# 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 . Fuzzy sets have applications in many fields such as information  and control . The thoery of fuzzy topological spaces was introduced and developed by Chang  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 . The concept of L-fuzzy normal spaces and Tietze extension theorem was introduced and studied in . The concept of soft fuzzy topological space was introduced by Ismail U. Triyaki . J. Tong  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 . 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.

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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
Author: Printer friendly Cite/link Email Feedback Vidhya, D.; Roja, E.; Uma, M.K. Scientia Magna Report 9INDI Mar 1, 2013 6579 Timelike parallel [p.sub.i]-equidistant ruled surfaces with a spacelike base curve in the Minkowski 3-space [R.sup.1.sub.3]. On right circulant matrices with Perrin sequence. Topological spaces