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Basically disconnectedness in soft L-fuzzy V spaces with reference to soft L-fuzzy BV open set.

[section]1. Introduction and preliminaries

The concept of fuzzy set was introduced by Zadeh [12]. Fuzzy sets have applications in many fields such as information [6] and control [5]. The thoery of fuzzy topological spaces was introduced and developed by Chang [3] and since then various notions in classical topology has been extended to fuzzy topological spaces. The concept of fuzzy basically disconnected space was introduced and studied in [7]. The concept of L-fuzzy normal spaces and Tietze extension theorem was introduced and studied in [10]. The concept of soft fuzzy topological space was introduced by Ismail U. Triyaki [8]. J. Tong [9] introduced the concept of B-set in topological space. The concept of fuzzy B-set was introduced by M.K. Uma, E. Roja and G. Balasubramanian [12]. In this paper, the new concepts of soft L-fuzzy topological space and soft L-fuzzy V space are introduced. In this connection, the concept of soft L-fuzzy BV basically disconnected space is studied. Besides giving some interesting properties, some characterizations are studied. Tietze extension theorem for a soft L-fuzzy BV basically disconnected space is established.

Definition 1.1. Let (X, T) be a topological space on fuzzy sets. A fuzzy set [lambda] of (X, T) is said to be

(i) fuzzy t-set if int[lambda] = intcl[lambda],

(ii) fuzzy B-set if [lambda] = [mu] [conjunction] [gamma]] where [mu] is fuzzy open and [gamma] is a fuzzy t-set.

Lemma 1.1. For a fuzzy set [lambda] of a fuzzy space X.

(i)1 - int[lambda] = cl(1 - [lambda]),

(ii)1 - cl[lambda] = int(1 - [lambda]).

Definition 1.2. Let X be a non-empty set. A soft fuzzy set(in short, SFS) A have the form A = ([lambda], M) where the function [lambda] : X [right arrow] I denotes the degree of membership and M is the subset of X. The set of all soft fuzzy set will be denoted by SF(X).

Definition 1.3. The relation [??] on SF(X) is given by ([mu], N) [??] ([lambda], M) [??] [mu](x) [less than or equal to] [lambda] (x), [for all]x [member of] X and M [subset or equal to] N:

Proposition 1.1. If ([[mu].sub.j], [N.sub.j]) [member of] SF(X), j [member of] J, then the family {([[mu].sub.j], [N.sub.j])|j [member of] J} has a meet, i.e. g.l.b. in (SF(X), [??]) denoted by [[??].sub.j [member of] J] ([[mu].sub.j], [N.sub.j]) and given by [[??].sub.j [member of] J] ([[mu].sub.j], [N.sub.j]) = ([mu], N) where

[mu](x) = [[conjunction].sub.j [member of] J][[mu].sub.j](x) [for all]x [member of] X

and

M = [intersection][M.sub.j] for j [member of] J.

Proposition 1.2. If ([[mu].sub.j], [N.sub.j]) [member of] SF(X), j [member of] J, then the family {([[mu].sub.j], [N.sub.j])|j [member of] J} has a join, i.e. l.u.b. in (SF(X), [??]) denoted by [[??].sub.j [member of] J] ([[mu].sub.j], [N.sub.j]) and given by [[??].sub.j [member of] J] ([[mu].sub.j], [N.sub.j]) = ([mu], N) where

[mu](x) = [[disjunction].sub.j [member of] J][[mu].sub.j](x), [for all]x [member of] X

and

M = [union][M.sub.j] for j [member of] J.

Definition 1.4. Let X be a non-empty set and the soft fuzzy sets A and C are in the form A = ([lambda], M) and C = ([mu], N). Then

(i) A [??] C if and only if [lambda](x) [less than or equal to] [mu](x) and M [subset or equal to] N for x [member of] X,

(ii) A = C if and only if A [??] C and C [??] A,

(iii) A [??] C = ([lambda], M) [??] ([mu], N) = ([lambda](x) [conjunction] [mu](x), M [intersection] N) for all x [member of] X,

(iv) A [??] C = ([lambda], M) [??] ([mu], N) = ([lambda] (x) [disjunction] [mu](x), M [union] N) for all x [member of] X.

Definition 1.5. For ([mu], N) [member of] SF(X) the soft fuzzy set ([mu], N)' = (1 - [mu], X \ N) is called the complement of ([mu], N):

Remark 1.1. (1 - [mu], X/N) = (1, X) - ([mu], N).

Proof. (1, X) - ([mu], N) = (1, X) [??] ([mu], N)' = (1, X) [??] (1 - [mu], X/N) = (1 - [mu], X/N).

Definition 1.6. Let S be a set. A set T [subset or equal to] SF(X) is called an SF-topology on X if

SFT1 (0, 0) [member of] T and (1, X) 2 T,

SFT2 ([[mu].sub.j], [N.sub.j]) [member of] T, j = 1, 2, ..., n [??] [[??].sup.n.sub.j=1] ([[mu].sub.j], [N.sub.j]) [member of] T,

SFT3 ([[mu].sub.j], [N.sub.j]) [member of] T, j [member of] J [[??].sub.j [member of] J] ([[mu.sub.]j], [N.sub.j]) [member of] T.

As usual, the elements of T are called open, and those of T' = {([mu], N)|([mu], N)' [member of] T} closed. If T is an SF-topology on X we call the pair (X, T) an SF-topological space(in short, SFTS).

Definition 1.7. The closure of a soft fuzzy set ([mu], N) will be denoted by [bar.([mu], N)]: It is given by

[bar.([mu], N)] = [??]{(v, L)|([mu], N) [??] (v, L) [member of] T'}.

Likewise the interior is given by

[([mu], N).sup.[omicron]] = [??]{(v, L)|(v, L) [member of] T, (v, L) [??] ([mu], N)}.

Note 1.1. (i) The soft fuzzy closure [bar.([mu], N)] is denoted by SFcl([mu], N).

(ii) The soft fuzzy interior [([mu], N).sup.[omicron]] is denoted by SFint([mu], N).

Proposition 1.3. Let [phi] : X [right arrow] Y be a point function.

(i) The mapping [[phi].sup.[??]] from SF(X) to SF(Y) corresponding to the image operator of the difunction (f, F) is given by

[[phi].sup.[??]]([mu], N) = (v, L) where v(y) = sup{[mu](x)|y = [phi](x)} and L = {[phi](x)|x [member of] N and v([phi](x)) = [mu](x)}.

(ii) The mapping [[phi].sup.[??]] from SF(X) to SF(Y) corresponding to the inverse image of the difunction (f, F) is given by

[[phi].sup.[??]]((v, L) = (v [omicron] [phi], [[phi].sup.-1][L]).

Definition 1.8. Let (X, T) be a fuzzy topological space and let [lambda] be a fuzzy set in (X, T). [lambda] is called fuzzy [G.sub.[delta]] if [lambda] = [[conjunction].sup.[infinity].sub.l=1] [[lambda].sub.i] where each [[lambda].sub.i] [member of] T, i [member of] I.

Definition 1.9. Let (X, T) be a fuzzy topological space and let [lambda] be a fuzzy set in (X, T). [lambda] is called fuzzy [F.sub.[sigma]] if [lambda] = [[disjunction].sup.[infinity].subi=1] [[lambda].sub.i] where each [[bar.[lambda]].sub.i] [member of] T, i [member of] I.

Definition 1.10. 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]] set is fuzzy open.

Definition 1.11. An intutionistic fuzzy set U of an intutionistic fuzzy topological space (X, T) is said to be an intutionistic fuzzy compact relative to X if for every family {[U.sub.j]: j [member of] J} of intutionistic fuzzy open sets in X such that U [subset or equal to] [[union].sub.j [member of] ]J [U.sub.j], there is a finite subfamily {[U.sub.j]: j = 1, 2, ..., n} of intutionistic fuzzy open sets such that U [subset or equal to] [[union].sup.n.sub.j=1] [U.sub.j].

Definition 1.12. The L-fuzzy real line R(L) is the set of all monotone dsecreasing elements [lambda] [member of] [L.sup.R] satisfying [disjunction]{[lambda](t): t [member of] R} = 1 and [conjunction]{[lambda](t): t [member of] R} = 0, after the identification of [lambda], [mu] 2 [L.sup.R] iff [lambda](t+) = [conjunction]{[lambda](s): s < t} and [lambda](t-) = [disjunction] {[lambda](s): s > t}. The natural L-fuzzy topology on R(L) is generated from the basis {[L.sub.t], [R.sub.t]: t [member of] R}, where [L.sub.t][[lambda]] = [lambda](t-)' and [R.sub.t][[lambda]] = [lambda](t+): A partial order on R(L) is defined by [[lambda]] [less than or equal to] [[mu]] iff [lambda](t-) [less than or equal to] [mu](t-) and [lambda](t+) [less than or equal to] [mu](t+) for all t [member of] R.

Definition 1.13. The L-fuzzy unit interval I(L) is a subset of R(L) such that [[lambda]] [member of] I(L) if [lambda](t) = 1 for t < 0 and [lambda](t) = 0 for t > 1. It is equipped with the subspace L-fuzzy topology.

[section]2. Soft L-fuzzy topological space

In this paper, (L, [??],') stands for an infinitely distributive lattice with an order reversing involution. Such a lattice being complete has a least element 0 and a greatest element 1. A soft L-fuzzy set in X is an element of the set L x L of all functions from X to L x L i.e. ([lambda], M) : X [right arrow] L x L be such that ([lambda], M)(x) = ([lambda](x),M(x)) = ([lambda](x), [[chi].sub.M](x)) for all x [member of] X.

A soft L-fuzzy topology on X is a subset T of L x L such that

(i) ([0.sub.X], [0.sub.X]), ([1.sub.X], [1.sub.X]) [member of] T,

(ii) ([[mu].sub.j], [N.sub.j]) [member of] T, j = 1, 2, ... , n [??] [[??].sup.n.sub.j=1] ([[mu].sub.j], [N.sub.j]) [member of] T,

(iii) ([[mu].sub.j], [N.sub.j]) [member of] T, j [member of] J [??} [[intersection].sub.j[member of]J] ([[mu].sub.j], [N.sub.j]) [member of] T.

A set X with a soft L-fuzzy topology on it is called a soft L-fuzzy topological space. The members of T are called the soft L-fuzzy open sets in the soft L-fuzzy topological space.

A soft L-fuzzy set ([lambda], M) in X is called a soft L-fuzzy closed if ([lambda], M)' is the soft L-fuzzy open where ([lambda], M)' = (1 - [lambda], 1 M) = ([1.sub.X], [1.sub.X]) - ([lambda], M).

If ([lambda], M), ([mu], N): X [right arrow] L x L, we define ([lambda], M) [??] ([mu], N), [??] [lambda](x) [less than or equal to] [mu](x) and M(x) [less than or equal to] N(x) for all x [member of] X.

A function f from a soft L-fuzzy topological space X to a soft L-fuzzy topological space Y is called soft L-fuzzy continuous if [f.sup.-1]([mu], N) is soft L-fuzzy open in (X, T), for each soft L-fuzzy open set in (Y, S).

If (X, T) is a soft L-fuzzy topological space and A [subset or equal to] X then (A, [T.sub.A]) is a soft L-fuzzy topological space which is called a soft L-fuzzy subspace of (X, T) where

[T.sub.A] = {([lambda], M)/A : ([lambda], M) is a soft L-fuzzy set in X}.

The soft L-fuzzy real line R(L x L) is the set of all monotone decreasing soft L-fuzzy set ([lambda], M) : R(L x L) - L x L satisfying

[??]{([lambda], M)(t)/t [member of] R} = [??]}([lambda], [[chi].sub.M])(t)/t [member of] R} = ([1.sub.X], [1.sub.X]),

[??]{([lambda], M)(t)/t 2 R} = [??]{([lambda], [[chi].sub.M])(t)/t [member of] R} = ([0.sub.X], [0.sub.X]), after the identification of ([lambda], M), ([mu], N) : R(L x L) [right arrow] L x L if for every t [member of] R iff

([lambda], M)(t-) = ([mu], N)(t-),

and

([lambda], M)(t+) = ([mu], N)(t+),

where ([lambda], M)(t-) = [[??].sub.s] < t([lambda], M)(s) and ([lambda], M)(t+) = [[??].sub.s] > t([lambda], M)(s): The natural soft L-fuzzy topology on R(L x L) by taking a sub-basis {[L.sub.t], [R.sub.t]/t [member of] R} where

[L.sub.t][[lambda], M] = ([lambda], M)(t-)', [R.sub.t][[lambda], M] = ([lambda], M)(t+).

This topology is called the soft L-fuzzy topology for R(L x L). {[L.sub.t]/t [member of] R} and {[R.sub.t]/t [member of] R} are called the left and right hand soft L-fuzzy topology respectively.

A partial order on R(L x L) is defined by [[lambda], M] [??] [[mu], N], ([lambda], M)(t-) [??] ([mu], N)(t-) and ([lambda], M)(t+) [??] ([mu], N)(t+) for all t [member of] R. The soft L-fuzzy unit interval I(L x L) is a subset of R(L x L) such that [[lambda], M] [member of] I(L x L) if

([lambda], M)(t) = ([1.sub.X], [1.sub.X]) for t < 0,

and

([lambda], M)(t) = ([0.sub.X], [0.sub.X]) for t > 1:

It is equipped with the subspace soft L-fuzzy topology.

Definition 2.1. Let (X, T) be soft L-fuzzy topological space. For any soft L-fuzzy set ([lambda], M) on X, the soft L-fuzzy closure of ([lambda], M) and the soft L-fuzzy interior of ([lambda], M) are defined as follows:

SLFcl([lambda], M) = [??]{([mu], N) : ([lambda], M) [??] ([mu], N), ([mu], N) is a soft L-fuzzy closed set in X},

SLFint([lambda], M) = tf([mu], N) : ([lambda], M) w ([mu], N), ([mu], N) is a soft L-fuzzy open set in Xg:

Definition 2.2. Let T be a soft L-fuzzy topology on X. Then (X, T) is called soft L-fuzzy non-compact if [[??].sub.i [member of] I] ([[lambda].sub.i], [M.sub.i]) = (1, X), ([[lambda].sub.i], [M.sub.i]) be soft L-fuzzy set in T, i [member of] I, there is a finite subset J of I with [[??].sub.j [member of] J] ([[lambda].sub.j], [M.sub.j]) [not equal to] (1,X):

Definition 2.3. Let (X, T) be a soft fuzzy L-fuzzy topological space. Let ([lambda], M) be any soft L-fuzzy set. Then ([lambda], M) is said to be soft L-fuzzy compact set if every family {([[lambda].sub.j], [M.sub.j]) : j [member of] J} of soft L-fuzzy open sets in X such that ([lambda], M) [??] [[??].sub.j [member of] J] ([[lambda].sub.j], [M.sub.j]), there is a finite subfamily i [member of] I, there is a finite subfamily f([[lambda].sub.j], [M.sub.j]) : j = 1, 2, ..., n} of soft L-fuzzy open sets such that ([lambda], M) [??] [[??].sub.j [member of] J] ([lambda]j ,Mj):

Definition 2.4. Let (X, T) be a soft L-fuzzy topological space. Let ([lambda], M) be any soft L-fuzzy set. Then ([lambda], M) is said to be a soft L-fuzzy t-open set if SLFint([lambda], M) = SLFint(SLFcl([lambda], M)).

Definition 2.5. Let (X, T) be a soft L-fuzzy topological space. Let ([lambda], M) be any soft L-fuzzy set. Then ([lambda], M) is said to be a soft L-fuzzy B open set (in short, SLFBOS) if ([lambda], M) = ([mu], N) [??] ([gamma],L)) where ([mu], N) is a soft L-fuzzy open set and ([gamma],L) is a soft L- fuzzy t-open set. The complement of soft L-fuzzy B-open set is a soft L-fuzzy B closed set (in short, SLFBCS).

[section]3. Soft L-fuzzy BV basically disconnected space

Definition 3.1. Let (X, T) be a soft L-fuzzy topological space and a soft L-fuzzy non-compact spaces. Let C be a collection of all soft L-fuzzy set which are both soft L-fuzzy closed and soft L-fuzzy compact sets in (X, T). Let

[([gamma],L).sup.-] = f([lambda], M) [member of] C: ([lambda], M) [??] ([gamma],L) [not equal to] ([0.sub.X], [0.sub.X]), ([gamma],L) is a soft L-fuzzy open set},

[([delta], P).sup.+] = f([lambda], M) [member of] C: ([lambda], M) [??] ([delta], P) = ([0.sub.X], [0.sub.X]),

([delta], P) is a soft L-fuzzy compact set in (X, T)}.

Then the collection V = {([lambda], M) : ([lambda], M) [member of] [([gamma],L).sup.-]} [??] {([mu], N) : ([mu], N) [member of] [([delta], P).sup.+]} is said to be soft L-fuzzy V structure on (X, T) and the pair (X, V) is said to be soft L-fuzzy V space.

Notation 3.1. Each member of soft L-fuzzy V space is a soft L-fuzzy Vopen set. The complement of soft L-fuzzy Vopen set is a soft L-fuzzy Vclosed set.

Definition 3.2. Let (X, V) be a soft L-fuzzy V space. For any soft L-fuzzy set ([lambda], M) on X, the soft L-fuzzy V closure of ([lambda], M) and the soft L-fuzzy V interior of ([lambda], M) are defined as follows:

SLFVcl([lambda], M) = [??]{([mu], N) : ([lambda], M) [??] ([mu], N), ([mu], N) is a soft L-fuzzy V closed set in Xg,

SLFVint([lambda], M) = [??]{([mu], N) : ([lambda], M) [??] ([mu], N), ([mu], N) is a soft L-fuzzy V open set in X}.

Definition 3.3. Let (X, V) be a soft L-fuzzy V space. Let ([lambda], M) be any soft L-fuzzy set in X. Then ([lambda], M) is said to be a soft L-fuzzy tV open set if SLFVint([lambda], M) = SLFVint(SLFVcl([lambda], M)).

Definition 3.4. Let (X, V) be a soft L-fuzzy V space. Let ([lambda], M) be any soft L-fuzzy set in X. Then ([lambda], M) is said to be a soft L-fuzzy BV open set (in short, SLFBVOS) if ([lambda], M) = ([mu], N) [??] ([gamma],L)) where ([mu], N) is a soft L-fuzzy V open set and ([gamma], L) is a soft L- fuzzy tV open set. The complement of soft L-fuzzy BV open set is a soft L-fuzzy BV closed set (in short, SLFBVCS).

Definition 3.5. Let (X, V) be a soft L-fuzzy V space. A soft L-fuzzy set ([lambda], M) is said to be soft L-fuzzy V[G.sub.[delta]] set (in short, SLFV[G.sub.[delta]]) if ([lambda], M) = [[??].sup.[infinity].sub.i=1] ([[lambda].sub.i], [M.sub.i]), where each ([[lambda].sub.i], [M.sub.i]) [member of] V. The complement of soft L-fuzzy V[G.sub.[delta]] set is said to be soft L-fuzzy V[F.sub.[sigma]] (in short, SLFV[F.sub.[sigma]]) set.

Remark 3.1. Let (X, V) be a soft L-fuzzy V space. For any soft L-fuzzy set ([lambda], M),

(i) which is both soft L-fuzzy BV open and soft L-fuzzy V[F.sub.[sigma]]. Then ([lambda], M) is said to be soft L-fuzzy BV open [F.sub.[sigma]] (in short, SLFBVO[F.sub.[sigma]]).

(ii) which is both soft L-fuzzy BV closed and soft L-fuzzy V[G.sub.[delta]]. Then ([lambda], M) is said to be soft L-fuzzy BV closed [G.sub.[delta]] (in short, SLFBVC[G.sub.[delta]]).

(iii) which is both soft L-fuzzy BV open [F.sub.[sigma]] and soft L-fuzzy BV closed [G.sub.[delta]]. Then ([lambda], M) is said to be soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] (in short, SLFBVCOGF).

Definition 3.6. Let (X, V) be a soft L-fuzzy V space. For any soft L-fuzzy set ([lambda], M) in X, the soft L-fuzzy BV closure of ([lambda], M) and the soft L-fuzzy BV interior of ([lambda], M) are defined as follows:

SLFBVcl([lambda], M) = [??]{([mu], N) : ([lambda], M) [??] ([mu], N), ([mu], N) is a soft L-fuzzy BV closed },

SLFBVint([lambda], M) = [??]{([mu], N) : ([lambda], M) [??] ([mu], N), ([mu], N) is a soft L-fuzzy BV open set }.

Proposition 3.1. Let (X, V) be a soft L-fuzzy V space. For any soft L-fuzzy set ([lambda], M) in X, the following statements are valid.

(i) SLFBVint([lambda], M) [??] ([lambda], M) [??] SLFBVcl([lambda], M),

(ii) (SLFBVint([lambda], M))' = SLFBVcl([lambda], M)',

(iii) (SLFBVcl([lambda], M))' = SLFBVint([lambda], M)'.

Definition 3.7. Let (X, V) be a soft L-fuzzy V space. Then (X, V) is said to be soft L-fuzzy BV basically disconnected if the soft L-fuzzy BV closure of every soft L-fuzzy BV open [F.sub.[sigma]] set is a soft L-fuzzy BV open set.

Proposition 3.2. Let (X, V) be a soft L-fuzzy V space, the following conditions are equivalent:

(i) (X, V) is a soft L-fuzzy BV basically disconnected space,

(ii) For each soft L-fuzzy BV closed [G.sub.[delta]] set ([lambda], M), SLFBVint([lambda], M) is soft L-fuzzy BV closed,

(iii) For each soft L-fuzzy BV open [F.sub.[sigma]] set ([lambda], M),

SLFBVcl([lambda], M) + SLFBVcl(SLFBVcl([lambda], M))' = ([1.sub.X], [1.sub.X]),

(iv) For every pair of soft L-fuzzy BV open [F.sub.[sigma]] sets ([lambda], M) and ([mu], N) with SLFBV cl([lambda], M) + ([mu], N) = ([1.sub.X], [1.sub.X]), we have SLFBVcl([lambda], M) + SLFBVcl([mu], N) = ([1.sub.X], [1.sub.X]).

Proof. (i)) [??] (ii). Let ([lambda], M) be any soft L-fuzzy BV closed [G.sub.[delta]] set in X. Then ([lambda], M)' is soft L-fuzzy BV open [F.sub.[sigma]]. Now,

SLFBVcl([lambda], M)' = (SLFBVint([lambda], M))'.

By (i), SLFBVcl([lambda], M)' is soft L-fuzzy BV open. Then SLFBVint([lambda], M) is soft L-fuzzy BV closed.

(ii)) [??] (iii). Let ([lambda], M) be any soft L-fuzzy BV open [F.sub.[sigma]] set. Then

SLFBVcl([lambda], M) + SLFBVcl(SLFBVcl([lambda], M))' (1) = SLFBVcl([lambda], M) + SLFBVcl(SLFBVint([lambda], M)'

Since ([lambda], M) is a soft L-fuzzy BV open [F.sub.[sigma]] set. Now, ([lambda], M)' is a soft L-fuzzy BV closed [G.sub.[delta]] set. Hence by (ii), SLFBVint([lambda], M)' is soft L-fuzzy BV closed. Therefore, by (1)

SLFBVcl([lambda], M) + SLFBVcl(SLFBVcl([lambda], M))' = SLFBVcl([lambda], M) + SLFBVcl(SLFBVint([lambda], M)') = SLFBVcl([lambda], M) + SLFBVint([lambda], M)' = SLFBVcl([lambda], M) + (SLFBVcl([lambda], M))' = SLFBVcl([lambda], M) + ([1.sub.X], [1.sub.X]) - SLFBVcl([lambda], M) = ([1.sub.X], [1.sub.X])

Therefore, SLFBVcl([lambda], M) + SLFBVcl(SLFBVcl([lambda], M))' = ([1.sub.X], [1.sub.X]).

(iii) [??] (iv). Let ([lambda], M) and ([mu], N) be soft L-fuzzy BV open [F.sub.[sigma]] sets with SLFBVcl([lambda], M) + ([mu], N) = ([1.sub.X], [1.sub.X]). (2)

By (iii),

([1.sub.X], [1.sub.X]) = SLFBVcl([lambda], M) + SLFBVcl(SLFBVcl([lambda], M))' = SLFBVcl([lambda], M) + SLFBVcl(([1.sub.X], [1.sub.X]) - SLFBVcl([lambda], M)) = SLFBVcl([lambda], M) + SLFBVcl([mu], N):

Therefore, SLFBVcl([lambda], M) + SLFBVcl([mu], N) = ([1.sub.X], [1.sub.X]).

(iv)) [??] (i). Let ([lambda], M) be a soft L-fuzzy BV open [F.sub.[sigma]] set. Put ([mu], N) = (SLFBVcl([lambda], M))' = ([1.sub.X], [1.sub.X]) - SLFBmathcalV cl([lambda], M). Then SLFBV cl([lambda], M) + ([mu], N) = ([1.sub.X], [1.sub.X]). Therefore by (iv), SLFBVcl([lambda], M) + SLFBVcl([mu], N) = ([1.sub.X], [1.sub.X]). This implies that SLFBVcl([lambda], M) is soft L-fuzzy BV open and so (X, V) is soft L-fuzzy BV basically disconnected.

Proposition 3.3. Let (X, V) be a soft L-fuzzy V space. Then (X, V) is soft L-fuzzy BV basically disconnected if and only if for all soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] sets ([lambda], M) and ([mu], N) such that ([lambda], M) [??] ([mu], N), SLFBVcl([lambda], M) [??] SLFBV int([mu], N).

Proof. Let ([lambda], M) and ([mu], N) be any soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] sets with ([lambda], M) [??] ([mu], N). By (ii) of Proposition 3.2, SLFBVint([mu], N) is soft L-fuzzy BV closed. Since ([lambda], M) is soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]], SLFBVcl([lambda], M) [??] SLFBVint([mu], N).

Conversely, let ([mu], N) be any soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] then SLFBVint([mu], N) is soft L-fuzzy BV open [F.sub.[sigma]] in X and SLFBVint([mu], N) [??] ([mu], N). Therefore by assumption, SLFBVcl(SLFBVint([mu], N)) [??] SLFBVint([mu], N). This implies that SLFBVint([mu], N) is soft L-fuzzy BV closed [G.sub.[delta]]. Hence by (ii) of Proposition 3.2, it follows that (X, V) is soft L-fuzzy BV basically disconnected.

Remark 3.2. Let (X, V) be a soft L-fuzzy BV basically disconnected space. Let {([[lambda].sub.i], [M.sub.i]), ([[mu].sub.i], [N.sub.i])'/i [member of] N} be collection such that ([[lambda].sub.i], [M.sub.i])'s and ([[mu].sub.i], [N.sub.i])'s are soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] sets and let ([lambda], M) and ([mu], N) be soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] sets. If

([[lambda].sub.i], [M.sub.i]) [??] ([lambda], M) [??} ([[mu].sub.j], [N.sub.j]) and ([[lambda].sub.i], [M.sub.i]) [??] ([mu], N) [??] ([[mu].sub.j], [N.sub.j]),

for all i, j [member of] N, then there exists a soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] set ([gamma],L) such that SLFBVcl ([[lambda].sub.i], [M.sub.i]) [??] ([gamma],L) [??] SLFBVint([[mu].sub.j], [N.sub.j]) for all i, j [member of] N.

Proof. By Proposition 3.3, SLFBV cl([[lambda].sub.i], [M.sub.i]) [??] SLFBVcl([lambda], M) u SLFBVint([mu], N) [??] SLFBVint([[mu].sub.j], [N.sub.j]) for all i, j ([[mu].sub.j], [N.sub.j]) N. Therefore, ([gamma],L) = SLFBVcl([lambda], M) u SLFBVint([mu], N) is a soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] set satisfying the required conditions.

Proposition 3.4. Let (X, V) be a soft L-fuzzy BV basically disconnected space. Let [{[[lambda].sub.l], [M.sub.l]}.sub.l [member of] Q] and [{[[mu].sub.l], [N.sub.l]}.sub.l [member of] Q] be monotone increasing collections of soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] sets of (X, V) and suppose that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] whenever [q.sub.1] < [q.sub.2] (Q is the set of all rational numbers). Then there exists a monotone increasing collection [{[gamma]l, [L.sub.l]}.sub.l [member of] Q] of soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] sets of (X, V) such that SLFBVcl[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a subset of L x L in X such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By Proposition 3.3, the countable collections {SLFBVcl([[lambda].sub.q], [M.sub.q])} and {SLFBVint([[mu].sub.q], [N.sub.q])} satisfy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By Remark 3.2, there exists a soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] set ([[delta].sub.1], [P.sub.1]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we get ([S.sub.2]).

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.sub.j](i, j < n) as well as

([[lambda].sub.q], [M.sub.q]) [??] SLFBVcl([PSI]) [??] ([[mu].sub.q'], [N.sub.q']),

and

([[lambda].sub.q], [M.sub.q]) [??] SLFBVint([PHI]) [??] ([[mu].sub.q'], [N.sub.q']),

whenever q < [q.sub.n] < q'. This shows that the countable collections [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] together with [PSI] and [PHI] fulfil the conditions of Remark 3.2. Hence, there exists soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] set ([[delta].sub.n], [P.sub.n]) such that SLFBVcl([[delta].sub.n], [P.sub.n]) [??] ([[mu].sub.q], [N.sub.q]) if [q.sub.n] < q, ([[lambda].sub.q], [M.sub.q]) [??] SLFBVint([[delta].sub.n], [P.sub.n]) if q < [q.sub.n], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], SLFBVcl([[delta].sub.n], [P.sub.n]) [??] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now, setting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we obtain the soft L-fuzzy sets [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.

[section]4. Properties and characterizations of SLFBV basically disconnected spaces

Definition 4.1. Let (X, V) be a soft L-fuzzy V space. A function f : X [right arrow] R(L x L) is called lower (upper) soft L-fuzzy BV continuous if [f.sup.-1]([R.sub.t])([f.sup.-1]([L.sub.t])) is soft L-fuzzy BV open [F.sub.[sigma]] (soft L-fuzzy BV open [F.sub.[sigma]]/soft L-fuzzy BV closed [G.sub.[delta]]), for each t [member of] R.

Proposition 4.1. Let (X, V) be a soft L-fuzzy V space. For any soft L-fuzzy set ([lambda], M) in X and let f : X [right arrow][right arrow] R(L x L) be such that f(x)(t) =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for all x [member of] X and t [member of] R. Then f is lower (upper) soft L-fuzzy BV continuous iff ([lambda], M) is soft L-fuzzy BV open [F.sub.[sigma]](soft L-fuzzy BV open [F.sub.[sigma]]/soft L-fuzzy BV closed [G.sub.[delta]]).

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

implies that f is lower soft L-fuzzy BV continuous iff ([lambda], M) is soft L-fuzzy BV open [F.sub.[sigma]].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

implies that f is upper soft L-fuzzy BV continuous iff ([lambda], M) is soft L-fuzzy BV closed [G.sub.[delta]].

Definition 4.2. The soft L-fuzzy characteristic function of a soft L-fuzzy set ([lambda], M) in X is a map [[chi].sub.([lambda], M)] : X [right arrow] L x L defined by

[[chi].sub.([lambda], M)](x) = ([lambda], M)(x) = ([lambda](x), [[chi].sub.M](x)),

for each x [member of] X.

Proposition 4.2. Let (X, V) be a soft L-fuzzy V space. Let ([lambda], M) be any soft L-fuzzy set in X. Then [[chi].sub.([lambda], M)] is lower (upper) soft L-fuzzy BV continuous iff ([lambda], M) is soft L-fuzzy BV open [F.sub.[sigma]](soft L-fuzzy BV open [F.sub.[sigma]]/soft L-fuzzy BV closed [G.sub.[delta]]).

Proof. The proof follows from Definition 4.2 and Proposition 4.1.

Definition 4.3. Let (X, V) be a soft L-fuzzy V space. A function f : (X, V) [right arrow] R(L x L) is said to be strongly soft L-fuzzy BV continuous if [f.sup.-1]([R.sub.t]) is soft L-fuzzy BV open [F.sub.[sigma]]/ soft L-fuzzy BV closed [G.sub.[delta]] and [f.sup.-1]([L.sup.t]) is soft L-fuzzy BV open [F.sub.[sigma]]/soft L-fuzzy BV closed [G.sub.[delta]] set for each t [member of] R.

Notation 4.1. The collection of all strongly soft L-fuzzy BV continuous functions in soft L-fuzzy V space with values R(L x L) is denoted by [SC.sub.BV].

Proposition 4.3. Let (X, V) be a soft L-fuzzy V space. Then the following conditions are equivalent:

(i) (X, V) is a soft L-fuzzy BV basically disconnected space,

(ii) If g, h : X [right arrow] R(L x L) where g is lower soft L-fuzzy BV continuous, h is upper soft L-fuzzy BV continuous, then there exists f [member of] [SC.sub.BV](X, V) such that g [??] f [??] h,

(iii) If ([lambda], M)', ([mu], N) are soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] sets such that ([mu], N) [??] ([lambda], M), then there exists strongly soft L-fuzzy BV continuous functions f : X [right arrow] R(L x L) such that ([mu], N) [??] ([L.sub.1])'f v [R.sub.0]f [??] ([lambda], M).

Proof. (i) [??] (ii). Define ([[xi].sub.k], [E.sub.k]) = [L.sub.k]h and ([[eta].sub.k], [C.sub.k]) = [R.sub.k]' g, k [member of] Q. Thus we have two monotone increasing families of soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] sets of (X, V). Moreover ([[xi].sub.k], [E.sub.k]) [??] ([[eta].sub.s], [C.sub.s]) if k < s. By Proposition 3.4, there exists a monotone increasing family [{([v.sub.k], [F.sub.k])}.sub.k [member of] Q] of soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] sets of (X, V) sets such that SLFBVcl([[xi].sub.k], [E.sub.k]) [??] ([v.sub.s], [F.sub.s]) and ([v.sub.k], [F.sub.k]) [??] SLFBVint([[eta].sub.s], [C.sub.s]) whenever k < s. Letting ([[phi].sub.t], [D.sub.t]) = [[??].sub.k < t]([v.sub.k], [F.sub.k])' for all t [member of] R, we define a monotone decreasing family {([[phi].sub.t], [D.sub.t]/t [member of] R} is a subset of L x L. Moreover, we have SLFBVcl([[phi].sub.t], [D.sub.t]) [??] SLFBVint([[phi].sub.s], [D.sub.s]) whenever s < t. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly, [[??].sub.t [member of] R]([[phi].sub.t], [D.sub.t]) = ([0.sub.X], [0.sub.X]). Now define a function f : X [right arrow] R(L x L) possessing the required properties. Let f(x)(t) = ([[phi].sub.t], [D.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 strongly soft L-fuzzy BV continuous. Observe that

[[??].sub.s > t]([[phi].sub.s], [D.sub.s]) = [[??].sub.s > t]SLFBVint([[phi].sub.s], [D.sub.s]),

and

[[??].sub.s < t]([[phi].sub.s], [D.sub.s]) = [[??].sub.s < t]SLFBVcl([[phi.sub.]s], [D.sub.s]).

Then [f.sup.-1]([R.sub.t]) = [R.sub.t] [omicron] f = [R.sub.t]([[phi].sub.t], [D.sub.t])(x) = [[??].sub.s > t]([[phi].sub.s], [D.sub.s]) = [[??].sub.s > t]SLFBVint([[phi].sub.s], [D.sub.s]) is soft Lfuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]]. And [f.sup.-1]([L'.sup.t]) = [[??].sub.s < t]([[phi].sub.s], [D.sub.s]) = [[??].sub.s < t]SLFBVcl([[phi].sub.s], [D.sub.s]) is soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]]. Therefore, f is strongly soft L-fuzzy BV continuous. To conclude the proof it remains to show that g [??] f [??] h. That is, [g.sup.-1]([L'.sub.t]) [??] [f.sup.-1]([L'.sup.t]) [??] [h.sup.-1]([L'.sub.t]) and [g.sup.-1]([R.sub.t]) [??] [f.sub.-1]([R.sub.t]) [??] [h.sub.-1]([R.sub.t]) for each t [member of] R. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, (ii) is proved.

(ii) [??] (iii). Suppose that ([lambda], M) is soft L-fuzzy BV closed [G.sub.[delta]] and ([mu], N) is soft L-fuzzy BV open [F.sub.[sigma]] such that ([mu], N) [??] ([lambda], M). Then [[chi].sub.([mu], N)] [??] [[chi].sub.([lambda], M)], where [[chi].sub.([mu], N)], [[chi].sub.([lambda], M)] are lower and upper soft L-fuzzy BV continuous functions, respectively. Hence by (ii), there exists a strong soft L-fuzzy BV continuous function f : X [right arrow] R(L x L) such that [[chi].sub.([mu], N)] [??] f [?? [[chi].sub.([lambda], M)]. Clearly, f(x) [member of] R(L x L) for all x [member of] X and ([mu], N) = [L'.sub.1][[chi].sub.([mu], N)] [??] [L'.sub.1] f [??] [R.sub.0]f [??} [R.sub.0] [[chi].sub.([lambda], M)] = ([lambda], M). Therefore, ([mu], N) [??] [L'.sub.1] f [??] [R.sub.0] f [??] ([lambda], M).

(iii))(i). [L'.sub.1] f and [R.sub.0] f are soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] sets. By Proposition 3.3, (X, V) is a soft L-fuzzy BV basically disconnected space.

[section]5. Tietze extension theorem

Definition 5.1. Let (X, V) be a soft L-fuzzy V space and A [subset or equal to] X then (A, V=A) is a soft L-fuzzy V space which is called a soft L-fuzzy V subspace of (X, V) where V=A = {([lambda], M)=A : ([lambda], M) [member of] V}.

Remark 5.1. Let X be a non-empty set and let A 1/2 X. Then the characteristic function * of A is a map [[chi].sup.*.sub.A] A = ([[chi].sub.A], [[chi].sub.A]) : X [right arrow] {([1.sub.X], [1.sub.X]), ([0.sub.X], [0.sub.X])g is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proposition 5.1. Let (X, V) be a soft L-fuzzy BV basically disconnected space and let A [subset] X be such that [[chi].sup.*.sub.A] A is a soft L-fuzzy BV open [F.sub.[sigma]] set in X. Let f : (A, V/A) [right arrow] I(L x L) be strong soft L-fuzzy BV continuous. Then f has a strong soft L-fuzzy BV continuous extension over (X, V).

Proof. Let g, h : X [right arrow] R(L x L) be such that g = f = h on A and g(x) = ([0.sub.X], [0.sup.X]]), h(x) = ([1.sub.X], [1.sub.X]) if x [not member of] A, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where ([[mu].sub.t], [N.sub.t]) is soft L-fuzzy BV open [F.sub.[sigma]] and is such that ([[mu].sub.t], [N.sub.t])/A = [R.sub.t]f and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where ([[lambda].sub.t], [M.sub.t]) is soft L-fuzzy BV closed open [G.sub.[delta]][F.sub.[sigma]] and is such that ([[lambda].sub.t], [M.sub.t])/A = [L.sub.t]f. Thus, g is lower soft L-fuzzy BV continuous and h is upper soft L-fuzzy BV continuous with g [??] h. By Proposition 4.3, there is a strong soft L-fuzzy BV continuous function F : X [right arrow] I(L x L) such that g [??] F [??] h. Hence F [congruent to] f on A.

References

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[3] C. L. Chang, Fuzzy Topological Spaces, J. Math. Anal. Appl., 24(1968), 182-190.

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[6] M. Sugeno, An Introductory Survey of Fuzzy Control, Information Sciences, 36(1985), 59-83.

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[8] Ismail U. Triyaki, Fuzzy Sets Over The Poset I, Haccttepe J. of Math. and Stat., 37(2008), 143-166.

[9] Tomasz Kubiak, Extending Continuous L-Real Functions Theorem, Math. Japonica, 31(1986), N0. 6, 875-887.

[10] Tomasz Kubiak, L-Fuzzy Normal Spaces and Tietze Extension Theorem, J. Math. Anal. Appl., 25(1987), 141-153.

[11] J. Tong, On Decompositon of Continuity in Topological Spaces, Acta Math. Hungar, 54(1989), 51-55.

[12] M. K. Uma, E. Roja and G. Balasubramanian, A New Characterization of Fuzzy Extremally Disconnected Spaces, Atti Sem. Mat. Fis. Univ. Modenae Reggio Emilia, L III(2005), 289-297.

[13] L. A. Zadeh, Fuzzy Sets, Information and Control, 8(1965), 338-353.

D. Vidhya ([dagger]), E. Roja ([double dagger]) and M. K. Uma #

Department of Mathematics, Sri Sarada College for Women, Salem-16, Tamil Nadu, India

Email: vidhya.d85@gmail.com
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