In this paper, we have introduced a new class of sets in generalized topological spaces called µ ß generalized a closed sets. Also we have investigated some of their basic properties. Kowsalya M | Jayanthi D "µ-β-generalized α-closed sets in generalized topological spaces" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: https://www.ijtsrd.com/papers/ijtsrd11665.pdf Paper URL: http://www.ijtsrd.com/mathemetics/applied-mathamatics/11665/-µ-β-generalized-α-closed-sets-in-generalized-topological-spaces/kowsalya-m
@ IJTSRD | Available Online @ www ISSN No: 245 Inte R µ-β-generalized α-clo Kowsalya M MSc Mathematics, Avinashilingam U Coimbatore, Tamil Nadu, Ind ABSTRACT In this paper, we have introduced a ne in generalized topological spaces generalized α-closed sets. Also we hav some of their basic properties. Keywords: Generalized topology, topological spaces, µ-α-closed sets, µ α-closed sets. 1. Introduction The concept of generalized topolog introduced by A. Csaszar [1] in 2 introduced many µ-closed sets like sets, µ-pre closed sets, µ-β-closed se sets, µ-regular closed sets etc., i topological spaces. In 2016 Srija. S an [4] introduced g u -semi closed set i topological spaces. In this paper, we ha a new class of sets in generalized topo called µ-β-generalized α-closed sets. investigated some of their basic p produced many interesting theorems. 2. Preliminaries Definition 2.1: [1] Let X be a non collection µ of subsets of X is a genera (or briefly GT) on X if it satisfies the fo i. Ø, X ∊ µ and ii. If {M i : i ∊ I} ⊆ µ, then ∪ i ∊ I M i w.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr 56 - 6470 | www.ijtsrd.com | Volum ernational Journal of Trend in Sc Research and Development (IJT International Open Access Journ osed sets in generalized topolog University, dia Jayanthi Assistant Professor of Mathem University, Coimbatore, ew class of sets called µ-β- ve investigated , generalized µ-β-generalized gical spaces is 2002. He also µ-semi closed ets, µ-α-closed in generalized nd Jayanthi. D in generalized ave introduced ological spaces Also we have properties and nempty set. A alized topology ol lowing: i ∊ µ If µ is a GT on X, then (X, µ) topological space (or briefly G of µ are called µ-open sets and called µ-closed sets. Definition 2.2: [1] Let (X, µ) X. Then µ-closure of A, den intersection of all µ-closed sets Definition 2.3: [1] Let (X, µ) X. Then µ-interior of A, den union of all µ-open sets contai Definition 2.4: [1] Let (X, µ) of X is said to be i. µ-semi-closed if i µ (c µ (A)) ⊆ A ii. µ-pre-closed if c µ (i µ (A)) ⊆ A iii. µ-α-closed if c µ (i µ (c µ (A))) ⊆ iv. µ-β-closed if i µ (c µ (i µ (A))) ⊆ v. µ-regular closed if A= c µ (i µ (A Definition 2.5: [3] Let (X, µ) of X is said to be i. µ-regular generalized closed i A ⊆ U, where U is µ-regular o ii. µ-generalized closed set if c µ (A U, and U is µ-open in X, iii. µ-generalized α-closed set if α ⊆ U, and U is µ-open in X. r 2018 Page: 2318 me - 2 | Issue – 3 cientific TSRD) nal gical spaces i D matics, Avinashi lingam Tamil Nadu, India ) is called a generalized GTS), and the elements d their complement are be a GTS and let A ⊆ noted by c µ (A), is the s containing A. be a GTS and let A ⊆ noted by i µ (A), is the ined in A. be a GTS. A subset A A A ⊆A A A)) be a GTS. A subset A if c µ (A) ⊆ U whenever open in X, A) ⊆ U whenever A ⊆ αc µ (A) ⊆ U whenever A
Transcript
@ IJTSRD | Available Online @ www.ijtsrd.com
ISSN No: 2456
InternationalResearch
µ-β-generalized α-closed sets in generalized topological spaces
Kowsalya M MSc Mathematics, Avinashilingam University,
Coimbatore, Tamil Nadu, India
ABSTRACT In this paper, we have introduced a new class of sets in generalized topological spaces called µgeneralized α-closed sets. Also we have investigated some of their basic properties. Keywords: Generalized topology, generalized topological spaces, µ-α-closed sets, µα-closed sets. 1. Introduction
The concept of generalized topological spaces is introduced by A. Csaszar [1] in 2002. He also introduced many µ-closed sets like µsets, µ-pre closed sets, µ-β-closed sets, µsets, µ-regular closed sets etc., in generalized topological spaces. In 2016 Srija. S and Jayanthi. D [4] introduced gu-semi closed set in generalized topological spaces. In this paper, we have introduced a new class of sets in generalized topological spaces called µ-β-generalized α-closed sets. Alsinvestigated some of their basic properties and produced many interesting theorems. 2. Preliminaries Definition 2.1: [1] Let X be a nonempty set. A collection µ of subsets of X is a generalized topology (or briefly GT) on X if it satisfies the fol
i. Ø, X ∊ µ and ii. If {Mi: i ∊ I} ⊆ µ, then ∪i ∊ I Mi
The concept of generalized topological spaces is introduced by A. Csaszar [1] in 2002. He also
closed sets like µ-semi closed closed sets, µ-α-closed
regular closed sets etc., in generalized l spaces. In 2016 Srija. S and Jayanthi. D
semi closed set in generalized topological spaces. In this paper, we have introduced a new class of sets in generalized topological spaces
closed sets. Also we have investigated some of their basic properties and
[1] Let X be a nonempty set. A collection µ of subsets of X is a generalized topology (or briefly GT) on X if it satisfies the following:
i ∊ µ
If µ is a GT on X, then (X, µ) is called a generalized topological space (or briefly GTS), and the elements of µ are called µ-open sets and their complement are called µ-closed sets. Definition 2.2: [1] Let (X, µ) be a GTS and let A X. Then µ-closure of A, denoted by cintersection of all µ-closed sets containing A. Definition 2.3: [1] Let (X, µ) be a GTS and let A X. Then µ-interior of A, denoted by iunion of all µ-open sets contained in A. Definition 2.4: [1] Let (X, µ) be a GTS. A subset A of X is said to be
i. µ-semi-closed if iµ(cµ(A)) ⊆ Aii. µ-pre-closed if cµ(iµ(A)) ⊆ A
iii. µ-α-closed if cµ(iµ(cµ (A))) ⊆iv. µ-β-closed if iµ(cµ(iµ (A))) ⊆ v. µ-regular closed if A= cµ(iµ (A))
Definition 2.5: [3] Let (X, µ) be a GTS. A subset A of X is said to be
i. µ-regular generalized closed if cA ⊆ U, where U is µ-regular open in X,
ii. µ-generalized closed set if cµ(A) U, and U is µ-open in X,
iii. µ-generalized α-closed set if αc⊆ U, and U is µ-open in X.
Apr 2018 Page: 2318
6470 | www.ijtsrd.com | Volume - 2 | Issue – 3
Scientific (IJTSRD)
International Open Access Journal
closed sets in generalized topological spaces
Jayanthi D Assistant Professor of Mathematics, Avinashilingam
University, Coimbatore, Tamil Nadu, India
If µ is a GT on X, then (X, µ) is called a generalized topological space (or briefly GTS), and the elements
open sets and their complement are
[1] Let (X, µ) be a GTS and let A ⊆ closure of A, denoted by cµ(A), is the
closed sets containing A.
[1] Let (X, µ) be a GTS and let A ⊆ interior of A, denoted by iµ(A), is the
open sets contained in A.
[1] Let (X, µ) be a GTS. A subset A
A A ⊆A
A (A))
[3] Let (X, µ) be a GTS. A subset A
regular generalized closed if cµ(A) ⊆ U whenever regular open in X,
(A) ⊆ U whenever A ⊆
closed set if αcµ(A) ⊆ U whenever A
International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
The complement of µ-semi-closed (respectively µ-pre-closed, µ-α-closed, µ-β-closed, µ-regular closed, etc.,) is called µ-semi-open (respectively µ-pre-open, µ-α-open, µ-β-open, µ-regular open, etc.,) in X. 3. µ-β-generalized α-closed sets in generalized
topological spaces In this section we have introduced µ-β-generalized α-closed sets in generalized topological spaces and studied some of their basic properties. Definition 3.1: A subset A of a GTS (X, µ) is called a µ-β-generalized α-closed set (briefly µ-βGαCS) if αcµ(A) ⊆ U whenever A ⊆ U and U is µ-β-open in X.
Example 3.2: Let X= {a, b, c} and let µ= {Ø, {a}, {b}, {a, b}, X}. Then (X, µ) is a GTS. Now µ-βO(X) = {Ø, {a}, {b}, {a, b}, {b, c}, {a, c}, X}. Then A = {c} is a µ-βGαCS in (X, µ).
Theorem 3.3: Every µ-closed set in (X, µ) is a µ-βGαCS in (X, µ) but not conversely.
Proof: Let A be a µ-closed set in (X, µ) then cµ(A) = A. Now let A ⊆ U where U is µ-β-open in (X, µ). Then αcµ(A) ⊆ cµ(A) = A ⊆ U, by hypothesis. Therefore αcµ(A) ⊆U. This implies, A is a µ-βGαCS in (X, µ).
Example 3.4: Let X= {a, b, c, d} and let µ = {Ø, {a}, {c}, {a, c}, X}. Then (X, µ) is a GTS. Now µ-βO(X) = {Ø, {a}, {c}, {a, b}, {a, c}, {a, d}, {b, c}, {c, d}, {a, b, c}, {b, c, d}, {a, c, d}, {a, b, d}, X}. Then A = {d} is a µ-βGαCS in (X, µ), but not a µ-closed set as cµ (A) = cµ ({d}) = {b, d} ≠ A.
Theorem 3.5: Every µ-α-closed set in (X, µ) is a µ- βGαCS in (X, µ).
Proof: Let A be a µ-α-closed in (X, µ), then cµ(iµ(cµ(A))) ⊆ A. Now let A ⊆ U where U isµ-β-open in (X, µ). Then αcµ (A) = A ∪ cµ(iµ(cµ(A))) = A ∪ A = A ⊆ U, by hypothesis. Therefore αcµ(A) ⊆ U and hence A is a µ-βGαCS in (X, µ).
Theorem 3.6: Every µ-regular closed set in (X, µ) is a µ-βGαCS in (X, µ) but not conversely.
Proof: Let A be a µ-regular closed set in (X, µ), then cµ(iµ(A)) = A. Now let A⊆ U where U is a µ-β-open
set in (X, µ). Then αcµ(A) = A ∪ cµ(iµ (cµ(A))) = A ∪ cµ(iµ(cµ(cµ(iµ(A))))) = A ∪ cµ(iµ(cµ (iµ(A)))) = A ∪ cµ(iµ(A)) = A ⊆ U, by hypothesis. Therefore αcµ(A) ⊆ U. This implies, A is a µ-βGαCS in (X, µ).
Example 3.7: Let X= {a, b, c} and µ = {Ø, {a}, {b}, {a, b}, X}. Then (X, µ) is a GTS. Now µ-βO(X) = {Ø, {a}, {b}, {a, b}, {b, c}, {a, c}, X}. Then A = {c} is a µ-βGαCS in (X, µ), but not a µ-regular closed set as cµ(iµ(A)) = cµ(iµ({c}) = Ø ≠ A.
Remark 3.8: A µ-pre-closed is not a µ- β-GαCS in (X, µ) in general.
Example 3.9: Let X = {a, b, c} and µ= {Ø, {a, b}, X}. Then (Χ, µ) is a GTS. Now µ-βO(X) = {Ø, {a}, {b}, {a, b}, {b, c}, {a, c}, X}. Then A= {a} is a µ-pre-closed in (X, µ) as cµ(iµ(A)) = cµ(iµ({a})) = Ø ⊆ A, but not a µ-βGαCS in (X, µ) as αcµ (A) = X
{a, b} = U.
Remark 3.10: A µ-β-closed is not a µ-βGαCS in (X, µ) in general.
Example 3.11: Let X = {a, b, c} and µ = {Ø, {a, b}, X}. Then (Χ, µ) is a GTS. Now µ-βO(X) = {Ø, {a}, {b}, {a, b}, {b, c}, {a, c}, X}. Then A = {a} is a µ-β-closed in (X, µ) as iµ(cµ(iµ(A)) = iµ(cµ(iµ({a})) = Ø ⊆ A, but not a µ-βGαCS as in Example 3.9.
In the following diagram, we have provided relations between various types of closed sets.
µ-closed set
µ-α-closed set µ-β-closed set
µ-βGαCS
µ-regular closed set µ-pre-closed set
Theorem 3.12: Let A and B be µ-βGαCSs in (X, µ) such that cµ(A) = αcµ(A) and cµ(B) = αcµ (B), then A ∪ B is a µ-βGαCS in (X, µ).
Proof: Let A∪ B ⊆ U, where U is a µ-β-open set in X. Then A ⊆ U and B ⊆ U. since A and B are µ-
International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
βGαCS, αcµ(A) ⊆ U and αcµ(B) ⊆ U. Now, αcµ(A∪B) ⊆ cµ(A∪B) = cµ(A) ∪ cµ(B) = αcµ(A) ∪ αcµ(B) ⊆ U ∪ U = U. Hence, A∪B is a µ-βGαCS in (X, µ).
Theorem 3.13: If a subset A of X is a µ-βGαCS and A ⊆ B ⊆ αcµ (A), then B is a µ-βGαCS in X.
Proof: Let A be a µ-βGαCS such that A ⊆ B ⊆ αcµ(A). Suppose U is a µ-β-open set of X such that B ⊆ U, by assumption, we have αcµ(A) ⊆ U as A ⊆ B ⊆ U. Now αcµ(B) ⊆ αcµ(αcµ(A)) = αcµ(A) ⊆ U. That is αcµ(B) ⊆ U. Therefore, B is a µ-βGαCS in X. Theorem 3.14: In a GTS X, for each x ∊ X, {x} is µ-β-closed or its complement X – {x} is a µ-βGαCS in (X, µ).
Proof: Suppose that {x} is not a µ-β-closed set in (X, µ). Then X – {x} is not a µ-β-open set in (X, µ). The only µ-β-open set containing X – {x} is X. Thus X – {x} ⊆ X and so αcµ (X – {x}) ⊆ αcµ (X) = X. Therefore αcµ (X – {x}) ⊆ X and so X– {x} is a µ-βGαCS in (X, µ).
Theorem 3.15: If A is both a µ-β-open set and µ-βGαCS in (X, µ), then A is a µ-α-closed set.
Proof: Let A be a µ-β-open set and µ-βGαCS in (X, µ). Then, αcµ(A) ⊆ A as A ⊆ A. we have A ⊆ αcµ (A). Therefore, A = αcµ(A). Hence A is a µ-α-closed set in (X, µ).
Theorem 3.16: Every subset of X is a µ-βGαCS in X iff every µ-β-open set in X is µ-α-closed in X.
Proof: Necessity: Let A be a µ-β-open set in X and by hypothesis, A is a µ-βGαCS in X. Hence by Theorem 3.15, A is a µ-α-closed set in X.
Sufficiency: Let A be a subset of X and U be a µ-β-open set such that A ⊆ U, then by hypothesis, U is µ-α-closed. This implies that αcµ(U) = U and αcµ(A) ⊆ αcµ (U) = U. Hence αcµ(A) ⊆ U. Thus A is a µ-βGαCS set in (X, µ).
Theorem 3.17: A subset A of X is a µ-βGαCS if and only if αcµ (A) – A contains no non-empty µ-β-closed set in X.
Proof: Let A be a µ-βGαCS in X. Suppose F is a non-empty µ-β-closed set such that F ⊆ αcµ (A) – A.
Then F ⊆ αcµ(A) ∩ Ac. Therefore F ⊆ αcµ(A) and F ⊆ Ac. This implies A ⊆ Fc. Since Fc is a µ-β-open set, by definition of µ-βGαCS, αcµ(A) ⊆ Fc. That is F ⊆ (αcµ(A))c. Hence F ⊆ αcµ(A) ∩ (αcµ(A))c = Ø. That is F= Ø, which is a contradiction. Thus αcµ(A) – A contains no non-empty µ-β-closed set in X.
Conversely, assume that αcµ (A) – A contains no non-empty µ-β-closed set in X. Let A ⊆ U, where U is a µ-β-open set in X. Suppose that αcµ(A) ⊈ U, then αcµ(A) ∩ Uc is a non-empty closed subset of αcµ(A) – A, which is a contradiction. Therefore αcµ(A) ⊆ U and hence A is a µ-βGαCS in X.
Theorem 3.18: If A ⊆ Y ⊆ X and A is a µ-βGαCS in X, then A is a µ-βGαCS relative to Y.
Proof: Given that A ⊆ Y ⊆ X and A is a µ-βGαCS in X. Let A ⊆ Y ∩ U, where U is µ-β-open set in X. since A is a µ-βGαCS, A ⊆ U implies αcµ(A) ⊆ U. This implies Y ∩ αcµ (A) ⊆ Y ∩ U and αcµ(A) ⊆ Y ∩ U. That is A is a µ-βGαCS relative to Y.
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4. Srija, s, and Jayanthi, D., gu- semi closed set in generalized topological spaces, International Journal of Scientific Engineering and Application Science (IJSEAS)-volume-2, issue-4, April 2016, 292-294.
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