =
Cl∗ (Int(X)).
(b) ⇒ (a) Let A is b − I-open and strong BI -set. Then, A ⊂ Int(Cl∗ (A)) ∪
Cl∗ (Int(A)) = Int(Cl∗ (U ∩ V )) ∪ Cl∗ (Int(U ∩ V )), where U is open and V is t − Iset and Int(Cl∗ (V )) = Cl∗ (Int(V )). Hence A ⊂ (Int(Cl∗ (U )) ∩ Int(Cl∗ (V ))) ∪
(Cl∗ (Int(U )) ∩ Cl∗ (Int(V )))
A ⊂ U ∩ (Int(CI ∗ (V ))) ∪ CI ∗ (Int(V )))
A ⊂ U ∩ (Int(CI ∗ (V )) ∪ CI ∗ (Int(V )))
A ⊂ U ∩ Int(CI ∗ (V ))
A ⊂ U ∩ Int(V ) = Int(A).
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[b-I-open sets and decom.of continuity...]
So A is open.
Remark 3. The notion of b−I-openness is different from that of strong BI -sets.
Because
(i) In Example 2.3, A = {b, c} is not b−I-open. But Int(Cl∗ (A)) = Cl∗ (Int(A)) =
Int(A) = {b}. So A is a strong BI -set.
(ii) In Example 2.1, A = {b, d} is b − I-open. But Int(Cl∗ (A)) = X 6= Int(A).
So A is not a strong BI -set.
Decomposition of Continuity
Definition 5. A function f : (X, τ ) → (Y, σ) is said to be b-continuous [1] (resp.
strong B−continuous [3]) if for each open set V of (Y, σ), f −1 (V ) is b−open (resp.
strong B-set)in (X, τ ).
Definition 6. A function f : (X, τ , I) → (Y, σ) is said to be b − I-continuous
(resp. semi-I-continuous [4], pre-I-continuous [3] strong BI -continuous) if for each
open set V of (Y, σ), f −1 (V ) is b-open (resp. semi-I-open, pre-I-open, strong BI -set)
in (X, τ ).
Proposition 6. If a function f : (X, τ , I) → (Y, σ) is semi-I-continuous (pre-Icontinuous), then f is b − I-continuous.
Proof. This is an immediate consequences of Proposition 2.2. (b) and (c).
Proposition 7. If a function f : (X, τ , I) → (Y, σ) is b − I-continuous, then f
is b-continuous.
Proof. This is an immediate consequences of Proposition 2.2 (e).
Proposition 8. If a function f : (X, τ , I) → (Y, σ) is strong B-continuous, then
f is strong BI -continuous.
Proof. This is an immediate consequences of Proposition 3.1 (b).
Theorem 1. For a function f : (X, τ , I) → (Y, σ) the following conditions are
equivalent:
(a) f is continuous,
(b) f is b − I-continuous and strong BI -continuous.
Proof. This is an immediate consequences of Proposition 3.2.
References
[1]. Dimitrije Andrijevi ć, On b-open Sets, MATHEMAT, 48 (1996), 59-64.
[2] M. E. Abd El-Monsef, E. F. Lashien and A. A. Nasef, On I-open sets and
I-continuous function, Kyungpook Math. J., 32 (1992), 21-30.
[3] J. Dontchev, Strong B-sets and another decomposition of continuity, Acta
Math. Hungar., 75 (1997), 259-265.
[4] J. Dontchev, Idealization of Ganster-Reilly decomposition theorems, Math.
GN/9901017, 5 Jan 1999 (Internet).
[5] E. Hatir and T. Noiri, On decompositions of continuity via idealization, Acta
Math. Hungar, 96 (4) (2002), 341-349.
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[A.C.Guler and G.Aslim]
[6] E. Hayashi, Topologies defined by local properties, math. Ann., 156 (1964),
205-215.
[7] D. Jankovic̆ and T. R. Hamlett, New topologies from old via ideals, Amer.
Math. Monthly, 97 (1990), 295-310.
[8] O. N j ȧstad, On some clasess of nearly open sets, Pacific J. Math., 15 (1965),
961-970.
[9] P. Samuels, A topology formed from a given topology and ideal, J. London
Math. Soc., 10 (1975), 409-416.
[10] R. Vaidyanathaswamy, Proc. Indian Acad Sci., 20 (1945),51-61
Gulhan ASLIM and Aysegul CAKSU GULER
Ege University, Department of Mathematics
35100 Izmir/TURKEY
e-mail : [email protected] and.
e-mail : [email protected].
Received November 04, 2004; Revised January 18, 2005.
Translated by Mamedova V.A.