Download ON δ-SEMIGENERALIZED CLOSED SETS IN TOPOLOGY

International Journal of Applied Mathematics and Applications
3(2), December 2011, pp. 137-143
© Global Research Publications
ON -SEMIGENERALIZED CLOSED SETS IN TOPOLOGY
M. RAJAMANI & S. PADMANABAN
ABSTRACT: In this paper, we introduce a new class of sets called δ-Semigeneralized Closed
Sets, briefly δ-sg-closed. We study some of its basic properties and investigate its relation to
sg-closed sets and δ-g-closed sets. We also introduce δ-sg-continuity and δ-sg-irresoluteness
by using δ-sg-closed sets and study some of their fundamental properties.
Mathematical Subject Classification: 54A05.
Keywords and Phrases: δ-closed set, δ-semi closure, δ-semigeneralized closed set.
1. INTRODUCTION
Levine [7] initiated the study of generalized closed sets, i.e., the sets whose closure
belongs to every open superset, and defined the notation of T1/2 space to be one in
which the closed sets and generalized closed sets coincide. By using δ-closure operator,
Dontchev and Ganster [4] introduced and studied the concept of δg-closed sets which
is a slightly stronger form of g-closedness, properly placed between δ-closedness and
g-closedness and introduced the notation of T3/4 spaces as the spaces where every
δg-closed set is δ-closed. Dontchev et al. [5] introduced and studied the notation of
gδ-closed and δg*-closed sets and used these sets to give characterizations of atmost
Hausdorff spaces. Jin Han Park, Dae Seob Song and Reza Saadati [12] introduced and
studied the concept of gδs-closed sets, which is slightly weaker form of δg-closedness
and δ-semiclosedness and slightly stronger form of δgs-closeness.
In this paper, we introduce a new class of sets called δ-Semigeneralized Closed
Sets, briefly δ-sg-closed. We study some of its basic properties and investigate its relation
to sg-closed sets and δ-g-closed sets. We also introduce δ-sg-continuity and
δ-sg-irresoluteness by using δ-sg-closed sets and study some of their fundamental
properties.
2. PRELIMINARIES
Throughout this paper, spaces (X, τ) and (Y, τ) always mean topological spaces on
which no separation axioms are assumed unless explicitly stated. Let A be a subset of a
space (X, τ). We denote closure and interior of A by cl (A) and int (A), respectively.
A subset A is said to be semi-open [6], if there exists an open set U such that U ⊂ A ⊂ Cl(U),
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or equivalently if A ⊂ Cl (Int (A)). The family of all semi-open sets of (X, τ) is denoted
by SO (X, τ). For each x ∈ X, the family of all semi-open sets containing x is denoted
by SO (X, x). The complement of a semi-open set is said to be semi-closed [6]. The
intersection of all semi-closed sets containing A is called the semi-closure [3] of A and
denoted by scl (A). The semi-interior of A denoted by sint (A), is the union of all
semi-open sets containing A. A subset A is called semi-regular [9] if it is both semi-open
and semi-closed. i.e., τ = τs.
The δ-interior [17] of a subset A of X is the union of all regular open sets of X
contained in A and is denoted by δ-int (A). The subset A is called δ-open [17] if
A = δ-int (A). i.e., a set is δ-open if it is the union of regular open sets. The complement
of δ-open set is δ-closed [17]. Alternatively, a set A ⊂ (X, τ) is called δ-closed [17] if
A = δ-cl (A), where δ-cl (A) = {x ∈ X : int (cl (U)) ∩ A ≠ φ, U ∈ τ and x ∈ U}.
The family of δ-open sets forms a topology on X and is denoted by τδ. It is well
known that τδ = τs.
A subset A of X is called preopen [8], if A ⊂ int (cl (A)). The complement of a
preopen sets is called preclosed. A subset A of X is called δ-semiopen [14], if
A ⊂ cl (δ-int (A)). The complement of a δ-semi open sets is called δ-semiclosed. A
subset A of X is called δ preopen [15], if A ⊂ int (δ-cl (A)). The complement of a
δ-preopen sets is called δ-preclosed. The intersection of all δ-semiclosed sets containing
A is called δ-semi-closure [14] of A and is denoted by δ-scl(A). Dually, the δ semi-interior
of A is defined to be union of all δ-semiopen sets contained in A and is denoted by
δ-sint (A). Note that δ-scl (A)= A ∪ int (δ-cl (A)) and δ-sint (A) = A ∩ cl (δ-int (A)).
A subset A of a space (X, τ) is said to be
(a) generalized closed [7] (briefly g-closed) if cl (A) ⊂ U whenever A ⊂ U and U
is open in X.
(b) generalized semi closed [1] (briefly gs-closed) if scl (A) ⊂ U whenever A ⊂ U
and U is open in X.
(c) δ-generalized semi closed [13] (briefly δgs-closed) if δ-scl (A) ⊂ U whenever
A ⊂ U and U is δ-open in X.
(d) δ-generalized closed [4] (briefly δg-closed) if δ-cl (A) ⊂ U whenever A ⊂ U
and U is open in X.
(e) generalized δ-closed [5] (briefly gδ-closed) if cl (A) ⊂ U whenever A ⊂ U and
U is δ-open in X.
(f)
semi generalized closed [2] (briefly sg-closed) if scl (A) ⊂ U whenever A ⊂ U
and U is semi open in X.
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A topological space (X, τ) is a semi-T1/2 [2] if and only if every sg-closed set is
semi-closed or equivalently the space (X, τ) is a semi-T1/2 [16] if and only if every
singleton is semi-open or semi-closed.
A function f : (X, τ) → (Y, σ) is said to be pre semi-open(resp. pre semi-closed) [11]
if f (U) ∈ SO (Y, σ) for every U ∈ SO (X, τ). A function f : (X, τ) → (Y, σ) is called
sg-continuous [16] if f –1(V) is a sg-closed set of (X, τ) for every closed set F of (Y, σ). A
function f : (X, τ) → (Y, σ) is said to be quasi-irresolute [10] (resp. strongly irresolute)
if for each x ∈ X and each V ∈ SO(Y, f(x)) there exists U ∈ SO(X, x) such that f(U) ⊂ scl(V).
3. -SEMIGENERALIZED CLOSED SETS
Definition 3.1: A subset A of a topological space (X, τ) is called δ-Semigeneralized
Closed (briefly δ-sg-closed) if δ-scl (A) ⊂ U when A ⊂ U and U ∈ SO (X, τ).
The complement of a δ-Semigeneralized Closed Set is called δ-Semigeneralized
open Sets (briefly δ-sg-open).
Example 3.2: Let X = {a, b, c} and let τ = {Φ, {a}, X} then SC (X) = {Φ, {b}, {c},
{b, c}, X} and δ-sg-cl (X) = {Φ, {b, c}, X}. If A = {b} then it is semiclosed but it is not
δ-sg-closed.
Example 3.3: Let X = {a, b, c} and let τ = {Φ, {a}, {b, c}, X} then SC (X) = {Φ, {a},
{b, c}, X} and δ-sg-cl (X) = P (X).
If A = {b} then it is δ-sg-closed but it is not semiclosed.
Result 3.4: Note that semi closedness is independent of δ-sg-closedness as it is
showed in the example 3.2 and example 3.3.
Lemma 3.5: Every δ-sg-closed set is sg-closed but not conversely.
Proof: It is true that scl (A) ⊂ δ-scl (A) for every subset A of (X, τ).
Example 3.6: Let X = {a, b, c}and let τ = {Φ, {a}, {a, b}, X} then δ-sg-cl (X) =
{Φ, {b, c}, X} and sg-c (X) = {Φ, {b}, {c}, {b, c}, X}. If A = {c} then it is sg-closed but
not δ-sg-closed.
Example 3.7: Let X = {a, b, c} and let τ = {Φ, {a}, X} then δ-sg-cl(X) = {Φ, {b, c}, X}
and δ-g-C (X) = {Φ, {b}, {c}, {a, b}, {b, c}, {a, c}, X}. If A = {a, b} then it is δ-g-closed
but not δ-sg-closed.
Example 3.8: Let X = {a, b, c} and let τ = {Φ, {a}, {b}, {a, b}, X} then δ-sg-cl (X)
= {Φ, {a}, {b}, {c}, {b, c}, {a, c}, X} and δ-g-cl (X) = {Φ, {c}, {b, c}, {a, c}, X}. If
A = {b} then it is δ-sg-closed but not δ-g-closed.
Remark 3.9: It should be noted that δ-g-closedness is independent of δ-sg-closedness
as it is shown in the above example 3.7 and 3.8.
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Lemma 3.10: Every δ-semi-closed set is δ-sg-closed but not conversely.
Proof: Let A ⊂ X be δ-semi-closed. Then A = δ-scl (A). Let A ⊂ U and U ∈ SO (X, t).
It follows that δ-scl (A) ⊂ U. This means that A is δ-sg-closed.
Example 3.11: Let X = {a, b, c} and let τ = {Φ, {a}, {b, c}, X} then δ-sc (X) =
{Φ, {a}, {b, c}, X} and δ-sg-cl (X) = {Φ, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, X}. If
A = {b} then it is δ-sg-closed but not δ-s-closed.
Lemma 3.12: Let A be semi-open subset of a topological space (X, τ). The set A is
δ-sg-closed if and only if A is sg-closed.
Proof: It is true that scl (A) = δ-scl (A) for every A ∈ SO (X, τ).
Theorem 3.13: A set A ⊂ (X, τ) is δ-sg-open if and only if F ⊂ δ-sint (A) whenever
F is semi-closed in X and F ⊂ A.
Proof: Necessary part: Let A be δ-sg-open and F ⊂ A where F is semi-closed. It is
obvious that Ac is contained in Fc. This implies that δ-scl (Ac) ⊂ Fc. Hence δ-scl (Ac) =
(δ-sint (A))c ⊂ Fc, i.e., F ⊂ δ-sint (A).
Sufficient part: If F is a semi-closed set with F ⊂ δ-sint (A),whenever F ⊂ A, then it
follows that Ac ⊂ Fc and (δ-sint (A))c ⊂ Fc, i.e, δ-scl (Ac) ⊂ Fc. Therefore Ac is δ-sg-closed
and therefore A is δ-sg-open. Hence the proof.
Theorem 3.14: A topological space (X, τ) is a semi T1/2 space if and only if every
δ-sg-closed set is semi-closed.
Proof: Necessary part: Let A ⊂ X be a δ-sg-closed. By lemma 3.5, A is sg-closed.
Since X is a semi-T1/2 space, A is semi closed. Sufficient part: Let x ∈ X. If {x} is not
semi-closed, then {x}c is not δ-semi open and thus only superset of {x}c is X. Trivially
{x}c is δ-sg-closed. By hypothesis {x}c is semi-closed or equivalently {x} is semi open.
Hence X is a semi-T1/2 space.
Lemma 3.15: Let A be a δ-sg-closed subset of (X, τ), then
(i)
δ-scl (A)\A does not contain a non empty semi-closed set
(ii) δ-scl (A)\ A is δ-sg-open
Proof: (i) Let F be a semi-closed set such that F ⊂ δ-scl (A)\A. Since Fc is semi-open
and A ⊂ Fc, it follows that δ-scl (A) ⊂ Fc, i.e., F ⊂ (δ-scl (A))c. This implies that
F ⊂ (δ-scl (A))c ∩ δ-scl (A) = φ. (ii) If A is δ-sg-closed and F is a semi closed set such
that F ⊂ δ-scl (A)\A, then by (i), F is empty and therefore F ⊂ δ-sint (δ-cl (A)\A). By
theorem 3.13 , δ-scl (A)\A is δ-sg open.
Lemma 3.16: If A is a δ-sg-closed set of a topological space (X, τ) such that
A ⊂ B ⊂ δ-scl (A), then B is also a δ-sg-closed set of (X, τ).
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Proof: Let O be a semi-open set of (X, τ) such that B ⊂ O. Then A ⊂ O. Since A is
δ-sg-closed, δ-scl (A) ⊂ O. δ-scl (B) ⊂ δ-scl (δ-scl (A)) = δ-scl (A) ⊂ O. Therefore B is
also a δ-sg-closed set of (X, τ).
4. -SEMIGENERALIZED CONTINUOUS FUNCTIONS
Definition 4.1: A function f : (X, τ) → (Y, σ) is called
(i)
δ-Semigeneralized Continuous (briefly δ-sg-continuous) if f –1(V) is δ -sg-closed
in (X, τ) for every semi-closed set V of (Y, σ).
(ii) δ-Semigeneralized irresolute (briefly δ-sg-irresolute) if f –1(V) is δ-sg-closed
in (X, τ) for every δ-sg-closed set V of (Y, σ).
Theorem 4.2: If f : (X, τ) → (Y, σ) is a bijective pre semi-open and δ-sg-Continuous
function, then f is δ-sg-irresolute.
Proof: Let V be a δ-sg-closed set of (Y, σ). Assume that f –1(V) ⊂ O, where
O ∈ SO (X, τ). Clearly V ⊂ f (O). Since f (O) ∈ SO (Y, σ) and V is a δ-sg-closed set, we
have δ-scl (V) ⊂ f (O) and therefore f –1(δ-scl (V)) ⊂ O. By fact that f is δ-sg-Continuous
and δ-scl (V) is semi-closed, it follows that δ-scl ( f –1(δ-scl (V)) ⊂ O and thus
δ-scl ( f –1(V)) ⊂ O. This means that f –1(V) is δ-sg-closed set in (X, τ) and therefore f is
δ- sg-irresolute.
Corollary 4.3: If f : (X, τ) → (Y, σ) is a bijective pre semi-open δ-sg-Continuous
function and (X, τ) is semi- T½ , then (Y, σ) is semi- T½ .
Proof: It suffices to apply theorem 3.14 and theorem 4.2.
Proposition 4.4: If f : (X, τ) → (Y, σ) is δ-sg-continuous, then f is sg-continuous.
Proof: It is an immediate consequence of Lemma 3.5.
The following example shows that the converse of proposition is not true.
Example 4.5: Let X = Y = {a, b, c}, τ = {Φ, {a}, {a, b}, X} and σ = {Φ, {a, b}, X}.
Let f : (X, τ) → (Y, σ) be a idendity function. Then f is semi-continuous and hence
sg-continuous but as it is shown in example 3.6. A = {c} is not sg-closed which means
that f is not δ-sg-Continuous.
Definition 4.6: A function f : (X, τ) → (Y, σ) is said to be δ-sg-closed if for every
semi-closed set V of (X, τ), f (V) is δ-sg-closed in (Y, σ).
Recall that a function f : (X, τ) → (Y, σ) is said to be irresolute, if f –1(V) ∈ SO (X, τ)
for every V ∈ SO (Y, σ).
Theorem 4.7: If f : (X, τ) → (Y, σ) is irresolute and δ-sg-closed, then f (A) is
δ-sg-closed in Y for a δ-sg-closed set A of X.
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Theorem 4.8: A function f : (X, τ) → (Y, σ) is quasi-irresolute and pre-semi-closed,
then for every δ-sg-closed set F of (Y, σ), f –1(F) is δ-sg-closed set of (X, τ).
Proof: Suppose that F is a δ-sg-closed set F of (Y, σ). Assume f –1(F) ⊂ O, where
O ∈ SO (X, τ).
By hypothesis, f is a quasi-irresolute and therefore
f (δ-scl ( f –1(F))  Oc) ⊂ δ-scl ( f ( f –1( F)))  f ( f –1(Fc)) ⊂ δ-scl (F)\F.
By the fact that f is pre-semi-closed, it follows that δ-scl (F)\F contains a semi-closed
subset f (δ-scl f –1(F))  Oc). Now we have f (δ-scl ( f –1(F))  Oc ) = Φ by lemma 3.15.
This implies that δ-scl ( f –1(F)) ⊂ O. This shows that f –1(F) is a δ-sg-closed set of (X, τ).
Theorem 4.9: If a topological space (X, τ) is semi- T 1 and f : (X, τ) → (Y, σ) is
2
surjective, quasi-irresolute and pre-semi-closed, then (Y, σ) is semi- T 1 .
2
Proof: Assume that A is a δ-sg-closed subset of (Y, σ). Then by theorem 4.8 we
have f –1(A) is δ-sg-closed subset of (X, τ). By theorem 3.14, f –1(A) is semi-closed and
hence, A is semi-closed. It follows that (Y, σ) is semi- T 1 .
2
Remark 4.10:
(1) If f : (X, τ) → (Y, σ) is δ-sg-irresolute and g : (Y, σ) → (Z, µ) is δ-sg-continuous,
then g ο f is δ-sg-continuous.
(2) If f : (X, τ) → (Y, δ) and g : (Y, σ) → (Z, µ) are δ-sg-irresolute, then g ο f is
δ-sg-irresolute.
REFERENCES
[1]
Arya S. P., and Nour T., (1990), Characterizations of s-Normal Spaces, Indian J. Pure Appl. Math.,
21, 717-719.
[2]
Bhattacharyya P., and Lahiri B. K., (1987), Semi-Generalized Closed Sets in Topology, Indian J.
of Math., 29(3), 375-382.
[3]
Crossley S. G., and Hildebrand S. K., (1971), Semi-Closure, Texas J. Sci., 22, 99-112.
[4]
Dontchev J., and Ganster M., (1996), On δ-Generalized Closed Set and T3/4-Spaces, Mem. Fac.
Sci. Kochi Univ. Ser. A (Math), 17, 15-31.
[5]
Dontchev J., Arokiarani I., and Balachandran K., (2000), On Generalized δ-closed Sets and Almost
Weakly Hausdorff Spaces, Questions Answers Gen Topology, 18(1), 17-30.
[6]
Levine N., (1963), Semi-open Sets and Semi-Continuity in Topological Spaces, Amer. Math.
Monthly, 70, 36-41.
[7]
Levine N., (1970), Generalized Closed Sets in Topology, Rend. Circ. Mat. Palermo, 19, 89-96.
[8]
Mashhour A. S., Abd El-Monsef M. E., and El-Deeb S. N., (1982), On Precontinuous and Week
Precontinuous Functions, Proc. Math. Phy. Soc. Egypt, 53, 47-53.
ON δ-SEMIGENERALIZED CLOSED SETS
IN
TOPOLOGY
143
[9]
Maio G. Di., and Noiri T., (1987), On s-Closed Spaces, Indian J. Pure Appl. Math., 18(3), 226-233.
[10]
Maio G. Di., and Noir T., (1988), Weak and Strong Forms of Irresolute Functions, Rend Circ.
Mat. Palermo (2) Suppl, 18, 255-273.
[11]
Miguel Calidas, and Saeid Jafari, (2003), On δ-semigeneralised Closed Sets in Topology,
Kyungpook Math. J., 43, 135-148.
[12]
Park J. H., Song D. S., and Saadati R., (2007), On Generalized δ-semi Closed Sets in Topological
Spaces, Chaos Solitons and Fractals, 33, 1329-1338.
[13]
Park J. H., Song D. S., and Lee B. Y., (2007), On δgs-closed Sets and Almost Weakly Hausdorff
spaces, Honam Mathematical J., (2007)29,597-615.
[14]
Park J. H., Lee B. Y., and Son M. J., (1997), On δ-semiopen Sets in Topological Space, J. Indian
Acad. Math., 19, 59-67.
[15]
Raychaudhuri S., and Mukherjee N., (1993), On δ-almost Continuity and δ-preopen Sets, Bull.
Inst. Math. Acad. Sinica, 21, 357-66.
[16]
Sundaram P., Maki H., and Balachandran K., (1991), Semi-Generalized Continuous Maps and
Semi-T1/2 Spaces, Bull. Fukuoka Univ. Ed. Part III, 40, 33-40.
[17]
Velič ko N. V., (1968), H-Closed Topological Spaces, Amer. Math. Soc. Transl., 78, 103-18.
M. Rajamani
Department of Mathematics, NGM College,
Pollachi-642001, Tamilnadu, India
E-mail: [email protected]
S. Padmanaban
Department of Science and Humanities,
Karpagam Institute of Technology,
Coimbatore-641105, Tamilnadu, India.
E-mail: [email protected]