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International Journal of Advance Foundation and Research in Computer (IJAFRC)
Volume 2, Issue 1, January 2015. ISSN 2348 – 4853
g#- Closed Sets in Topological Spaces
N. Sowmya*, M. Elakkiya, Asst.Prof N. Balamani
Department of Mathematics, Avinashilingam Institute for Home Science and Higher Education for
Women, Coimbatore-43, TamilNadu, India.
e-mail: [email protected]* , [email protected]
ABSTRACT
The aim of this paper is to introduce and study the concept of a new class of sets called
closed sets in topological spaces. Also we study some basic properties of g#- closed sets.
g#-
Index Terms : sg-open sets, g- open sets, g# -open sets, g#- closed sets.
I.
INTRODUCTION
Levine [4] introduced generalized closed sets [briefly g-closed] and studied their basic properties. Veera
Kumar [8,9] introduced - closed sets in topological spaces and g#-closed sets in topological spaces and
studied their properties. The aim of this paper is to define a new class of closed sets called g#- closed
sets and obtain some basic properties.
II. PRELIMINARIES
Definition 2.1: A subset A of a topological space (X,τ) is called
(i) Semi-closed set [3] if int(cl(A))⊆A.
(ii) -closed set [7] if cl(int(cl(A)))⊆A.
(iii) Generalized closed set [4] (briefly g-closed) if cl(A)⊆U, whenever A⊆U and U is open in X.
(iv) Semi-generalized closed set [2](briefly sg-closed) if scl(A)⊆U whenever A⊆U and U is semi-open
set in X.
(v) -generalized closed set [6](briefly g-closed) if cl(A)⊆U whenever A⊆U and U is open set in X.
(vi) Generalized -closed set [5](briefly g -closed) if cl(A)⊆U whenever A⊆U and U is -open set in X.
(vii) Generalized semi-closed [1](briefly gs-closed) if scl(A)⊆U whenever A⊆U and U is open set in X.
(viii) g*s-closed set [10] if scl(A)⊆U whenever A⊆U and U is gs-open set in X.
The complement of the above mentioned closed sets are their respective open sets.
Definition 2.2: A subset A of a topological space (X,τ) is called
(i)
-closed set [8] if scl(A)⊆U whenever A⊆U and U is sg-open set in X.
#
(ii)
g -closed set [9] if cl(A)⊆U whenever A⊆U and U is g-open set in X.
The complement of the above mentioned closed sets are their respective open sets.
III.
g#- CLOSED SETS IN TOPOLOGICAL SPACES
Definition 3.1: A subset A of a topological space (X,τ) is called a g#-closed set if
cl(A)⊆U whenever
A⊆U and U is g#-open set in X.
67 | © 2014, IJAFRC All Rights Reserved
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International Journal of Advance Foundation and Research in Computer (IJAFRC)
Volume 2, Issue 1, January 2015. ISSN 2348 – 4853
Theorem 3.2: If a subset A of a topological space (X,τ) is closed then it is g#-closed set in X.
Proof: Let A be a closed set of X. Let U be a g#-open set containing A in X. Since A is closed, A⊆ cl(A)⊆ U.
Since cl(A) ⊆ cl(A),we have cl(A) ⊆ U. Thus A is
g#- closed set in X.
The converse of the above theorem need not be true as seen from the following example.
Example 3.3: Let X={a,b,c} with topology τ={φ,{a,b},X}.The sets {b,c} and {c,a} are
not closed set in X.
Theorem 3.4: If a subset A of a topological space (X,τ) is g#- closed set then it is
g#- closed sets but
g#- closed set in X.
Proof: Let A be a g#- closed set of X. Let U be an g-open set containing A in X. Since every g#- open set is
g-open and A is g#- closed, cl(A) ⊆U. Since cl(A) ⊆ cl(A),we have cl(A) ⊆U. Thus A is g#- closed set
in X.
The converse of the above theorem need not be true as seen from the following example.
Example 3.5: Let X={a,b,c} with topology τ={φ,{c},{a,b},X}.The sets {a},{b},{b,c},{c,a} are
sets, but not g#- closed set in X.
g#- closed
Theorem 3.6: If a subset A of a topological space (X,τ) is -closed set then it is g#- closed set in X.
Proof: Let A be a -closed set of X. Let U be a sg-open set containing A in X. Since every g#-open set is
sg-open and since A is -closed, we have scl(A)⊆U. We know that cl(A)⊆scl(A), we get cl(A)⊆U. Thus A
is g#- closed set in X.
The converse of the above theorem need not be true as seen from the following example.
Example 3.7: Let X={a,b,c} with topology τ={φ,{c},{a,b},X}.The sets {a},{b},{b,c},{c,a} are
sets, but not -closed set in X.
g#-closed
Theorem 3.8: If a subset A of a topological space (X,τ) is g-closed set then it is g#- closed set in X.
Proof: Let A be an g-closed set of X . Let U be an open set containing A in X. Since every open set is g#open and since A is g-closed,we have cl(A)⊆U. We know that cl(A)⊆ cl(A).Hence cl(A)⊆U. Thus A
is g#- closed set in X.
The converse of the above theorem need not be true as seen from the following example.
Example 3.9: Let X={a,b,c} with topology τ={ φ,{a},{b},{a,b},X}. The sets {a} and {b} are g#- closed sets
but not g-closed set in X.
Theorem 3.10: If a subset A of a topological space (X,τ) is sg-closed set then it is g#- closed set in X.
Proof: Let A be a sg-closed set of X . Let U be a semi-open set containing A in X. Since every g#-open set is
semi-open set and since A is sg-closed, we have scl(A)⊆U. We know that cl(A)⊆scl(A).Hence
68 | © 2014, IJAFRC All Rights Reserved
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International Journal of Advance Foundation and Research in Computer (IJAFRC)
Volume 2, Issue 1, January 2015. ISSN 2348 – 4853
we get cl(A)⊆U.Thus A is g#- closed set in X.
The converse of the above theorem need not be true as seen from the following example.
Example 3.11: Let X={a,b,c} with topology τ={ φ,{a},{a,b},X}. The set {c,a} is
closed set in X.
g#-closed set but not sg
Theorem 3.12: If a subset A of a topological space (X,τ) is g*s-closed set then it is g#- closed set in X.
Proof: Let A be a g*s-closed set of X. Let U be a gs-open set containing A in X. Since every g#- open set is
gs-open and since A is g*s-closed, we have have scl(A)⊆U.We know that cl(A)⊆scl(A),we get
cl(A)⊆U.Thus A is g#- closed set in X.
The converse of the above theorem need not be true as seen from the following example.
Example 3.13: Let X={a,b,c} with topology τ={ φ,{c},{a,b},X}. The sets {a} and {b} are
but not g*s-closed set in X.
g#- closed sets
Theorem 3.14: A set A is g#- closed if and only if cl(A)-A contains no non-empty g#- closed set in X.
Proof: Let A be a g#- closed set of X. Suppose, let cl(A)-A contains a g#- closed set (say) F. That is
F⊆ cl(A)-A⟹F⊆ cl(A)⋂AC. Hence F⊆ cl(A) and F⊆AC⟹A⊆FC. Hence FC is g#- open set and A is g#closed set, we have cl(A)⊆FC⟹F⊆ ( cl(A))C. Hence we get F⊆ ( cl(A))⋂ ( cl(A))C= φ.
Therefore F=φ.
Conversely, let cl(A)-A contains no non-empty g#-closed set of X. Let A⊆U where U is g#- open set in X.
Suppose let cl(A)⊈U, UC ⋂ cl(A)≠ φ, then UC ⋂ cl(A) is a non-empty closed set of cl(A)-A. This
implies that cl(A)-A contains a non-empty closed which is a contradiction. Hence cl(A)⊆U. Thus A is
g#- closed set in X.
Theorem 3.15: Suppose that B⊆A⊆X, B is g#-closed set relative to A and that A is both g#-open and
g#-closed subset of X, then B is g#-closed set relative to X.
Proof: Let B⊆U and U be an open set in X. It is given that B⊆A⊆X. Therefore B⊆U and B⊆A⟹B⊆A⋂U.
Since B is g#-closed set relative to A, cl(B)⊆U. That is A⋂ cl(B)⊆A⋂U ⟹ A⋂ cl(B)⊆U. Thus
(A⋂ cl(B))⋃( cl(B))C⊆U⋃( cl(B))C⟹A⋃( cl(B))C⊆U⋃( cl(B))C.Since A is
g#-closed set in X,
cl(A)⊆ U⋃ ( cl(B))C. Also B⊆A⟹ cl(B)⊆ cl(A). Thus cl(B)⊆ U⋃ ( cl(B))C. Hence cl(B)⊆U
and hence B is g#-closed set relative to X.
Theorem 3.16: Let A⊆Y⊆X and suppose that A is g#-closed set in X, then A is g#-closed set relative to Y.
Proof: Let A be g#-closed set in X and let A⊆Y⋂U where U is g#-open set in X. Since A is g#-closed,
cl(A)⊆U. That is Y⋂ cl(A)⊆Y⋂U, where Y⋂ cl(A) is the -closure of A in Y. Hence
cly(A)⊆Y⋂U.
#
Thus A is g -closed set relative to Y.
Remark 3.17: The union of two g#- closed sets need not be a g#-closed set in X.
69 | © 2014, IJAFRC All Rights Reserved
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International Journal of Advance Foundation and Research in Computer (IJAFRC)
Volume 2, Issue 1, January 2015. ISSN 2348 – 4853
Example3.18: Let X={a,b,c} with topology τ={ φ,{a},{b},{a,b},X}. The sets {a}, {b}, {c}, {b,c},{c,a} are
closed sets. The union of the sets {a} and {b} is not a g#-closed set in X.
IV.
g#-
g#-OPEN SETS IN TOPOLOGICAL SPACES.
Definition 4.1: A subset A of a topological space (X,τ) is said to be a g#-Open set if its complement is
g#-closed set in X.
Remark 4.2: The intersection of two g#- open sets need not be a g# -open set in X.
Example 4.3: Let X={a,b,c} with topology τ={ φ,{a},{b},{a,b},X}.The sets {a}, {b}, {a,b}, {b,c},{c,a} are
g#-open sets. The intersection of the sets {b,c} and {c,a} is not a g#-open set in X.
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