Download Closed and closed set in supra Topological Spaces

534
Int. J Comp Sci. Emerging Tech
Vol-2 No 4 August, 2011
Closed and
closed set in supra
Topological Spaces
I. Arockiarani * and M.Trinita Pricilla**
*Department of Mathematics,Nirmala College for Women, Coimbatore – 641 046.
**Department of Mathematics,Jansons Institute of technology, Karumathampatti,India
[email protected]
Abstract -
The aim of this paper is to define and
investigate a weaker class of supra
closed set in
supra topological spaces. New characterizations of
supra
closed set are established. We do a
comparative study of supra
closed set with the existing sets together with
closed set,
c1 µ
closed and supra
suitable examples. We further discuss the concept of
supra
and supra
continuity and obtained
their applications.
Keywords and Phrases:
Definition: 1.2 [8]
The supra closure of a set A is denoted by
C1µ(A),and defined as
The supra interior of a set A is denoted
by intµ(A),and defined as
Intµ
closed set,
closed set and
closed set
1. Introduction and Preliminaries
Introduction
Definition: 1.3 [8]
Let (
) be a topological space and µ be a
supra topology on
we call µ a supra topology
associated with
The notion of
closed sets was introduced by
T.Noiri and O.R.Syed [ 9].Sr.I.Arockiarani and Jeenu
korian [1] introduced
and
closed sets
in topological spaces.In 1983, A.S.Mashhour et al [8]
introduced the supra topological spaces and studied
S-S continuous functions and S* - continuous
functions. In 2010, O.R.Sayed and Takashi Noiri [10]
introduced supra b - open sets and supra b continuity on topological spaces. In this paper, we
use
closed and
closed set as a tool to
introduce the concept of supra
supra
closed and
closed set. We discuss the concept of
supra
continuity and supra
continuity and also obtained their applications.
1.1. Preliminaries
Definition: 1.1 [ 8]
A subfamily µ of x is said to be a supra
topology on X if i)

ii) If
for all i
J, then
.( ,µ) is called a supra topological space. The
elements of µ are called supra open sets in ( µ) and
complement of supra open set is called supra closed
sets and it is denoted by µc.
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Definition: 1.4
Let ( µ) be a supra topological space. A set
A is called supra semi - open set if A  C1µ (Int µ(A)
). The complement of supra semi - open set is supra
semi - closed set.
Definition: 1.5
Let
µ) be a supra topological space. A
set A of X is called supra generalized - closed set
(simply gµ - closed) if C1µ(A)  U whenever A 
U and U is supra open. The complement of supra
generalized closed set is supra generalized open set.
Definition: 1.6
Let
µ) be a supra topological space. A
set A of X is called supra semi - generalized closed
set (simply sgµ - closed) if SC1µ (A)  U and U is
supra semi - open. The complement of supra semi generalized closed set is supra semi - generalized
open set.
Definition: 1.7
Let
µ) be a supra topological space. A set
A of X is called supra generalized- semi closed set
(simply gsµ - closed) if SC1µ (A)  U whenever A 
U and U is supra - open. The complement of supra
generalized- semi closed set is supra generalized
semi - open set.
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Int. J Comp Sci. Emerging Tech
2. Supra
Vol-2 No 4 August, 2011
Closed and Supra
Closed Set
Definition: 2.1
A Subset A of (X, ) is said to be supra 
- closed in (X,) if Scl (A)  Int (U) whenever A 
U and U is supra - open.
Consider F  AC then F  Cl (F) X– SCl (A).
Therefore F  SCl (A)  [X– SCl (A)] = .Hence F
= .
Proposition: 2.9
A subset A is supra S – closed then SCl
(A) – A does not contain non-empty supra -open
and supra  - closed set.
Proof: Similar to theorem 2.8
Definition: 2.2
A Subset A of (X,) is said to be supra S closed in (X,) if Scl (A)  Int cl (U) whenever A
 U and U is supra - open.
Definition: 2.3
A Subset A of (X,) is said to be supra  closed in (X, ) if Scl (A)  Int (U) whenever A 
U and U is supra - open.
Definition: 2.4
A Subset A of (X,) is said to be supra S
- closed in (X,) if Scl (A)  Int cl (U) whenever
A  U and U is supra - open.
Definition: 2.5
A Subset A of (X, ) is said to be supra
regular open if A = Int(Cl (A)) and supra regular
closed if A = cl(Int (A)) . The finite union of supra
regular open set is said to be supra - open.
Theorem: 2.6
If D (E)  DS (E) for each subset E of
supra topological space (X,) then the union of two
supra  - closed sets is supra  - closed.
Proof: Let A and B be two supra  - closed subsets
in (X, ) then SCl (A)  Int (U) ; SCl (B) 
Int (U). Let U be supra  - open such that A  B 
U. Then we have SCl (A  B) = SCl (A)  SCl
(B). Thus SCl (A  B)  Int (U). Hence A  B is
supra - Closed.
Proposition: 2.7
Every supra open and supra semi-closed
subset (X,) is supra  - closed.
Proof: Let A be supra open and supra semi-closed
subset of (X,) where A  U and U is supra  open. Then SCl (A)  Int (U), Since A is Supra –
open and supra semi – closed. Therefore A is supra
 - closed.
Proposition: 2.8
A subset A is supra  - Closed then SCl
(A) – A does not contain a non empty supra  closed subset.
Proof: Let A be supra  - closed set. Suppose F  
is a supra  - closed set of SCl (A) – A and F  SCl
(A) – A. This implies F  SCl (A) and F  AC.
Proposition: 2.10
If a subset A of (X,) is supra  - open and
supra  - closed then it is supra semi-closed.
Proof: Let A be supra  - open and supra  - closed
then SCl (A)  A. Therefore A = SCl (A). Hence
A is supra semi-closed.
Theorem: 2.11
A subset A is supra regular open iff A is
supra  - open, supra -open and supra  - closed.
Proof: Suppose A is supra  - open and supra  closed then by proposition 2.10, A is supra semi –
closed, so Int Cl (A)  A. Then A  Int (Cl(A)).
This implies that A = Int Cl (A). Therefore A is
supra regular open.
Conversely, Let A be supra regular open then A
is supra  - open, supra  - open and supra  closed.
Remark: 2.12
Every supra  - closed set is supra S – closed.
Proof: Let A  U and U is supra  - open.Let A be
supra  - closed then SCl (A)  Int Cl (A).
Therefore A is supra S – closed.
Remark: 2.13
But the converse is not true by the following
example:
Let X = {a, b, c,d};
{, , {a},{b},{a,b}} .
{a,b} is supra
closed but it is not supra
closed.
Remark: 2.14
We have the following relationship between supra
closed set and supra
closed set and
other related sets.
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Int. J Comp Sci. Emerging Tech
Vol-2 No 4 August, 2011
4.
closed
closed
Every supra s - continuous function is
supra s - continuous function.
Proof: It is obvious.
closed
closed
gsµ - closed
gµ - closed
3. Supra  - Continuity and Supra s Continuity
Let f: (X, )  (Y, ) be a function from a
supra topological space (X, ) into a supra
topological space
(Y,).
Example: 3.6
Let X = {a, b, c,d};
Hence f is supra S – continuous but it is not
supra  - continuous and supra  - continuous.
Also f is supra S – continuous but it is not supra  continuous.
Remark: 3.7
We have the following relationship between supra
continuity; supra
continuity and
other related sets.
continuity
continuity
continuity
Definition: 3.2
A function f : (X, )  (Y, ) is said to be
supra  - irresolute (resp supra S – irresolute) if f-1
(V) is supra  - closed (resp supra S – closed) in
(X, ) for every supra - closed (resp supra S –
closed) set V of (Y, ).
Definition: 3.4
A function f: (X, )  (Y, ) is said to be supra
 - irresolute (resp supra S – irresolute) if f-1 (V)
is supra  - closed (resp supra S – closed) in (x,
) for every supra  - closed (resp Supra s –
closed) set V of (Y, ).
Proposition: 3.5
1. Every supra  - continuous function is
supra S – continuous function.
2. Every supra  - continuous function is supra
s - continuous function.
3. Every supra  - continuous function is supra
 - continuous function.
, {a},{b},{a,b}};
Y = {p,q,r}; σ={,Y , {p},{q},{p,q}}
Define a function f: (X,)  (Y,) such that f(a) =
p; f(b) = r ; f(c) = q = f(d).
Definition: 3.1
A function f : (X, )  (Y, ) is said to be
supra  - continuous (resp supra S - continuous) if
f-1 (V) is supra  - closed (resp supra S – closed) in
(X, ) for every supra closed set V of (Y, ).
Definition: 3.3
A function f : (X, )  (Y, ) is said to be
supra  - continuous (resp supra S –continuous)
if f-1 (V) is supra  - closed (resp supra S –
closed) in (X, ) for every supra closed set V of
(Y,).
{,
continuity
gsµ - continuity
sgµ - continuity
gµ - continuity
4. Applications
Definition: 4.1
A supra topological space (X, µ) is
1. Supra πΩ – T 1 if every supra πΩs-closed set is
2
2.
supra semi-closed in (X, µ)
Supra πΩ – Ts if every supra πΩs–closed set is
supra closed in (X, µ).
Proposition: 4.2
Let (X, µ) be a supra topological space.
1. For every x εX, { x } is supra π-closed or its
complement
X-{ x } is supra πΩ –closed in
(X, µ).
2. For every x εX, { x } is supra open and supra πclosed or its complement X – { x } is supra
πΩSclosed in (X,µ)
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Int. J Comp Sci. Emerging Tech
Proof:
1. Suppose { x } is not supra π-closed in (X, µ) then
X–{ x } is not supra π-open and the only supra πopen set containing X–{ x } is X. Since supra πclosed set is supra semi-closed then Sclµ {X–
{ x }} X= Intµ (x). Therefore X–{ x } is supra
πΩ-closed in (X,µ).
2. Suppose { x } is not supra open and let U be supra πopen set such that X–{ x } U.If U = X then Sclµ
{X – { x }}  Intµ (C1µ (U)) = U.
If U = X – { x } then Intµ ( Clµ (U)) = Int (C1µ (X
–{ x }] = Intµ { x } = X.Hence SC1µ {X –{ x }} 
Intµ (C1(U)). Therefore X–{ x } is supra πΩSclosed.
Proposition: 4.3
Let (X, µ) be a supra topological space
1. For every x εX, { x }is supra semi-closed or its
complement X–{ x } is supra πΩ –closed in
(X,µ).
2. For every x εX, { x } is supra open and supra
semiclosed or its complement X – { x } is
supra πΩSclosed in (X,µ).
Proof: It is obvious.
Theorem: 4.4
For a supra topological space (X, µ) if every
supra πΩ-closed set is supra semi-closed in(X, µ)
then for each x εX, { x } is supra semi-open or supra
semi-closed in (X µ).
Proof: Suppose that for a point x εX, { x }is not
supra semi-closed in (X, µ).By proposition 4.3, X–
{ x } is supra πΩ-closed in (X, µ).By assumption X–
{ x } is supra semi-closed in (X, µ)and hence { x } is
supra semi-open. Therefore each singleton set is
supra semi-open or supra semi-closed in (X, µ).
Vol-2 No 4 August, 2011
Theorem: 4.6
Let f: (X, µ)  (Y, σ) & g: (Y, σ)  (Z,γ)
be two function then
1.
gof is supra πΩ-continuous (resp Supra πΩS
– continuous) if g is continuous and f is
supra πΩ-continuous (resp supra πΩScontinuous).
2.
gof is supra πΩ-irresolute (resp supra πΩSirresolute) if f and g are supra πΩ-irresolute
(resp πΩS-irresolute).
3.
got is supra πΩ-continuous if g is supra πΩcontinuous (resp supra πΩS-contiuous) and f
is supra Ω-irresolute (resp supra πΩSirresolute).
4.
Let (Y, σ) be a supra πΩ-TS space, then gof
is continuous if f is continuous and g is
supra πΩS-continuous.
5.
Let f be supra πΩS-continuous then f is
continuous (resp supra semi-continuous) if
(X, µ) is supra πΩ-TS (resp supra πΩ-T1/2).
Proof: It is obvious.
Theorem: 4.7
Let f: (X, µ)  (Y, σ) be a function.
1.
Let f be an supra πΩS-irresolute and closed
surjection if (X, µ) is an supra
πΩ-TS space then (Y, σ) is also supra πΩ-TS
2.
Let f be an supra πΩS-irresolute and semiclosed surjection. If (X, µ) is an supra πΩ-TS
space then (Y, σ) is also supra πΩ-T1/2
3.
Let f be an supra πΩS-irresolute and pre
semi-closed surjection. If (X, µ) is an supra
πΩ-T1/2 space then (Y, σ) is also supra πΩT1/2
Proof: It is obvious.
References:
[1] I.Arockiarani and Jeenu kurian, on
Theorem: 4.5
For a topological space (X, µ) the following
properties hold:
1. If (X, µ)is supra πΩ-TS then for each x εX, the
singleton { x } is supra open or supra semiclosed.
2. If (X, µ) is supra πΩ-TS then it is supra πΩ-T1/2
Proof:
1. Suppose that x εX, { x }is not supra semiclosed. By proposition4.3, then X–{ x } is
supra πΩ-closed in (X, µ). Hence X–{ x } is
supra πΩs-closed in (X, µ).Then X–{ x } is
supra closed in (X, µ). Thus { x } is supra
open in (X, µ).
2. Let X be supra πΩ-TS space then every
supra πΩS-closed set is supra closed. Thus
every supra πΩS- closed set is supra semiclosed. Since every supra closed set is supra
semi-closed. Therefore X is supra πΩ-T1/2
space.
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Vol-2 No 4 August, 2011