Download SEMI-GENERALIZED SEPARATION AXIOMS IN TOPOLOGY*

International Journal of Mathematics and Computing Applications
ISSN: 0976-6790
Vol. 3, Nos. 1-2, January-December 2011, pp. 17-25
© International Science Press
SEMI-GENERALIZED SEPARATION AXIOMS
IN TOPOLOGY*
Govindappa Navalagi
Abstract: In this paper, we introduce and study a new class of separation axioms called
semi-generalized separation axioms using semi-generalized open sets due to Bhattacharya
and Lahari. The connections between these separation axioms and other existing
well-known related semi separation axioms are also investigated.
2000 Math.Subject Classification:Primary: 54A05, 54C08, 54D10;
Key Words and Phrases: Semiopen sets, semiclosed sets, sg-closed sets, sg-open sets,
semicontinuous functions
1. INTRODUCTION
N. Levine [13] introduced the concept of semiopen sets in topology. In [14] Levine
generalized the concept of closed sets to generalized closed sets. Bhattacharya and
Lahiri [4] generalized the concept of closed sets to semi generalized closed sets via.
Semiopen sets. The complement of a semiopen (resp. g-closed, semi generalized
closed) set is called a semiclosed [5] (resp. g-open [14], semi generalized open [4])
set. Recently, there is a vast progression in the field of generalized closed sets. In
[15] Maheshwari and Prasad have defined the concepts of semi-Ti, i = 0, 1, 2 spaces.
After then there are many works on semi separation axioms. In this paper, we
introduce the generalized forms of semi separation axioms using the concepts of
semi generalized open sets called semi generalized - Ti (briefly denoted by sg-Ti)
spaces. Also, we define the concepts of ψ-open sets in topology to define the another
class of separation axioms called ψ-separation axioms which is weaker than the
class of semi-Ti i = 0, 1, 2 axioms spaces and stronger than sg-Ti axioms. Among
other things, we study their basic properties and relative preservation properties of
these spaces.
2. PRELIMINARIES
Through this paper, we denote X the topological space on which no separation axioms
are assumed unless explicitly stated. Let A be the subset of the space X. We denote
Department of Mathematics, KLE Society, G.H. College, Haveri-581110, Karnataka, India.
E-mail: [email protected]
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clA and intA respectively closure and the interior of the set A. We, below list the
definitions and results which are useful in the sequel.
Definition 2.1: Let A be a subset of a space X then A is said to be:
(i) a semiopen if A ⊂ clintA [13], (ii) semi-preopen set [1] if A ⊂ cl int clA, (iii)
a δ-closed [19] if A = clδ(A) where clδ (A) = {x ∈ X — intclA ∩ A ≠ 0 , x ∈ U and U
is open set} (iv) a generalized -closed (i.e., g-closed) set [14] if clA ⊂ U whenever A
⊂ U and U is open set and (v) a δ-generalized closed (i.e.δ-g-closed) set [10] if cl
δ(A) ⊂ A whenever A ⊂ U and U is open set in X.
The complement of a semiopen set (resp. semipreopen, δ-closed and g-closed
and δ-g-closed) set is called semiclosed [5] (resp. semipreclosed [1], δ-open [19]
and g-open [14] and δ-g-closed [10]) set.
Definition 2.2: The semiclosure [5] (resp. semipreclosure [1]) of a subset A of X
is the intersection of all semiclosed (resp. semipreclosed) sets that contains A and is
denoted by sclA (resp. spclA). The union of all semiopen subsets of X is called the
semi interior [6] of A and is denoted by sint A. Semi generalized closure of a subset A
of a space X is the intersection of all sg-closed sets containing A and is denoted by
sgcl(A) [16]. A point x of a space X is called a semi generalized limit point. Written as
sg-limit point) of a subset A of X, if for each sg-open set U containing x, A ∩ (U–{x})
≠ 0 and the set of all sg-limit points of A, denoted by sgd(A), is called semi generalized
derived set of A [16].
Definition 2.3: Let A be a subset of a space X then A is said to be (i) a semi
generalized closed (i.e. sg-closed) set [4] if sclA ⊂ U whenever A ⊂ U and U is
semiopen set, (ii) a generalized semiclosed (i.e., gs-closed) set [2] if sclA ⊂ U
whenever A ⊂ U and U is open set in X, (iii) a generalized semipreclosed (i.e.,
gsp-closed) set [9] (iv) a ψ-closed set [18] if sclA ⊂ U whenever A ⊂ U and U is
sg-open set.
Definition 2.4: A space X is called a (i) T1/2 space [14] if every g-closed set is
closed, (ii) semi –T1/2 space [4] if every sg-closed set is semiclosed, (iii) semi-TD
space [12] if every singleton set is either open or nowhere dense set and (iv) semipreT1/2 space [9] if gsp-closed set is semipreclosed set.
Definition 2.5: A function f : (X, τ) → (Y, σ) is said to be :
(i) semi-continuous [13] if f–1(V) is semiopen set in (X, τ) for every open set V of
(Y, σ), (ii) g-continuous [3] if f–1(V) is g-closed in (X, τ) for every closed set V of
(Y, σ), (iii) sg-continuous [17] if f–1(V) is sg-closed in (X, τ) for every closed set
V of
(Y, σ), (iv) irresolute [6] if f–1(V) is semiopen in (X, τ) for every semiopen set
V of
Semi-generalized Separation Axioms in Topology*
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(Y, σ), (v) sg-irresolute [17] if f–1(V) is sg-closed in (X, τ) for every sg-closed set
V of (Y, σ), (vi) pre-semi-open [6] if f(U) is semiopen in (Y, σ) for every semiopen set
U of (X, τ) (vii) ψ-continuous [6] if f–1(V) is a ψ-closed set of (X, τ) for every closed
set V of (Y, σ) and (viii) ψ-irresolute [18] if f–1(V) is ψ-closed set of (X, τ) for every
ψ-closed set V of (Y, σ).
3. SEMI-GENERALIZED SEPARATION AXIOMS
In this section, we define and study some new separation axioms using sg-open sets
which are weaker than semi separation axioms.
We, recall the following.
Definition 3.1 [11a]: A space X is called semi generalized –T0 (briefly written as
sg-T0) iff to eahc pair of distinct points x, y of X, there exists a sg-open set containing
one but not the other.
Clearly, every semi-T0 space is sg-T0 space since every semi open set is sg-open
set but converse is not true [11a].
We, characterize sg-To spaces in the following.
noindent THEOREM 3.2: If in any topological space X, semi generalized
closures of distinct points are distinct, then X is sg-T0.
Proof: Let x, y ∈ X, x ≠ y imply sgcl{x} ≠ sgcl{y}. Then there exists a point z ∈
X such that z belongs one of two sets, say, sgcl{y} but not to sgcl{x}. If we suppose
that z ∈ sgcl{x}, then z ∈ sgcl{y} ⊂ sgcl{x}, which is contradiction. So, y ∈
X-sgcl{x}, where X-sgcl{x} is sg-open set which does not contain x. This shows that
X is sg-T0.
Theorem 3.3: In any topological space X, semi generalized closures (sgclosures)
of distinct points are distinct.
Proof: Let x, y ∈ X with x ≠ y. We show that sgcl{x} ≠ sgcl{y} : Consider the set
In view of the above two theorem, we conclude the following:
Theorem 3.4: Every topological space is sg-To.
We, recall the following.
DEFINITION 3.5: [18a]: A topological space X is semi-Co(resp. α-Co) if, for x,
y ∈ X with x ≠ y, there exists G ∈ SO(X) (resp. G ∈ α(X)) such that sclG (resp. αclG) contains one of x and y, but not the other.
Clearly, every semi-Co space is semi-To but converse is not true see, Remark-2.3
[18a] and every α-Co space is semi-Co but converse is not true see, Remark-3.3
[18a]. Thus, in view of the above discussions, we have the following.
Definition 3.6: Every α-Co space ⇒ semi-Co ⇒ semi-To ⇒ sg-To but converses
are not true in general.
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We, define the following.
Definition 3.7: A space X is called semi generalized –T1 (briefly written as
sg-T1) iff to each pair of distinct points x, y of X, there exists a pair of sg-open sets,
one containing x but not y, and the other containing y but not x.
Clearly, every semi-T1 space is sg-T1 space since every semiopen set is sg-open
set.
We define the following.
Definition 3.8: A subset A of a space X is called a semi generalized neighbourhood
(i.e. sgnbd.) of a point x of X if there exists sg-open set U containing x such that A ⊂
U.
Definition 3.9: The union of all sg-open sets which are contained in A is called
the semi generalized interior of A and is denoted by sgintA.
Since the union of sg-open sets is sg-open and hence sgintA is sg-open.
Now, we prove the following property of sgnbd. of A.
Lemma 3.10: A subset of a space X is sg-open iff it is a sgnbd. of each of its
points.
Easy proof is omitted.
Definition 3.11: A point x of X is called a semi generalized interior point (i.e.
sg-interior point) of A ⊂ X iff x ∈ A.
Next, we prove easily the following lemma.
Lemma 3.12: Let X be a space and A ⊂ X, x ∈ X. Then x is a sg-interior point of
A iff A is a sgnbd. of x.
Theorem 3.13: For a topological space X, each of the following statement is
equivalent:
(a) X is sg-T1
(b) Each one pointic set is sg-closed set in X
Proof: follows by Lemma 3.5 above.
From the definition of sg-limit point and sgd(A), the following are proved in
[16].
Lemma 3.14: If A is a subset of a space X then sgcl(A) = A [sgd(A).
Lemma 3.15: A point x ∈ sgcl(A) iff every sg-open set containing x contains the
point of A.
Now, we prove the following.
Semi-generalized Separation Axioms in Topology*
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Theorem 3.16: If X is sg-T1 and p ∈ sgd(A) for some subset A of X, then every
sgnbd. of p contains infinitely many points of A.
Theorem 3.17: In a sg-T1 space X, sgd(A) is sg-closed for any subset A of X.
Next, we give the invariant property of sg-T1 spaces in the following.
Theorem 3.18: Let f : X → Y be an injective and sg-irresolute mapping. If Y is
sg-T1 then X is sg-T1.
Proof is easy and hence omitted.
We, define the following.
Definition 3.19: A space X is calle semi generalized – T2 space (briefly written
as sg-T2 space) iff to each pair of distinct points x, y of X there exists a pair of
disjoint sg-open sets, one containing x and the other containing y.
Clearly, every semi-T2 space is sg-T2 space since every semiopen set is sg-open
set.
Theorem 3.20: If f:X → Y is injective and sg-irresolute and Y is sg-T2 space then
X is sg-T2.
Easy proof is omitted.
Next, we define the following.
Definition 3.21: A space X is called semi generalized -Ro(i.e. written as sg-Ro)
iff for each sg-open set G and x ∈ G implies sgcl{x} ⊂ G.
Clearly, every semi-Ro space is sg-Ro.
Definition 3.22: A space X is called semi generalized – R1 (i.e. written as sg-R1)
space iff for x, y ∈ X with sgcl{x} ≠ sgcl{y}, there exist disjoint sg-open sets U and
V such that sgcl{x} ⊂ U and sgcl{y} ⊂ V.
Clearly, every semi-R1 space is sg-R1.
Clearly, every sg-R1 space is sg-Ro space.
Next, we give the following.
Theorem 3.23: The following are equivalent.
(a) X is sg-T2 space
(b) X is sg-R1 and sg-T1 space
(c) X is sg-R1 and sg-To.
Proof is easy and hence omitted.
4. -SEPARATION AXIOMS
In this section, we define and study some new separation axioms by defining
ψ-open sets which are stronger than semi generalized separation axioms.
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Definition 4.1: A subset A of X is called ψ-open set of X if F ⊂ sintA whenever
F is sg-closed and F ⊂ A.
Clearly, every open set, semiopen set is ψ-open and every ψ-open set is sg-open
set.
We, define the following.
Definition 4.2: A subset A of a space X is called a ψ-neighbourhood of a point x
of X if there exists ψ-open set U containing x such that A ⊂ U.
Definition 4.3: The union of all ψ-open sets which are contained in A is called
the ψ-interior of A and is denoted by ψ-intA.
Since the union of ψ-open sets is ψ-open and hence ψ-intA is ψ-open.
Now, we prove the following property of ψ-nbd. of A.
Lemma 4.4: A subset of a space X is ψ-open iff it is a ψ-nbd. of each of its
points.
Easy proof is omitted.
Definition 4.5: A point x of X is called a ψ-interior point of A ⊂ X iff x ∈ A.
Next, we prove easily the following lemma.
Lemma 4.6: Let X be a space and A ⊂ X, x ∈ X. Then x is a ψ - interior point of
A iff A is a ψ-nbd. of x.
Definition 4.7: The ψ-closure of a subset A of X is the intersection of all
ψ-closed sets that contains A and is denoted by ψ-clA.
Definition 4.8: A point x of a space X is called a ψ-limit point of a subset A of X,
if for each ψ-open set U containing x, A ∩ (U– {x}) ≠ 0
Definition 4.9: The set of all ψ-limit points of A, denoted by ψ-d(A), is called
ψ-derived set of A
Definition 4.11: A space X is called ψ – T0 iff to eahc pair of distinct points x, y
of X, there exists a ψ-open set containing one but not the other.
Clearly, every semi-T0 space is ψ-T0 space and every ψ-T0 is sg-T0 since every
semi-open set is ψ-open and every ψ-open set is sg-open set.
Theorem 4.12: If in any topological space X, ψ-closures of distinct points are
distinct, then X is ψ-T0:
We, define the following.
Definition 4.13: A space X is called ψ-T1 iff to each pair of distinct points x, y of
X, there exists a pair of ψ-open sets, one containing x but not y, and the other containing
y but not x.
Semi-generalized Separation Axioms in Topology*
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Clearly, every semi-T1 space is ψ-T1 space and ψ-T1 space is sg-T1 space since
every semiopen set is ψ-open and every ψ-open set is sg-open set.
Theorem 4.14: For a topological space X, each of the following statement are
equivalent:
(a) X is ψ–T1
(b) Each one pointic set is ψ-closed set in X
Proof follows by Lemma 3.5 above.
From the definition of ψ-limit point and ψ-d(A), the following can be easily
proved.
Lemma 4.15: If A is a subset of a space X then ψ-cl(A) = A ∪ ψ-d(A).
Lemma 4.16: A point x ∈ ψ-cl(A) iff every ψ-open set containing x contains the
point of A.
Now, we prove the following.
Theorem 4.17: If X is ψ-T1 and p ∈ ψ-d(A) for some subset A of X, then every
ψ-nbd. of p contains infinitely many points of A.
Theorem 4.18: In a ψ-T1 space X, ψ-d(A) is sg-closed for any subset A of X.
Next, we give the invariant property of ψ-T1 spaces in the following.
Theorem 4.19: Let f : X → Y be an injective and ψ-irresolute mapping. If Y is
ψ-T1 then X is ψ-T1.
Proof is easy and hence omitted.
We, define the following.
Definition 4.20: A space X is called ψ – T2 space iff to each pair of distinct
points x, y of X there exists a pair of disjoint ψ-open sets, one containing x and the
other containing y.
Clearly, every semi-T2 space is ψ-T2 and ψ-T2 space is sg-T2 space since every
semiopen set is ψ-open and every ψ-open set is sg-open set.
Theorem 4.21: If f:X → Y is injective and ψ-irresolute and Y is ψ-T2 space then
X is ψ-T2.
Easy proof is omitted.
Next, we define the following.
Definition 4.22: A space X is called ψ-Ro iff for each ψ-open set G and x ∈ G
implies ψ-cl{x} ⊂ G.
Clearly, every semi-Ro space is ψ-Ro and every ψ-Ro space is sg-Ro.
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Definition 4.23: A space X is called ψ-R1 space iff for x, y ∈ X with ψ-cl{x} ≠
ψ-cl{y}, there exist disjoint ψ-open sets U and V such that ψ-cl{x} ⊂ U and ψ-cl{y}
⊂ V.
Clearly, every semi-R1 space is ψ-R1 and every ψ-R1 space is sg-R1.
Clearly, every ψ-R1 space is ψ-Ro space.
Next, we give the following.
Theorem 4.24: The following are equivalent.
(a) X is ψ-T2 space
(b) X is ψ-R1 and ψ-T1space
(c) X is ψ-R1 and ψ - To.
Proof is easy and hence omitted.
REFERENCES
M.E Abd El-Monsef, S.N. El-Deeb, and R.A. Mahmoud, “β-open and β-continuous
Mappings”, Bull. Fac. Sci. Assiut Univ., 12, (1983), 70-90.
[2] D. Andrijevic, “Semipreopen Sets”, Mat. Vesnik, 38(1), (1986), 24-32.
[3] S.P. Arya, and T.M. Nour, “Characterizations of S-normal Spaces”, Indian J.Pure
Appl.Math., 21(8), (1990), 717-719.
[4] K.Balachandran, P. Sundaram, and H. Maki, “On Generalized Continuous Maps in
Topological Spaces”, Mem.Fac.Sci. Kochi Univ. Ser.A (Math.), 12, (1991), 5-13.
[5] P. Bhattacharya, and B.K. Lahiri, “Semi Generalized Closed Sets in Topology”, Indian
J. Math., 29(3), (1987), 375-382.
[6] N. Biswas, “On Characterizations of Semicontinuous Functions”, Atti. Accad. Naz.
Lincei Rend.Cl.Sci.Fis.Mat.Natur. (8)48, (1970), 399-402.
[7] S.G. Crossley, and H.K. Hildebrand, “Semitopological Properties”, Fund. Math.,
74, (1972), 233-254.
[8] R. Devi, K. Balachandran, and H. Maki, “Semi-generalized Homeomorphisms and
Generalized Semi-homeomorphisms in Topological Spaces”, Indian J.Pure and
Appl.Math., 26(3), (1995), 271-284.
[9] R. Devi, H. Maki, and K. Balachandran, “Semi-generalized Closed Maps and
Generalized Semi-closed Maps”, Mem.Fac.Sci.Kochi Univ.Ser.A (Math.), 14(1993),
41-54.
[10] J. Dontchev, “On Generalizing Semipreopen Sets”, Mem.Fac.Sci.Kochi Univ.Ser.A,
Math.16(1995), 35-48.
[11] J. Dontchev and M. Ganster, On δ-generalized Closed Sets and T 3/4-spaces,
Mem.Fac.Sci.Kochi Univ.Ser.A (Math.)17(1996),15-31.
[1]
Semi-generalized Separation Axioms in Topology*
25
[12] J. Dontchev, and H. Maki, “On sg-closed Sets and Semi-λ -closed Sets, Questions
Answers Gen. Topology, 15(1997), 259-266.
[13] M. Ganster, S. Jafari, and G.B. Navalagi, “Semi-g-regular and Semi-g-normal Spaces”,
(Communicated).
[14] D.S.Jankovic and I.L. Reilly, “On Semi Separation Properties”, Indian J.Pure
Appl.Math.,16(9), (1985), 957-964.
[15] N. Levine, “Semiopen Sets and Semicontinuity in Topological Spaces”,
Amer.Math.Monthly, 70 (1963), 36-41.
[16] ______, Generalized Closed Sets in Topolog”, Rend.Circ.Mat.Palermo, 19(2), (1970),
89-96.
[17] S.N. Maheshwari, and R. Prasad, “Some New Separation Axioms”, Ann.Soc.Sci.
Bruxelles, Ser.I., 89 (1975), 395-402.
[18] G.B. Navalagi, “Properties of sg-closed Sets and gs-closed Sets in Topology”,
(communicated).
[19] P. Sundaram, H. Maki, and K. Balachandran, “Semi-generalized Continuous Maps an
Semi-T1/2 Spaces”, Bull.Fukuoka Univ.Ed.Part III, 40(1991), 33-40.
[20] M.K.R.S.Veera Kumar, “Between Semi-closed Sets and Semi-pre Closed Sets”,
Rendi.Conti.Trieste (Italy), (Accepted).
[21] _______, “Some Separation Properties Using α-open sets”, Indian J. of Mathematics,
40(2), (1998), 143-147.
[22] N.V. Velicko, “H-closed topological spaces”, Amer.Math.Soc. Transl.78, (1968),
103-118.