A view on ordered intuitionistic Fuzzy smooth quasi uniform basically disconnected spaces.

[section]1. Introduction

The concept of Fuzzy set was introduced by Zadeh . Since then the concept has invaded nearly all branches of Mathematics. Chang  introduced and developed the theory of Fuzzy topological spaces and since then various notions in classical topology have been extended to Fuzzy topological spaces. Fuzzy sets have applications in many fields such as information  and control . Atanassov  generalised Fuzzy sets to intuitionistic Fuzzy sets. Cocker  introduced the notions of an intuitionistic Fuzzy topological space. Young Chan Kim and Seok Jong Lee [10,11] have discussed some properties of Fuzzy quasi uniform space. Tomasz Kubiak [7,8] studied L-Fuzzy normal spaces and Tietze extension Theorem and extending continuous L-Real functions. G. Thangaraj and G. Balasubramanian  discussed On Fuzzy pre-basically disconnected spaces. In this paper, a new class of intuitionistic Fuzzy smooth quasi uniform topological spaces called ordered intuitionistic Fuzzy smooth quasi uniform topological spaces is introduced. Tietze extention theorem for ordered intuitionistic Fuzzy smooth quasi uniform basically disconnected spaces has been discussed besides providing several other propositions.

[section]2. Preliminaries

Definition 2.1.  Let X be a non empty fixed set and I the closed interval [0,1]. An intuitionistic Fuzzy set (IFS). A is an object of the following form A = {< x, [[mu].sub.A](x) , [[gamma].sub.A](x) >: x [member of] X} where the function [[mu].sub.A]: X [right arrow] I and [[gamma].sub.A]: X [right arrow] I denote the degree of membership (namely [[mu].sub.A](x)) and the degree of non membership (namely [[gamma].sub.A](x)) for each element x [member of] X to the set A respectively and 0 [less than or equal to] [[mu].sub.A](x) + [[gamma].sub.A](x) [less than or equal to] 1 for each x [member of] X. Obviously, every Fuzzy set A on a nonempty set X is an IFS of the following form A = {<x, [[mu].sub.A](x), 1-[[mu].sub.A](x)>:x [member of] X}. For the sake of simplicity, we shall use the symbol A = <x, [[mu].sub.A](x) , [[gamma].sub.A](x)> for the intuitionistic Fuzzy set A = {<x, [[mu].sub.A](x), [[gamma].sub.A](x):x [member of] X>}. For a given non empty set X, denote the family of all intuitionistic Fuzzy sets in X by the symbol [[zeta].sup.X]

Definition 2.2.  Let X be a nonempty set and the IFSs A and B in the form A = {<x, [[mu].sub.A](x), [[gamma].sub.A](x)>:x [member of] X}, B = {<x, [[mu].sub.B](x), [[gamma].sub.B](x)>:x [member of] X}. Then

(i) A [subset or equal to] B iff [[mu].sub.A](x) [less than or equal to] [[mu].sub.B](x) and [[gamma].sub.A](x) [greater than or equal to] [[gamma].sub.B](x) for all x [member of] X,

(ii) [bar.A] = {<x, [[gamma].sub.A](x), [[mu].sub.A](x)>:x [member of] X},

(iii) A [intersection] B = {<x, [[mu].sub.A](x) [conjunction] [[mu].sub.B](x), [[gamma].sub.A](x) [disjunction] [[gamma].sub.B](x)>: x [member of] X},

(iv) A [union] B = {<x, [[mu].sub.A](x) [disjunction] [[mu].sub.B](x), [[gamma].sub.A](x) [conjunction] [[gamma].sub.B](x)>: x [member of] X}.

Definition 2.3 The IFSs [0.sub.~] and [1.sub.~] are defined by [0.sub.~] = {<x, 0,1>:x [member of] X} and [1.sub.~] = {<x, 1,0>:x [member of] X}.

Definition 2.4.  An intuitionistic Fuzzy topology (IFT) in Coker's sense on a non-empty set X is a family [tau] of IFSs in X satisfying the following axioms.

([T.sub.1])[0.sub.~],[1.sub.~] [member of] [tau];

([T.sub.2])[G.sub.1] [intersection] [G.sub.2] [member of] [tau] for any [G.sub.1], [G.sub.2] [member of] [tau];

([T.sub.3]) [union] [G.sub.i] [member of] [tau] for arbitrary family {[G.sub.i]/i [member of] I}[subset or equal to][tau].

In this paper by (X, [tau]) or simply by X we will denote the Cocker's intuitionistic Fuzzy topological space (IFTS). Each IFSs in [tau] is called an intuitionistic Fuzzy open set(IFOS) in X. The complement [bar.A] of an IFOS A in X is called an intuitionistic Fuzzy closed set (IFCS) in X.

Definition 2.5.  Let a and b be two real numbers in [0,1] satisfying the inequality a + b [less than or equal to] 1. Then the pair <a, b> is called an intuitionistic Fuzzy pair. Let <[a.sub.1], [b.sub.1]>, <[a.sub.2], [b.sub.2]> be any two intuitionistic Fuzzy pairs. Then define

(i) <[a.sub.1], [b.sub.1]> [less than or equal to] <[a.sub.2], [b.sub.2]> if and only if [a.sub.1] [less than or equal to] [a.sub.2] and [b.sub.1] [greater than or equal to] [b.sub.2],

(ii) <[a.sub.1], [b.sub.1]> = <[a.sub.2], [b.sub.2]> if and only if [a.sub.1] = [a.sub.2] and [b.sub.1] = [b.sub.2],

(iii) If {<[a.sub.i], [b.sub.i]/i [member of] J>} is a family of intuitionistic Fuzzy pairs, then [disjunction] <[a.sub.i], [b.sub.i]> = < [disjunction] [a.sub.i], [conjunction] [b.sub.i]> and [conjunction] <[a.sub.i], [b.sub.i]> = < [conjunction] [a.sub.i], [disjunction] [b.sub.i]>,

(iv) The complement of an intuitionistic Fuzzy pair <a, b> is the intuitionistic Fuzzy pair defined by <[bar.a,b]> = <b, a>,

(v) [1.sup.~] = <1,0> and [0.sup.~] = <0,1>.

Definition 2.6.  Let (X, T) be any Fuzzy topological space. (X, T) is called Fuzzy basically disconnected if the closure of every Fuzzy open [F.sub.[sigma]] is Fuzzy open.

Definition 2.7.  A function U: [[OMEGA].sub.X] [right arrow] L is said to be an L-Fuzzy quasi-uniformity on X if it satisfies the following conditions.

(i) U([f.sub.1] [??] [f.sub.2]) [greater than or equal to] U([f.sub.1]) [conjunction] U([f.sub.2]) for [f.sub.1], [f.sub.2] [member of] [[OMEGA].sub.X],

(ii) For f [member of] [[OMEGA].sub.X] we have [disjunction] {U([f.sub.1])/[f.sub.1][??][f.sub.1] [less than or equal to] f} [greater than or equal to] U(f),

(iii) If [f.sub.1] [greater than or equal to] f then U([f.sub.1]) [greater than or equal to] U(f),

(iv) There exists f [member of] [[OMEGA].sub.X] such that U(f) = 1. Then the pair (X, U) is said to be an L-Fuzzy quasi uniform space.

[section]3. Ordered intuitionistic Fuzzy smooth quasi uniform basically disconnected spaces

Definition 3.1. Let [[OMEGA].sub.X] denotes the family of all intuitionistic Fuzzy functions f: [[zeta].sup.X] [right arrow] [[zeta].sup.X] with the following properties.

(i) f([0.sub.~]) = [0.sub.~],

(ii) A [subset or equal to] f(A) for every A [member of] [[zeta].sup.X],

(iii) f([union] [A.sub.i]) = [union] f([A.sub.i]) for every [A.sub.i] [member of] [[zeta].sup.X], i [member of] J.

For f [member of] [[OMEGA].sub.X], the function [f.sup.-1] [member of] [[OMEGA].sub.X] is defined by [f.sup.-1](A) = [intersection] {B/f([[bar] B])[subset or equal to][[bar] A]}.

For f, g [member of] [[OMEGA].sub.X], we define, for all [Amember of][[zeta].sup.X], f&sqcap; g(A) = [intersection] {f([A.sub.1]) [union] g([A.sup.2])/[A.sub.1] [union] [A.sup.2] = A}, (f [omicron] g)(A) = f(g(A)).

Definition 3.2. Let (X, U) be an intuitionistic Fuzzy quasi uniform space. Define, for each r [member of] (0,1] = [I.sub.0], s [member of] [0,1) = [I.sub.1] with r + s [less than or equal to] 1 and A [member of] [[zeta].sup.X], (r, s)IFQ[I.sub.u](A) = [union] {B/f(B) [subset or equal to] A for some f [member of] [[OMEGA].sub.X] with U(f) > <r, s>}.

Definition 3.3. Let (X, U) be an intuitionistic Fuzzy quasi uniform space. Then the function [T.sub.u]: [[zeta].sup.X] [right arrow] [I.sub.0][x][I.sub.1] is defined by [T.sub.u](A) = [union] {<r, s>/(r, s)IFQ[I.sub.u](A) = A, r [member of] [I.sub.0], s [member of] [I.sub.1] with r + s [less than or equal to] 1}. Then the pair (X, [T.sub.u]) is called an intuitionistic Fuzzy smooth quasi uniform topological space. The members of (X, [T.sub.u]) are called an intuitionistic Fuzzy smooth quasi uniform open set.

Note 3.1. The complement of an intuitionistic Fuzzy smooth quasi uniform open set is an intuitionistic Fuzzy smooth quasi uniform closed set.

Definition 3.4. Let (X, [T.sub.u]) be an intuitionistic Fuzzy smooth quasi uniform topological space and A be an intuitionistic Fuzzy set. Then the intuitionistic Fuzzy smooth quasi uniform interior of A is denoted and defined by IFSQin[t.sub.u](A) = [union] {B/B [subset or equal to] A and B is an intuitionistic Fuzzy smooth quasi uniform open set where r [member of] [I.sub.0], s [member of] [I.sub.1] with r + s [less than or equal to] 1}.

Definition 3.5. Let (X, [T.sub.u]) be an intuitionistic Fuzzy smooth quasi uniform topological space and A be an intuitionistic Fuzzy set. Then the intuitionistic Fuzzy smooth quasi uniform closure of A is denoted and defined by IFSQc[l.sub.u](A) = [intersection] {B/B [contains or equal to] A and B is an intuitionistic Fuzzy smooth quasi uniform closed set where r [member of] [I.sub.0], s [member of] [I.sub.1] with r + s [less than or equal to] 1}.

Definition 3.6. Let (X, [T.sub.u]) be an intuitionistic Fuzzy smooth quasi uniform topological space and A be an intuitionistic Fuzzy set. Then A is said to be an intuitionistic Fuzzy smooth quasi uniform [G.sub.[delta]] set if A = [[intersection].sup.[infinity].sub.i = 1][A.sub.i] where each [A.sub.i] is an intuitionistic Fuzzy smooth quasi uniform open set, where r [member of] [I.sub.0], s [member of] [I.sub.1] with r + s [less than or equal to] 1. The complement of an intuitionistic Fuzzy smooth quasi uniform [G.sub.[delta]] set is an intuitionistic Fuzzy smooth quasi uniform [F.sub.[sigma]] set.

Note 3.2. Every intuitionistic Fuzzy smooth quasi uniform open set is an intuitionistic Fuzzy smooth quasi uniform [G.sub.[delta]] set and every intuitionistic Fuzzy smooth quasi uniform closed set is an intuitionistic Fuzzy smooth quasi uniform [F.sub.[sigma]] set.

Definition 3.7. Let (X, [T.sub.u]) be an intuitionistic Fuzzy smooth quasi uniform topological space and A be any intuitionistic Fuzzy set in (X, [T.sub.u]). Then A is said to be

(i) increasing intuitionistic Fuzzy set if x [less than or equal to] y implies A(x) [less than or equal to] A(y). That is, [[mu].sub.A](x) [less than or equal to] [[mu].sub.A](y) and [[gamma].sub.A](x) [greater than or equal to] [[gamma].sub.A](y);

(ii) decreasing intuitionistic Fuzzy set if x [less than or equal to] y implies A(x) [greater than or equal to] A(y). That is, [[mu].sub.A](x) [greater than or equal to] [[mu].sub.A](y) and [[gamma].sub.A](x) [less than or equal to] [[gamma].sub.A](y).

Definition 3.8. Let X be an ordered set. [T.sub.u] is an intuitionistic Fuzzy smooth quasi uniform topology defined on X. Then (X, [T.sub.u], [less than or equal to]) is said to be an ordered intuitionistic Fuzzy smooth quasi uniform topological space.

Definition 3.9. Let (X, [T.sub.u], [less than or equal to]) be an ordered intuitionistic Fuzzy smooth quasi uniform topological space and A be any intuitionistic Fuzzy set in (X, [T.sub.u], [less than or equal to]). Then we define

(i) [IFSQI.sub.u](A) = Intuitionistic Fuzzy smooth quasi uniform increasing closure of A = The smallest intuitionistic Fuzzy smooth quasi uniform increasing closed set containing in A.

(ii) [IFSQD.sub.u](A) = Intuitionistic Fuzzy smooth quasi uniform decreasing regular closure of A = The smallest intuitionistic Fuzzy smooth quasi uniform decreasing closed set containing in A.

(iii) IFSQ[I.sup.0.sub.u](A) = Intuitionistic Fuzzy smooth quasi uniform increasing interior of A = The greatest intuitionistic Fuzzy smooth quasi uniform increasing open set contained in A.

(iv) [IFSQD.sup.0.sub.u](A) = Intuitionistic Fuzzy smooth quasi uniform decreasing interior of A = The greatest intuitionistic Fuzzy smooth quasi uniform decreasing open set contained in A.

Proposition 3.1. Let (X, [T.sub.u], [less than or equal to]) be an ordered intuitionistic Fuzzy smooth quasi uniform topological space. Then for any two intuitionistic Fuzzy sets A and B in (X, [T.sub.u], [less than or equal to]) the following are valid.

(i) [[bar.IFSQI].sub.u](A)] = [IFSQD.sup.0.sub.u]([[bar.A]),

(ii) [bar.[IFSQD.sub.u](A)] = IFSQ[I.sup.0.sub.u]([bar.A]),

(iii) [bar.IFSQ[I.sup.0.sub.u](A)] = [IFSQD.sub.u]([bar.A]),

(iv) [bar.[IFSQD.sup.0.sub.u](A)] = [IFSQI.sub.u]([bar.A]).

Proof. Since [IFSQI.sub.u](A) is an intuitionistic Fuzzy smooth quasi uniform increasing closed set containing A, [[bar.IFSQI].sub.u](A)] is an intuitionistic Fuzzy smooth quasi uniform decreasing open set such that [bar.IFSQI.sub.u](A)] [subset or equal to] A. Let B be another intuitionistic Fuzzy smooth quasi uniform decreasing open set such that B [subset or equal to][bar.A]. Then [bar.B] is an intuitionistic Fuzzy smooth quasi uniform increasing closed set such that [bar.B] [contains or equal to] A. It follows that [IFSQI.sub.u](A) [subset or equal to] [bar.B]. That is, B [subset or equal to] [[bar.IFSQI].sub.u](A)]. Thus, [[bar.IFSQI].sub.u](A)] is the largest intuitionistic Fuzzy smooth quasi uniform decreasing open set such that [[bar.IFSQI].sub.u](A)] [subset or equal to][bar.A]. That is, [[bar.IFSQI].sub.u](A)] = [IFSQD.sup.0.sub.u]([bar.A]). The proof of (2), (3) and (4) are similar to (1).

Definition 3.10. Let (X, [T.sub.u], [less than or equal to]) be an ordered intuitionistic Fuzzy smooth quasi uniform topological space.

(i) An intuitionistic Fuzzy set A in (X, [T.sub.u], [less than or equal to]) which is both intuitionistic Fuzzy smooth quasi uniform increasing (decreasing) open and intuitionistic Fuzzy smooth quasi uniform increasing (decreasing) [F.sub.[sigma]] is defined by intuitionistic Fuzzy smooth quasi uniform increasing (decreasing) open [F.sub.[sigma]].

(ii) An intuitionistic Fuzzy set A in (X, [T.sub.u], [less than or equal to]) which is both intuitionistic Fuzzy smooth quasi uniform increasing (decreasing) closed and intuitionistic Fuzzy smooth quasi uniform increasing (decreasing) [G.sub.[delta]] is defined by intuitionistic Fuzzy smooth quasi uniform increasing (decreasing) closed [G.sub.[delta]].

(iii) An intuitionistic Fuzzy set A in (X, [T.sub.u], [less than or equal to]) which is both intuitionistic Fuzzy smooth quasi uniform increasing (decreasing) open [F.sub.[sigma]] and intuitionistic Fuzzy smooth quasi uniform increasing (decreasing) closed [G.sub.[delta]] is defined by intuitionistic Fuzzy smooth quasi uniform increasing (decreasing) closed open [G.sub.[delta]][F.sub.[sigma]].

Definition 3.11. Let (X, [T.sub.u], [less than or equal to]) be an ordered intuitionistic Fuzzy smooth quasi uniform topological space. Let A be any intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set in (X, [T.sub.u], [less than or equal to]). If [IFSQI.sub.u](A) is an intuitionistic Fuzzy smooth quasi uniform increasing open set in (X, [T.sub.u], [less than or equal to]), then (X, [T.sub.u], [less than or equal to]) is said to be upper intuitionistic Fuzzy smooth quasi uniform basically disconnected space. Similarly we can define lower intuitionistic Fuzzy smooth quasi uniform basically disconnected space.

Definition 3.12. An ordered intuitionistic Fuzzy smooth quasi uniform topological space (X, [T.sub.u], [less than or equal to]) is said to be ordered intuitionistic Fuzzy smooth quasi uniform basically disconnected space if it is both upper intuitionistic Fuzzy smooth quasi uniform basically disconnected space and lower intuitionistic Fuzzy smooth quasi uniform basically disconnected space.

Proposition 3.2. Let (X, [T.sub.u], [less than or equal to]) be an ordered intuitionistic Fuzzy smooth quasi uniform topological space. Then the following statements are equivalent:

(i) (X, [T.sub.u], [less than or equal to]) is an upper intuitionistic Fuzzy smooth quasi uniform basically disconnected space,

(ii) For each intuitionistic Fuzzy smooth quasi uniform decreasing closed [G.sub.[delta]] set A, then [IFSQD.sup.0.sub.u](A) is an intuitionistic Fuzzy smooth quasi uniform decreasing closed,

(iii) For each intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set A, we have [IFSQD.sub.u]([IFQD.sup.0.sub.u][bar.(((A)]))]) = [[bar.IFSQI].sub.u](A)],

(iv) For each intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set A and intuitionistic Fuzzy smooth quasi uniform decreasing set B in (X, [T.sub.u], [less than or equal to]) with [IFSQI.sub.u](A) = [bar.B], we have, [IFSQD.sub.u](B) = [[bar.IFSQI].sub.u](A)].

Proof. (i)[??](ii) Let A be any intuitionistic Fuzzy smooth quasi uniform decreasing closed [G.sub.[delta]] set. Then [bar.A] is an intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set and so by assumption (1), [IFSQI.sub.u]([bar.A]) is an intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set. That is, [IFSQD.sup.0.sub.u](A) is an intuitionistic Fuzzy smooth quasi uniform decreasing closed.

(ii)[??](iii) Let A be any intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set. Then [bar.A] is an intuitionistic Fuzzy smooth quasi uniform decreasing closed [G.sub.[delta]] set. Then by (2), [IFSQD.sup.0.sub.u]([bar.A]) is an intuitionistic Fuzzy smooth quasi uniform decreasing closed [G.sub.[delta]] set. Now, [IFSQD.sub.u]([IFSQD.sup.0.sub.u]([bar.A])) = [IFSQD.sup.0.sub.u]([bar.A]) = [[bar.IFSQI].sub.u](A)].

(iii)[??](iv) Let A be an intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set and B be an intuitionistic Fuzzy smooth quasi uniform decreasing open [F.sub.[sigma]] set such that [IFSQI.sub.u](A) = [bar.B]. By (3), [IFSQD.sub.u]([[bar.IFSQI].sub.u](A)]) = [[bar.IFSQI].sub.u](A)]. [IFSQD.sub.u](B) = [[bar.IFSQI].sub.u](A)].

(iv)[??](i) Let A be an intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set. Put B = [[bar.IFSQI].sub.u](A)]. Clearly, B is an intuitionistic Fuzzy smooth quasi uniform decreasing set. By (4) it follows that [IFSQD.sub.u](B) = [[bar.IFSQI].sub.u](A)]. That is, [[bar.IFSQI].sub.u](A)] is an intuitionistic Fuzzy smooth quasi uniform decreasing open [F.sub.[sigma]] set. Hence (X, [T.sub.u], [less than or equal to]) is an upper intuitionistic Fuzzy smooth quasi uniform basically disconnected space.

Proposition 3.3. Let (X, [T.sub.u], [less than or equal to]) be an ordered intuitionistic Fuzzy smooth quasi uniform topological space. Then (X, [T.sub.u], [less than or equal to]) is an upper intuitionistic Fuzzy smooth quasi uniform basically disconnected space if and only if for each A and B are intuitionistic Fuzzy smooth quasi uniform decreasing closed open [G.sub.[delta]][F.sub.[sigma]] such that A [subset or equal to] B we have, [IFSQD.sub.u](A) [subset or equal to] [IFSQD.sup.0.sub.u](A).

Proof. Suppose (X, [T.sub.u], [less than or equal to]) is an upper intuitionistic Fuzzy smooth quasi uniform basically disconnected space and let A be an intuitionistic Fuzzy smooth quasi uniform decreasing open [F.sub.[sigma]] set and B be an intuitionistic Fuzzy smooth quasi uniform decreasing closed [G.sub.[delta]] set such that A [subset or equal to]B. Then by (2) of Proposition 3.2, [IFSQD.sub.u]0(A) is an intuitionistic Fuzzy smooth quasi uniform decreasing closed set. Also, since A is an intuitionistic Fuzzy smooth quasi uniform decreasing open [F.sub.[sigma]] set and A [subset or equal to] B, it follows that A [subset or equal to]IFSQ[D..sup.0.sub.u](B). This implies that [IFSQD.sub.u](A) [subset or equal to][IFSQD.sup.0.sub.u](B).

Conversely, let B be any intuitionistic Fuzzy smooth quasi uniform decreasing closed open [G.sub.[delta]][F.sub.[sigma]] set. Then by Definition 3.4, [IFSQD.sup.0.sub.u](B) is an intuitionistic Fuzzy smooth quasi uniform decreasing open [F.sub.[sigma]] set and it is also clear that [IFSQD.sup.0.sub.u](B)[subset or equal to]B. Therefore by assumption, [IFSQD.sub.u]([IFSQD.sup.0.sub.u](B))[subset or equal to][IFSQD.sup.0.sub.u](B). This implies that [IFSQD.sup.0.sub.u](B) is an intuitionistic Fuzzy smooth quasi uniform decreasing closed set. Hence by (2) of Proposition 3.2, it follows that (X, [T.sub.u], [less than or equal to]) is an upper intuitionistic Fuzzy smooth quasi uniform basically disconnected space.

Remark 3.1. Let (X, [T.sub.u], [less than or equal to]) be an upper intuitionistic Fuzzy smooth quasi uniform basically disconnected space. Let {[A.sub.i], [[barB].sub.i]]/i [member of] N} be collection such that [A.sub.i]'s are intuitionistic Fuzzy smooth quasi uniform decreasing open [F.sub.[sigma]] sets and [B.sub.i] are intuitionistic Fuzzy smooth quasi uniform decreasing closed [G.sub.[delta]] sets. Let A and [[bar] B] be an intuitionistic Fuzzy smooth quasi uniform decreasing open [F.sub.[sigma]] set and intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set respectively. If [A.sub.i][subset or equal to]A [subset or equal to][B.sub.j] and [A.sub.i][subset or equal to]B[subset or equal to][B.sub.j] for all i, j [member of] N, then there exists an intuitionistic Fuzzy smooth quasi uniform decreasing closed open [G.sub.[delta]][F.sub.[sigma]] set C such that [IFSQD.sub.u]([A.sub.i]) [subset or equal to] C [subset or equal to] [IFSQD.sup.0.sub.u]([B.sub.j]) for all i, j [member of] N.

Proof. By Proposition 3.3, [IFSQD.sub.u]([A.sub.i]) [subset or equal to][IFSQD.sub.u](A) [intersection] [IFSQD.sup.0.sub.u](B)[subset or equal to][IFSQD.sup.0.sub.u]([B.sub.j]) for all i, j [member of] N. Letting C = [IFSQD.sub.u](A) [intersection] [IFSQD.sup.0.sub.u](B) in the above, we have C is an intuitionistic Fuzzy smooth quasi uniform decreasing closed open [G.sub.[delta]][F.sub.[sigma]] set satisfying the required conditions.

Proposition 3.3. Let (X, [T.sub.u], [less than or equal to]) be an ordered intuitionistic Fuzzy smooth quasi uniform basically disconnected space. Let [{[A.sub.q]}.sub.q [member of] Q] and [{[B.sub.q]}.sub.q [member of] Q] be monotone increasing collections of an intuitionistic Fuzzy smooth quasi uniform decreasing open [F.sub.[sigma]] sets and intuitionistic Fuzzy smooth quasi uniform decreasing closed [G.sub.[delta]] sets of (X, [T.sub.u], [less than or equal to]) respectively. Suppose that [A.sub.q1][subset or equal to][B.sub.q2] whenever [q.sub.1] < [q.sub.2] ( Q is the set of all rational numbers). Then there exists a monotone increasing collection [{[C.sub.q]}.sub.q [member of] Q] of an intuitionistic Fuzzy smooth quasi uniform decreasing closed open [G.sub.[delta]][F.sub.[sigma]] sets of (X, [T.sub.u], [less than or equal to]) such that [IFSQD.sub.u](A[q.sup.l])[subset or equal to]C[q.sub.2] and C[q.sub.1][subset or equal to] [IFSQD.sup.0.sub.u]([B.sub.q2]) whenever [q.sub.1]<[q.sub.2].

Proof. Let us arrange all rational numbers into a sequence {[q.sub.n]} (without repetitions). For every n [greater than or equal to] 2, we shall define inductively a collection {[C.sub.qi]/1 [less than or equal to] i<n}[subset][[zeta].sup.X] such that

[IFSQD.sub.u]([A.sub.q]) [subset or equal to][C.sub.qi], if q < [q.sub.i]; [C.sub.qi][subset or equal to] [IFSQD.sup.0.sub.u]([B.sub.q]) , if[q.sub.i] < q, for all i < n. ([S.sub.n])

By Proposition 3.3, the countable collections {ISFQ[D.sub.U]([A.sub.q])} and {[IFSQD.sup.0.sub.u]([B.sub.q])} satisfying [IFSQD.sub.u] (A[q.sup.l]) [subset or equal to] [IFSQD.sup.0.sub.u]([B.sub.q2]) if [q.sub.1] < [q.sub.2]. By Remark 3.1, there exists an intuitionistic Fuzzy smooth quasi uniform decreasing closed open [G.sub.[delta]][F.sub.[sigma]] set [D.sub.1] such that

[IFSQD.sub.u] (A[q.sup.l])[subset or equal to][D.sub.1][subset or equal to][IFSQD.sup.0.sub.u]([B.sub.q2]).

Letting C[q.sub.1] = [D.sub.1], we get ([S.sub.2]). Assume that intuitionistic Fuzzy sets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are already defined for i < n and satisfy ([S.sub.n]). Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] whenever [q.sub.i]<[q.sub.n]<[q.sup.j](i,j<n) , as well as [A.sub.q][subset or equal to] [IFSQD.sub.u](E) [subset or equal to][B.sub.q'] and [A.sub.q] [subset or equal to] [IFSQD.sup.0.sub.u](F) [subset or equal to] [B.sub.q'] whenever q < [q.sub.n] < [q.sup.l] This shows that the countable collection [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] together with E and F fulfil the conditions of Remark 3.1. Hence, there exists an intuitionistic Fuzzy smooth quasi uniform decreasing closed open [G.sub.[delta]][F.sub.[sigma]] set [D.sub.n] such that [IFSQD.sub.u]([D.sub.n]) [subset or equal to][B.sub.q], if [q.sub.n] < q; [A.sub.q][subset or equal to][IFSQD.sup.0.sub.u]([D.sub.n]) , if q < [q.sub.n]; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if [q.sub.n]<[q.sup.j], where 1 [less than or equal to] i, j [less than or equal to] n - 1. Letting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we obtain an intuitionistic Fuzzy sets [C.sub.q1], [C.sub.q2], [C.sub.q3], ..., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that satisfy ([S.sub.n + 1]). Therefore, the collection [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has the required property.

Definition 3.13. Let (X, [T.sub.u], [less than or equal to]) and (Y, [S.sub.V], [less than or equal to]) be an ordered intuitionistic Fuzzy smooth quasi uniform topological spaces and f: (X, T, [less than or equal to]) [right arrow] (Y, S, [less than or equal to]) be an intuitionistic Fuzzy function. Then f is said to be an (r, s) intuitionistic Fuzzy quasi uniform increasing (decreasing) continuous function if for any intuitionistic Fuzzy smooth quasi uniform open (closed) set A in (Y, [S.sub.v], [less than or equal to]), [f.sup.-1](A) is an intuitionistic Fuzzy smooth quasi uniform increasing (decreasing) open [F.sub.[sigma]] (closed [G.sub.[delta]]) set in (X, [T.sub.u], [less than or equal to]).

If f is both (r, s) intuitionistic Fuzzy quasi uniform increasing continuous function and (r, s) intuitionistic Fuzzy quasi uniform decreasing continuous function then it is called ordered (r, s) intuitionistic Fuzzy quasi uniform continuous function.

[section]4. Tietze extention theorem for ordered intuitionistic Fuzzy smooth quasi uniform basically disconnected space

An intuitionistic Fuzzy real line [R.sub.I](I) is the set of all monotone decreasing intuitionistic Fuzzy set [Amember of][[zeta].sup.R] satisfying

[union] {A(t):t [member of] R} = [0.sup.~],

[intersection] {A(t):t [member of] R} = [1.sup.~].

After the identification of intuitionistic Fuzzy sets A, B [member of] [R.sub.I](I) if and only if A(t-) = B(t-) and A(t+) = B(t+) for all t [member of] R where

A(t-) = [intersection] {A(s):s < t} and A(t+) = [union] {A(s):s > t}.

The natural intuitionistic Fuzzy topology on [R.sub.I](I) is generated from the basis {[L.sup.I.sub.t], [R.sup.I.sub.t]:t [member of] R} where [L.sup.I.sub.t], [R.sup.I.sub.t] are function from [R.sub.I](I) [right arrow] I[I.sub.I](I) are given by [L.sup.I.sub.t][A] = [[bar] A(t-)] and [R.sup.I.sub.t][A] = A(t+).

The intuitionistic Fuzzy unit interval I[I.sub.I](I) is a subset of [R.sub.I](I) such that [A] [member of] I[I.sub.I](I) if the member and non member of A are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

respectively.

Definition 4.1. Let (X, [T.sub.u], [less than or equal to]) be an ordered intuitionistic Fuzzy smooth quasi uniform topological space and f: X [right arrow] [R.sub.I](I) be an intuitionistic Fuzzy function. Then f is said to be lower (r, s) intuitionistic Fuzzy quasi uniform continuous function if [f.sup.-1]([R.sup.I.sub.t]) is an intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set or intuitionistic Fuzzy smooth quasi uniform decreasing closed [G.sub.[delta]] set, for t [member of] R.

Definition 4.2. Let (X, [T.sub.u], [less than or equal to]) be an ordered intuitionistic Fuzzy smooth quasi uniform topological space and f: X [right arrow] [R.sub.I](I) be an intuitionistic Fuzzy function. Then f is said to be upper (r, s) intuitionistic Fuzzy quasi uniform continuous function if [f.sup.-1]([L.sup.I.sub.t]) is an intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set or intuitionistic Fuzzy smooth quasi uniform decreasing closed [G.sub.[delta]] set, for t [member of] R.

Note 4.1. Let X be a non empty set and A [member of][[zeta].sup.X] Then [A.sup.~] = <[[mu].sub.A](x) , [[gamma].sub.A](x) > for every x [member of] X.

Proposition 4.1. Let (X, [T.sub.u], [less than or equal to]) be an ordered intuitionistic Fuzzy smooth quasi uniform topological space, A [member of][[zeta].sup.X] and f:X [right arrow] [R.sub.I](I) be such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and for all x [member of] X. Then f is lower (upper) (r, s) intuitionistic Fuzzy quasi uniform continuous function if and only if A is an intuitionistic Fuzzy smooth quasi uniform increasing (decreasing) open [F.sub.[sigma]] (closed [G.sub.[delta]]) set.

Proof. It suffices to observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus proved.

Definition 4.3. let X be any non empty set. An intuitionistic Fuzzy* characteristic function of an intuitionistic Fuzzy set A in X is a map [[PSI].sub.A]: X [right arrow] I[I.sub.I](I) defined by [[PSI].sub.A](x) = [A.sup.~] for each x [member of] X.

Proposition 4.2. Let (X, [T.sub.u], [less than or equal to]) be an ordered intuitionistic Fuzzy smooth quasi uniform topological space, A [member of][[zeta].sup.X] Then [[PSI].sub.A] is lower (upper) (r, s) intuitionistic Fuzzy quasi uniform continuous function if and only if A is an intuitionistic Fuzzy smooth quasi uniform increasing (decreasing) open [F.sub.[sigma]] (closed [G.sub.[delta]]) set.

Proof. Proof is similar to Proposition 4.1.

Proposition 4.3. Let (X, [T.sub.u], [less than or equal to]) be an ordered intuitionistic Fuzzy smooth quasi uniform topological space. Then the following are equivalent:

(i) (X, [T.sub.u], [less than or equal to]) is an upper intuitionistic Fuzzy smooth quasi uniform basically disconnected space.

(ii) If g, h: X [right arrow] [R.sub.I](I), g is an lower (r, s) intuitionistic Fuzzy quasi uniform continuous function, h is an upper (r, s) intuitionistic Fuzzy quasi uniform continuous function and g [subset or equal to] h, then there exists an (r, s) intuitionistic Fuzzy quasi uniform continuous function f: (X, [T.sub.u], [less than or equal to] ) [right arrow] [R.sub.I](I) such that g [subset or equal to] f [subset or equal to] h.

(iii) If [[bar] A] is an intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set and B is an intuitionistic Fuzzy smooth quasi uniform decreasing open [F.sub.[sigma]] set such that B[subset or equal to]A, then there exists an (r, s) intuitionistic Fuzzy quasi uniform increasing continuous function f: (X, T, [less than or equal to]) [right arrow] [R.sub.I](I) such that B[subset or equal to][f.sup.- 1]([bar.[L.sup.I.sub.1]])[subset or equal to][f.sup.-1]([R.sup.I.sub.0])[subset or equal to] A.

Proof. (i) [right arrow] (ii) Define [A.sub.r] = [h.sup.-1]([L.sup.I.sub.r]) and [B.sub.r] = [g.sup.- 1]([bar.R.sup.I.sub.r]), for all r [member of] Q(Q is the set of all rationals). Clearly, [{[A.sub.r]}.sub.r [member of] Q] and [{[B.sub.r]}.sub.r [member of] Q] are monotone increasing families of an intuitionistic Fuzzy smooth quasi uniform decreasing open [F.sub.[sigma]] sets and intuitionistic Fuzzy smooth quasi uniform decreasing closed [G.sub.[delta]] sets of (X, [T.sub.u], [less than or equal to]). Moreover [A.sub.r][subset or equal to][B.sub.s] if r < s. By Proposition 3.4, there exists a monotone increasing family [{[C.sub.r]}.sub.r [member of] Q] of an intuitionistic Fuzzy smooth quasi uniform decreasing closed open [G.sub.[delta]][F.sub.[sigma]] sets of (X, [T.sub.u], [less than or equal to]) such that [IFSQD.sub.u]([A.sub.r]) [subset or equal to][C.sub.s] and [C.sub.r][subset or equal to][IFSQD.sup.0.sub.u]([B.sub.s]) whenever r < s (r, s [member of] Q). Letting [V.sub.t] = [[intersection].sub.r < t][bar.Cr] for t [member of] R, we define a monotone decreasing family {[V.sub.t]|t [member of] R}[subset or equal to][[zeta].sup.X] Moreover we have [IFSQI.sub.u]([V.sub.t]) [subset or equal to] IFSQ[I.sup.0.sub.u]([V.sub.s]) whenever s < t. We have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly, [[intersection].sub.t [member of] R][V.sub.t] = [0.sub.~]. Now define a function f: (X, [T.sub.u], [less than or equal to]) [right arrow] [R.sub.I](I) possessing required conditions. Let f(x)(t) = [V.sub.t](x), for all x [member of] X and t [member of] R. By the above discussion, it follows that f is well defined. To prove f is an (r, s) intuitionistic Fuzzy quasi uniform increasing continuous function. Observe that [ [union] .sub.s > t][V.sub.s] = [[union].sub.s > t] IFSQ[I.sup.0.sub.u]([V.sub.s]) and [[intersection].sub.s < t][V.sub.s] = [[intersection].sub.s < t] [IFSQI.sub.u]([V.sub.s]). Then [f.sup.-1]([R.sup.I.sub.t]) = [[union].sub.s > t][V.sub.s] = [[union].sub.s > t] IFSQ[I.sup.0.sub.u]([V.sub.s]) is an intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set and [f.sup.-1]([bar.[L.sup.I.sub.t]]) = [[intersection].sub.s < t] [V.sub.s] = [[intersection].sub.s < t] [IFSQI.sub.u]([V.sub.s]) is an intuitionistic Fuzzy smooth quasi uniform increasing closed [G.sub.[delta]] set. Therefore, f is an (r, s) intuitionistic Fuzzy quasi uniform increasing continuous function. To conclude the proof it remains to show that g[subset or equal to] f [subset or equal to] h. That is, [g.sup.-1]([bar.[L.sup.I.sub.t]])[subset or equal to][f.sup.- 1]([bar.[L.sup.I.sub.t]])[subset or equal to][h.sup.-1]([bar.[L.sup.I.sub.t]]) and [g.sup.-1]([R.sup.I.sub.t])[subset or equal to][f.sup.-1]([R.sup.I.sub.t])[subset or equal to][h.sup.-1]([R.sup.I.sub.t]) for each t [member of] R.

We have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, the condition (ii) is proved.

(ii) [right arrow] (iii)[bar.A] is an intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set and B is an intuitionistic Fuzzy smooth quasi uniform decreasing open [F.sub.[sigma]] set such that B [subset or equal to] A. Then, [[PSI].sub.B][subset or equal to][[PSI].sub.A], [[PSI].sub.B] and [[PSI].sub.A] lower and upper (r, s) intuitionistic Fuzzy quasi uniform continuous function respectively. Hence by (2), there exists an (r, s) intuitionistic Fuzzy quasi uniform increasing continuous function f: (X, [T.sub.u], [less than or equal to]) [right arrow] I[I.sub.I](I) such that [[PSI].sub.B][subset or equal to]f[subset or equal to][[PSI].sub.A]. Clearly, f(x) [member of] [0,1] for all x [member of] X and B = [[PSI].sup.-1.sub.B]([bar.[L.sup.I.1]])[subset or equal to][f.sup.-1]([bar.[L.sup.I.1]])[subset or equal to][f.sup.-1]([R.sup.I.sub.0])[subset or equal to][[PSI].sup.- 1.sub.A]-1([R.sup.I.sub.0]) = A. Therefore, B[subset or equal to][f.sup.-1]([bar.[L.sup.I.1]])[subset or equal to][f.sup.-1]([R.sub.0]I) [subset or equal to] A.

(iii) [??] (i) Since [f.sup.-1]([bar.[L.sup.I.1]]) and [f.sup.-1]([R.sup.I.sub.0]) are intuitionistic Fuzzy smooth quasi uniform decreasing closed [G.sub.[delta]] and intuitionistic Fuzzy smooth quasi uniform decreasing open [F.sub.[sigma]] sets by Proposition 3.3, (X, [T.sub.u], [less than or equal to]) is an upper intuitionistic Fuzzy smooth quasi uniform basically disconnected space.

Note 4.2. Let X be a non empty set and A [subset]X. Then an intuitionistic Fuzzy set [[chi].sup.*.sub.A] is of the form < x, [[chi].sub.A](x) , 1-[[chi].sub.A](x)> where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proposition 4.4. Let (X, [T.sub.u], [less than or equal to]) be an upper intuitionistic Fuzzy smooth quasi uniform basically disconnected space. Let A [subset]X be such that [[chi].sup.*.sub.A] is an intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] in (X, [T.sub.u], [less than or equal to]). Let f: (A, [T.sub.u]/A)[right arrow] I [I.sub.I](I) be an (r, s) intuitionistic Fuzzy quasi uniform increasing continuous function Then f has an (r, s) intuitionistic Fuzzy quasi uniform increasing continuous extension over (X, [T.sub.u], [less than or equal to]).

Proof. Let g, h: X [right arrow] I[I.sub.I](I) be such that g = f = h on A and g(x) = <0,1> = [0.sup.~], h(x) = <1,0> = [1.sup.~] if x [not member of] A. For every t [member of] R, We have,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [B.sub.t] is an intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] such that [B.sub.t]/A = [f.sup.-1]([R.sup.I.sub.t]) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [D.sub.t] is an intuitionistic Fuzzy smooth quasi uniform increasing open [F.sub.[sigma]] set such that [D.sub.t]/A = [f.sup.-1]([L.sup.I.sub.t]) Thus, g is an lower (r, s) intuitionistic Fuzzy quasi uniform continuous function and h is an upper (r, s) intuitionistic Fuzzy quasi uniform continuous function with g [subset or equal to] h. By Proposition 4.3, there is an (r, s) intuitionistic Fuzzy quasi uniform increasing continuous function F:X [right arrow] I[I.sub.I](I) such that g [subset or equal to] F [subset or equal to]h. Hence F [equivalent to]f on A.

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S. Padmapriy ([dagger]), M. K. Uma ([double dagger]) and E. Roja ([double dagger])

Department of Mathematics, Sri Sarada College for Women, Salem, Tamil Nadu, India

E-mail: priyasathi17@gmail.com
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Author: Printer friendly Cite/link Email Feedback Padmapriya, S.; Uma, M.K.; Roja, E. Scientia Magna Report 9INDI Mar 1, 2013 7130 Some remarks on Fuzzy Bairs spaces. Computing the number of integral points in 4-dimensional ball. Fuzzy algorithms Fuzzy logic Fuzzy systems Topological spaces