Download On Submaximality in Intuitionistic

On Submaximality in Intuitionistic
Topological Space
Abstact: In this study, the minimal submaximal element of LIT(X) (the lattice of all
intuitionistic topologies for X, ordered by inclusion) are determined. After wards, a new
contractive property “intuitionistic mega-connectedness” is defined. By identifying those
members of LIT(X) which are intuitionistic mega-connected we conclude that submaximality
and mega-connectedness are not complementary intuitionistic topological invariants.
Keywords: Intuitionistic set; intuitionistic topology; intuitionistic submaximality and
mega-connectedness.
2000 MSC Classification: 03F55 - 54A40
1. Introduction: Submaximality was first defined and characterized by Bourbaki
(1961) (however see also Hewitt, 1943). Recently, submaximality has been studied by many
resarchers prominantly (Dontchev, 1995, Arhangel’skit& Colins, 1995; Douwen, 1993).
The family of all intuitionistic topologies definable for an infinite set X forms a
complete atomic and complemented lattice (ordered by inclusion) and is denoted LIT(X)
(Özçelik, Narlı, 1999). A topological invariant property P is called contractive (expansive)
when, for a given member  of LIT(X) with property P , any weaker (stronger) member of
LIT(X) has property P ; then, given a contractive (expansive) property P,a member  of
LIT(X) is said to be maximal P (minimal P) if and only if  has propert P but stronger
(weaker) member of LIT(X) has propert P.
An expansive property P and contractive property Q are called complementary when
the minimal P members of LIT(X) coincide with the maximal Q members.
Some examples of complementary topological invariants are; T1 and “all proper closed
sets are finite” ;Door and “filter-connected”;TD and nested; Disconnected and principal of
order two (Cameron, 1997; Larson, 1973; Kennedy&Cartan, 1996)
The main purpose of this article is to identify those members of LIT(X) which are
minimal submaximal by using the definition of intuitionistic submaximality [Özçelik, Narlı,
1999].
2.Preliminaries:Here we shall present the fundamental definitions. The following one
is obviously insprired by K.T Atanassow (1986) and is first given in (D.Çoker, 1995).
2.1.Defination: Let X be a nonempty set. An intuitionistic set (IS for short) A is an
object having the form
A=<X,A1,A2> (or A=<A1,A2>)
where A1 and A2 are disjoint subsets of X. The set A1 is called the set of members of A, while
A2 is called the set of nonmembers of A (Çoker, 1996).
2.2.Definition: Let X be a nonempty set and A and B be in the form A=<A1,A2>,
B=<B1,B2> respectively (Çoker, 1996). Furtermore, let {Ai: iI}be an arbitrary family of IS’s
in X, where Ai=<Ai(1),Ai(2)> then;
a) =<,X> , X=<X,>
b) AB iff A1B1 and A2B2
c) AB iff A1B1 and A2B2
d) Ac=<A2,A1> here Ac complementary of A
e) Ai =< Ai(1), Ai(2)> and Ai=< Ai(1), Ai(2)>
2.3.Definition: An intuitionistic topology (IT for short) on a nonempty set X is a
family  of IS’s in X containing ,X which is closed under orbitrary unions and finite
intersections. In this case the pair (X, ) is called an intuitionistic topological space (ITS for
short) and any IS in  is known as an intuitionistic open set (IOS for short) in X , the
complement of such an IOS in X is called an intuitionistic closed set (ICS for short) in X .
2.4.Example: Any topological space (X, ,) is obviusly an intuitionistic topological
space with the form;
={A: A} whenever we identify a subset A in X with its counter pair A=<A,Ac>.
2.5.Example: Let X={a,b,c,d} and consider the family = {,X,T1,T2, T3, T4, T5, T6}
where
T1=<{a},>,
T2=<{b},>,
T6=<{a,c},>.Then  is IT on X.
T3=<{c},>,
T4=<{a,b},>,
T5=<{b,c},>,
2.6.Defination: Let 1 and 2 be two IT on X. Then 1 is said to be contained in 2 if
G2 for each G1. In this case, we also say that 1 is coarser than 2, or 2 is finer than 1.
2.7.Defination: Let (X, ) be an ITS and A=<A1,A2> be an IS in X. Then the interior
and closure of A are defined by;
int(A)= {Gi: Gi is an IOS in X and GiA}
cl(A)= {Ki : Ki is an ICS in X and AKi}
2.8.Defination: Let (X, ) be an ITS and A=<A1,A2> be an IS in X. The set A is called
-dense in X iff cl(A)=X .
2.9.Example: Consider the ITS (X, ) where X={a,b,c,d,e} and
= {,X, <{a,b,c},{e}>,<{c,d},{e}>,<{c},{e}>,<{c},{d,e}>,<{a,b,c,d},{e}>}
if we let B=<{b,c},{d}>, then
int(B)=<{c},{d,e}> and
cl(B)=X . So B is a dense IS in X.
3.Submaximality
A topological space (X, ) is submaximal if every dense subset of X is  -open. In this
section the definition of submaximality for an ITS is given. Also some characterizations of
submaximality on IT are investigated.
3.1.Defination: Let (X, ) be an ITS. Then
LIT(X) is called submaximal
intuitionistic topology (IT-sub for short) if every -dense subset of X is an IOS in X.
3.2.Example: Let X={a,b}and the family
= {,X, <{a},>,<{b},>,<{a},{b}>,<,>,<,{b}>} is an IT-sub on X.
3.3.Remark: The following example shows that it is not neccesarily true to say that
the intuitionistic form of a submaximal space is submaximal.
3.4.Example: Let X={1,2} and = {,X,{1}}.Then the topological space (X, ) is
submaximal, the intuitionistic form of  is = {,X, <{1},{2}>} and A=<{1},> is a -dense
IS,but A is not an IOS. From this  is not an IT-sub on X.
3.5.Theorem: Submaximality is an expansive intuitionistic topological invariant.
Proof: Let 1,2 in LIT(X) such that 12 and ,1 is IT-sub. Take A as a 2-dense IS.
Then A is a 1-dense IS. Since 1 is an IT-sub so A is a 1-IOS. Then A is a 2-IOS so 2 is a
IT-sub.
3.6.Theorem: Let X be a non empty set and A be an IS. Then
M(A)={G: AG or GA} is a IT-sub member of LIT(X).
Proof: Let us take A=<A1 ,A2> . Then there are the following cases.
i) If B=<B1,B2> is any IS such that
B1A1 then either BM(A) or since <,A2> is an IOS cl(B)  < A2,  > X
ii) If B=<B1,B2> is any IS such that A1B1 and B2(B1)c then either BM(A) or
cl(B)  <A2B1, >X
iii) If B=<B1,B2> is any IS such that B1 A1 and B2 A2 then either BM(A) or
cl(B)  <A2B1, >X
iv) If B=<B1,B2> is any IS such that
A2 B1 then either BM(A) or
cl(B)  < B1, >X
v) If B=<B1,B2> is any IS such that
A1B1 and A2 B2 then
a) cl(B) <A2B1, >X
b) If A2=B2 then BM(A)
Therefore M(A) is submaximal IT on X
S.D. Mc CARTAN (1996) has shown that M(A) ={G:AG or GA,  AX } is a
minimal submaximal member of LT(X). The following example shows that M(A) is not a
minimal IT –sub member of LIT(X).
3.7.Example: Let X={1,2,3} and A=<{1},{3}> be an IS. Consider the IT-sub
members of LIT(X)
M(A)={<{1},{3}>,<{1},{2,3}>,<,{3}>,<,{2,3}>,<,{1,3}>,<{1},>,<{1,2},{3}>,
<{1,2}, >, <{1,3},>,<{1,2,3},>,<,{1,2,3}>} and
={<,{3}>,<,{2,3}>,<,{1,3}>,<{1,2},{3}>,<{1,2}, >,<{1,2,3},>,<,{1,2,3}>}
Since  is coarser than M(A), then M(A) is not minimal submaximal IT member of LIT(X).
3.8.Theorem: Let X be a nonempty set and A is any subset of X. The family
(A)={< ,B>: ABX }  {<C,D>: (A)cC, D(C)c } is a minimal IT-sub member of
LIT(X).
Proof : First we show that (A) is IT-sub on X . From this there are following cases.
Let E=<E1,E2> be an IS .
i) If E1A then < ,A>(A) so cl(E)  < ,A> X
ii) If E2A then either E(A)
or < ,AE1>(A) this implies that
cl(E)  <AE1,> X
iii) If A E1 then either E1=X in which case E= X(A) or cl(E)  < E1,> X
iv) If AE 1 and E 1A then either E(A) or cl(E)  <AE1,> X
This shows that (A) is a IT-sub.
Let  be an IT-sub and   (A). Any
E IS in the family {<C,D>: AcC,DCc} is
(A)-dense so E is -dense . Since  is IT-sub then E is an -IOS. On the other hand, all the
IS in the form <,X\{x}> where xAc must be -IOS, because if <,X\{x}> is not an -IOS
then the IS <X\{x},> is -dense, this contradicts whit the submaximality of . This implies
that all the IS in the family {<,B>: AB,BX}are -IOS. From this (A)= 
3.9.Defination:  LIT(X) is called megaconnected if there exist no IS of X which is
both IOS and is "sandwiched" (that is there exist non-empty proper ICS sets F1, F2 such that
F1A F2) between non-empty proper -ICS subsets of X.
3.10.Corallary: Mega-connectedness is contractive invariant.
Proof: This immediate from the definition 3.9.
3.11.Corallary: (A) is a megaconnected member of LIT(X).
Proof: Since there is no ICS different from  which contained any set in the family
{<,B>: BA1,BX} nor is there any ICS different from X which contains any set in the
family {<C,D>: AcC,DCc}this means that there is no IOS in (A) which is sandwiched,
so (A) is megaconnected.
The following example shows that (A) is not a maximal megaconnected member of
LIT(X).
3.12.Example: Let X ={1,2,3,4} and A={4} is a subset of X. Consider the families
(A)= {<,{4}>,<,{1,4}>,<,{2,4}>,<,{3,4}>,,{1,2,4}>,<,{1,3,4}>,<,{2,3,4}>,
<,X>,<{1,2,3},>,<X,>,<{1,2,3},{4}>} and
=(A) {<{1},{2,3,4}>,<{1},{4}>,<{1},{2,4}>,<{1},{3,4}>}
 is megaconnected member of LIT(X).
3.12.Result: From thorem 3.8, corallary 3.11 and example 3.12, submaximality and
megaconnectedness are not complementary intuitionistic topological invariants.
3.13.Theorem: Let X be a non-empty set and A a non-empty subset of X. The family
N(A)={<B,C>:BAc ,AC}{<D,E>: AcD, EDc} is megaconnected member of LIT(X).
Proof: Let LIT(X) such that N(A)   . Then  must contain at least one of the sets;
B=<F1,G1> where A G1
C=<F2,G2> where A F2
D=<F3,G3> where F3A
There are the following cages
i) If B then either BN(A) or since the IOS’s <Ac  G1, > and <, F1A> are
N(A)-open , then -open. This implies that the IS’s <, Ac  G1> B and
B<F1A, > are -ICS’s which means that the IOS B is sandwiched between
these ICS’s.
ii) If C  then either C=X or with the above argument C is sandwiched between
the ICS’s <, Ac  G2>C and C < F2, >
iii)If D then either D= or D is sandwiched between the ICS’s <,AcG3> D
and D  <,A>
This show that  LIT(X) is not megaconnected and N(A) is a maximal megaconnected
member of LIT(X).
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