Applications of ig , dg , bg-Closed type sets in topological ordered spaces

Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, and the like. Nachbin (1965) initiated the study of topological ordered spaces. Levine (1970) introduced the class of g-closed sets, a super class of sets in 1970. Veera Kumar (2000) introduced a new class of sets, called g*-closed sets in 2000, which is properly placed in between the class of closed sets and the class of g-closed sets. Veera Kumar (2002) introduced the concept of i-closed, d-closed and b-closed sets in 2001. Srinivasarao introduced ig-closed, dg-closed, bg-closed, ig*-closed, dg*closed and bg*-closed sets in 2014. In this paper, Srinivasarao discusses the possible applications of ig, dg and bg – closed type sets in topological ordered spaces. A topological ordered space is a triple (X, , ≤), where is a topology on X, Where X is a non-empty set and ≤ is a partial order on X.


INTRODUCTION
Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, and the like.Nachbin (1965) initiated the study of topological ordered spaces.Levine (1970) introduced the class of g-closed sets, a super class of sets in 1970.Veera Kumar (2000) introduced a new class of sets, called g*-closed sets in 2000, which is properly placed in between the class of closed sets and the class of g-closed sets.Veera Kumar (2002) introduced the concept of i-closed, d-closed and b-closed sets in 2001.Srinivasarao introduced ig-closed, dg-closed, bg-closed, ig*-closed, dg*closed and bg*-closed sets in 2014.In this paper, Srinivasarao discusses the possible applications of ig, dg and bg -closed type sets in topological ordered spaces.
A topological ordered space is a triple (X, , ≤), where is a topology on X, Where X is a non-empty set and ≤ is a partial order on X.

Definition 1
For any xX, {yX/x≤y} will be denoted by [x, ] and {yX/y≤x} will be denoted by [, x].A subset A of a topological ordered space (X, , ≤) is said to be increasing if A = i(A) where i(A) =  A a [a, ] (Veera Kumar, 2002).

Definition 2
For any xX, {yX/y≤x} will be denoted by [, x].A subset A of a topological ordered space (X, , ≤) is said to be Kumar, 2002).

PRELIMINARIES Definition 1
A subset A of a topological space (X,  ) is called 1) a generalized closed set (briefly g-closed) (Levine, 1970) if cl(A)  U whenever A  U and U is open in (X, ).
2) a g * -closed set (Veera Kumar, 2000) if cl(A)  U whenever A  U and U is g-open in (X, ).

Definition 2
A subset A of a topological space (X,  , ≤) (Veera Kumar,   2002; Srinivasarao, 2014) is called 1) an i-closed set if A is an increasing set and closed set.
2) a d-closed set if A is a decreasing set and closed set.
3) a b-closed set if A is both an increasing and decreasing set and a closed set.

Theorem 1: Every closed set is a g-closed set
The following example supports that a g-closed set need not be closed set in general (Veera Kumar, 2000).

Theorem 2: Every g*-closed set is a g-closed set
The following example supports that a g-closed set need not be a g*-closed set in general (Veera Kumar, 2000).
Theorem 4: Every d-closed set is a dg-closed set The following example supports that a dg-closed set need not be d-closed set in general (Srinivasarao, 2014).

Theorem 5: Every b-closed set is a bg-closed set
The following example supports that a bg-closed set need not be a b-closed set in general (Srinivasarao, 2014).
Let A = {c}.Clearly A is a bg-closed set but not a b-closed set (Srinivasarao, 2014).

Theorem 6: Every bg-closed set is an ig-closed set
The converse of the above theorem need not be true (Srinivasarao, 2014).This will be justified from the following example.
Let A = {c}.Clearly A is an ig-closed set but not a bg-closed set.

Theorem 7: Every bg-closed set is a dg-closed set
The converse of the above theorem need not be true (Srinivasarao, 2014).This will be justified from the following example.

Theorem 8: Every b-closed set set is an i-closed set
The converse of above theorem need not be true (Srinivasarao, 2014).This will be justified from the following example.

Theorem 9: Every b-closed set is a d-closed set
The converse of above theorem need not be true (Srinivasarao, 2014).This will be justified from the following example.
Clearly A is an ig-closed set but not a ig * -closed set.
Theorem 11: Every dg * -closed set is an dg-closed set The converse of above theorem need not be true (Srinivasarao, 2014).This will be justified from the following example.

Theorem 12: Every bg * -closed set is a bg-closed set
The converse of above theorem need not be true (Srinivasarao, 2014).This will be justified from the following example.

Theorem 13: Every bg * -closed set is an ig * -closed set
The converse of above theorem need not be true (Srinivasarao, 2014).This will be justified from the following example.

Theorem 14: Every bg * -closed set is an dg * -closed set
The converse of above theorem need not be true (Srinivasarao, 2014).This will be justified from the following example.

Theorem 15: Every i-closed set is an ig * -closed set
The converse of above theorem need not be true (Srinivasarao, 2014).This will be justified from the following example.(Srinivasarao, 2014).The class of all ig * -closed sets properly contains the class of all i-closed sets.

Theorem 16: Every d-closed set is a dg * -closed set
The converse of above theorem need not be true (Srinivasarao, 2014).This will be justified from the following example.

Theorem 17: Every b-closed set is a bg * -closed set
The converse of above theorem need not be true (Srinivasarao, 2014).This will be justified from the following example.Then every bg * -closed set is an ig-closed set (Srinivasarao, 2014).The converse of above theorem need not be true.This will be justified from the following example.
The class of all ig-closed sets properly contains the class of all bg * -closed sets.
Theorem 19: Every bg * -closed set is a dg-closed set The converse of above theorem need not be true (Srinivasarao, 2014).This will be justified from the following example.

APPLICATIONS OF g-CLOSED SETS
We introduce the following definitions.
Theorem 1: Every i T 1/2 space is b T 1/2 space

Proof
Let (X,  , ≤) be i T 1/2 space.Let A be bg-closed subset of X.Then A is an ig-closed set.Since (X,  , ≤) is an i T 1/2 space then 'A' is a closed set.Therefore every bg-closed set is a closed set.Hence (X,  ) is a b T 1/2 space.The converse of the above theorem need not be true.This will be justified from the following example.

Proof
Let (X,  , ≤) be d T 1/2 space.We show that (X,  , ≤) is a b T 1/2 space.Let A be bg-closed subset of X.Then A is a dg-closed subset of X.Since (X,  , ≤) is d T 1/2 space, we have A is a closed set.Thus every d T 1/2 space is b T 1/2 space.The converse of the above theorem need not be true.This will be justified from the following example.
Clearly A is dg-closed set but not a closed set.Hence (X, , 3 We thus introduce the following definitions.

Definition 2
The topological ordered space (X, ,  ) is called i) i T i,1/2 space if every ig-closed set is an i-closed set.
ii) d T d,1/2 space if every dg-closed set is an d-closed set.
iii) b T b,1/2 space if every bg-closed set is a b-closed set.iv) C T i space if every closed set is an i-closed set.

Proof
Let (X,  , ≤) be c T b space.We show that (X,  , ≤) is a c T i space.Let A be a closed set.Since (X,  , ≤) is c T b space, then A is a b-closed set.Then A is an i-closed set.Therefore every closed set is an i-closed set.Then (X,  , ≤) is a c T i space.
Hence every c T b space is a c T i space.The converse of above theorem need not be true.This will be justified from the following example.

Proof
Let (X,  , ≤) be c T b space.We show that (X,  , ≤) is a c T d space.Let A be a closed set.Since (X,  , ≤) is c T b space, then A is a b-closed set.Then A is a d-closed set.Therefore every closed set is a d-closed set.Then (X,  , ≤) is a c T d space.
Hence every c T b space is a c T d space.The converse of the above theorem need not be true.This will be justified from the following example.

Proof
Let (X,  , ≤) be c T b space.We show that (X,  , ≤) is a i T b space.Let A be an i-closed set.Then A is a closed set.Since (X,  , ≤) is c T b space, then A is a b-closed set.. Therefore every i-closed set is a b-closed set.Then (X,  , ≤) is an i T b space.Hence every c T b space is a i T b space.The converse of the above theorem need not be true.This will be seen in the following example.

Proof
Let (X,  , ≤) be c T b space.We show that (X,  , ≤) is a d T b space.Let A be a d-closed set then A is a closed set.Since (X,  , ≤) is c T b space then A is a b-closed set.Thus every d-closed set is a b-closed set.Thus every c T b space is d T b space.The converse of the above theorem need not be true.This will be justified from the following example.
Theorem 15: Every i T b space is an b T b,½ space

Proof
Let be (X,  , ≤) iT b space.Now we (X,  , ≤) is a b T b,1/2 space.Let A be a bg-closed set.Then A is an ig-closed set.
Since (X,  , ≤) is i T b space then A is a b-closed set.Therefore every bg-closed set is a b-closed set Hence every i T b space is an b T b , 1/2 space.The converse of the above theorem need not be true.This will be justified from the following example.

Proof
Let be (X,  , ≤) d T b space.Now we (X,  , ≤) is a b T b,1/2 space.Let A be a bg-closed set.Then A is a dg-closed set.
Since (X,  , ≤) is d T b space then A is a b-closed set.Therefore every bg-closed set is a b-closed set Hence every d T b space is an b T b , 1/2 space.The converse of the above theorem need not be true.This will be justified from the following example.

Example 24
Let X = {a, b, c}, , Theorem 17: Every i T b space is an i T 1/2 space Proof Let (X,  , ≤) be i T b space.we show that (X,  , ≤) is a i T 1/2 space.Let A be a ig-closed set.Since (X,  , ≤) is i T b space then A is a b-closed set.Then A is a closed set.Thus every iclosed set is a closed set.Thus every i T b space is i T 1/2 space.The converse of the above theorem need not be true.This will be justified from the following example.then A is a b-closed set.Then A is a closed set.Thus every bgclosed set is a closed set.Thus every b T b,1/2 space is b T 1/2 space.
The converse of the above theorem need not be true.This will be seen in the following example.

CONCLUSION
In this paper, we introduced i T i,1/2, d T d,1/2 , b T b,1/2, i T 1/2, d T 1/2, b T 1/2, new class of spaces using g-closed type sets in topological ordered spaces and studied various relationships between them.
4) ig-closed set if icl(A)  U whenever A  U and U is open in (X, ). 5) dg-closed set if dcl(A)  U whenever A  U and U is open in (X, ). 6) bg-closed set if bcl(A)  U whenever A  U and U is open in (X, ).

2 
sets are  , X, {b, c }. i-closed sets are  , X, {b, c} and closed sets are  , X .Here every closed set is an i-closed set.Let A = {b, c}.Clearly A is a closed set but not a b-closed set.Thus (X, , ≤ 1 ) is c T i space but not c T b space.Theorem 5: Every c T b space is a c T d space sets are  , X, {b, c }. d-closed sets are  , X, {b, c} and b-closed sets are  , X .Here every closed set is a d- closed set.Let A = {b, c}.Clearly A is a closed set but not a bclosed set.Thus (X, , 2  ≤ 1 ) is c T d space but not c T b space.Theorem 6: Every c T b space is a i T b space

1 
{a}, {b}, {a, b}} and ≤ 2 = {(a, a), (b, b),(c, c), (a, b), (c, b)}.Clearly (X, , ≤ 2 ) is a topological ordered space.Closed sets are  , X, {c}, {a, c}, {b, c }. i-closed sets are  , X .b-closed sets are  , X. Clearly every i-closed set is a b-closed set where as every closed set is not a b-closed set.Let A = {c} or {a, c} or {b, c}.Clearly A is a closed set but not a b-closed set.Thus (

1 1 
= { , X, {a}, {b}, {a,b}} and ≤ 2 = {(a, a), (b, b), (c, c), (a, b), (c, b)}.Clearly (X, , ≤ 2 ) is a topological ordered space.Here b-closed sets are  , X. d- closed sets are  , X, {c}, {b, c} and bg-closed sets are  , X. Let A = {c} or {b, c}.Clearly A is a d-closed set but not a bclosed set.Every bg-closed sets is a b-closed set where as every d-closed set is not a b-closed set.Thus (X, , 1  ≤ 1 ) is a b T b,1/2 space but not d T b space.