# neutrosophic pre-continuous multifunctions and almost pre-continuous multifunctions.

1 IntroductionThe fundamental concept of the fuzzy sets was first introduced by Zadeh in his classical paper [12] of 1965. The idea of "intuitionistic fuzzy sets" was first published by Atanassov [7] and many works by the same author and his colleagues appeared in the literature [15, 16]. The theory of fuzzy topological spaces was introduced and developed by Chang [6] and since then various notions in classical topology have been extended ta fuzzy topological spaces. In 1997, Coker [5] introduced the concept intuitionistic fuzzy multifunctions and studied their lower and upper intuitionistic fuzzy semi continuity from a topological space to an intuitionistic fuzzy topological space. F. Smarandache defined the notion of neutrosophic topology on the non-standard interval [13, 14, 18, 19, 20]. Also in various recent papers, F. Smarandache generalizes intuitionistic fuzzy sets (IFSs) and other kinds of sets to neutrosophic sets (NSs). Also, (Zhang, Smarandache, and Wang, 2005) introduced the notion of interval neutrosophic set which is an instance of neutrosophic set and studied various properties. Recently, Wadei Al-Omeri and Smarandache [9, 10, 14, 21, 22] introduce and study a number of the definitions of neutrosophic continuity, neutrosophic open sets, and obtain several preservation properties and some characterizations concerning neutrosophic functions and neutrosophic connectedness. the theory of multifunctions plays an important role in functional analysis and fixed point theory. It also has a wide range of applications in artificial intelligence, economic theory, decision theory,non-cooperative games.

The concepts of the upper and lower pre-continuous multifunctions was introduced in [17].In this paper we introduce and study the neutrosophic version of upper and lower pre-continuous multifunctions. Inspired by the research works of Smarandache [13, 2], we introduce and study the notions of neutrosophic upper pre-continuous and neutrosophic upper pre-continuous multifunctions in this paper. Further, we present some characterizations and properties.

This paper is arranged as follows. In Section 2, we will recall some notions which will be used throughout this paper. In Section 3, neutrosophic upper pre-continuous (resp. neutrosophic lower pre-continuous) are introduced and investigate its basic properties. In Section 4, we study upper almost neutrosophic pre-continuous (lower almost neutrosophic pre-continuous) and study some of their properties. Finally, the applications are vast and the researchers in the field are exploring these realms of research and proved.

2 Preliminaries

Definition 2.1. [4] Let R be a non-empty fixed set. A neutrosophic set (NS for short) S is an object having the form S = {<r, [[mu].sub.S](r), [[sigma].sub.S](r), [[gamma].sub.S](r)>: r [member of] R}, where [[mu].sub.S](r), [[sigma].sub.S](r), and [[gamma].sub.S](r) are represent the degree of member ship function, the degree of indeterminacy, and the degree of non-membership, respectively, of each element r [member of] R to the set S.

Neutrosophic sets in S will be denoted by S, [lambda], [psi], W, B, G, etc., and although subsets of R will be denoted by R, B, T, B, [p.sub.0], r, etc.

A neutrosophic set S = {<r, [[mu].sub.S](r), [[sigma].sub.S](r), [[gamma].sub.S](r)> : r [member of] R} can be identified to an ordered triple <[[mu].sub.S](r), [[sigma].sub.S](r), [[gamma].sub.S](r)> in [??][0.sup.-], [1.sup.+][??] on R.

Remark 2.2. [4] A neutrosophic set S is an object having the form S = {r, [[mu].sub.S](r), [[sigma].sub.S](r), [[gamma].sub.S](r)} for the NS S= {<r, [[mu].sub.S](r), [[sigma].sub.S](r), [[gamma].sub.S](r)> : r [member of] R}.

Definition 2.3. [1] Let S = <[[mu].sub.S](r), [[sigma].sub.S](r), [[gamma].sub.S](r)> be an NS on R. Maybe the complement of the set S(C(S), for short) definitionned as follows.

(i) C(S) = {<r, 1 - [[mu].sub.S](r), 1 - [[gamma].sub.S](r)> : r [member of] R},

(ii) C(S) = {<r, [[gamma].sub.S](r), [[sigma].sub.S](r), [[mu].sub.S](r)> : r [member of] R}

(iii) C(S) = {<r, [[gamma].sub.S](r), 1 - [[sigma].sub.S](r), [[mu].sub.S](r)> : r [member of] R}

Definition 2.4. [4] Let r be a non-empty set, and GNSs S and B be in the form S = {r, [[mu].sub.S](r), [[sigma].sub.S](r), [[gamma].sub.S](r)}, B = {r, [[mu].sub.B](r), [[sigma].sub.B](r), [[gamma].sub.B](r)}. Then (S [??] B) definitionned as follows.

(i) Type 1: S [??] B [??] [[mu].sub.S](r) [less than or equal to] [[mu].sub.B](r), [[sigma].sub.S](r) [greater than or equal to] [[sigma].sub.B](r), and [[gamma].sub.S](r) [less than or equal to] [[gamma].sub.B](r) or

(ii) Type 2: S [??] B [??] [[mu].sub.S](r) [less than or equal to] [[mu].sub.B](r), [[sigma].sub.S](r) [greater than or equal to] [[sigma].sub.B](r), and [[gamma].sub.S](r) [greater than or equal to] [[gamma].sub.B](r).

Definition 2.5. [4] Let {[S.sub.j] : j [member of] J} be an arbitrary family of an NSs in R. Then

(i) [intersection][S.sub.j] definitionned as:

-Type 1: [mathematical expression not reproducible]

-Type 2: [mathematical expression not reproducible]

(ii) [union][S.sub.j] definitionned as:

-Type 1: [mathematical expression not reproducible]

-Type 2: [mathematical expression not reproducible]

Definition 2.6. [2] A neutrosophic topology (NT for short) and a non empty set R is a family T of neutrosophic subsets of R satisfying the following axioms

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

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

(iii) [union] [G.sub.i] [member of] T, [for all] {[G.sub.i]|j [member of] J}[??] T.

The pair (R, T) is called a neutrosophic topological space (NTS for short).

Definition 2.7. [4] Let S be an NS and (R, T) an NT where S = {r, [[mu].sub.S](r), [[sigma].sub.S](r), [[gamma].sub.S](r)}. Then,

(i) NCL(S) = [intersection]{K : K is an NCS in R and S [??] K}

(ii) NInt(S) = [union]{G : G is an NOS in R and G [??] S}

It can be also shown that NCl(S) is an NCS and NInt(S) is an NOS in R. We have

(i) S is in R iff NCl(S).

(ii) S is an NCS in R iff NInt(S) = S.

Definition 2.8. [4] Let S = {[[mu].sub.S](r), [[sigma].sub.S](r), [[gamma].sub.S](r)} be a neutrosophic open sets and B = {[[mu].sub.B](r), [[sigma].sub.B](r), [[gamma].sub.B](r)} a neutrosophic set on a neutrosophic topological space (R, T). Then

(i) S is called neutrosophic regular open iff S = NInt(NCl(S)).

(ii) The complement of neutrosophic regular open is neutrosophic regular closed.

Definition 2.9. [9] Let S be an NS and (R, T) an NT. Then

(i) Neutrosophic semiopen set (NSOS) if S [??] NCl(NInt(S)),

(ii) Neutrosophic preopen set (NPOS) if S [??] NInt(NCl(S)),

(iii) Neutrosophic [alpha]-open set (N[alpha]OS) if S [??] NInt(NCl(NInt(S)))

(iv) Neutrosophic [beta]-open set (N[beta]OS) if S [??] NCl(NInt(NCl(S)))

Definition 2.10. [11] Let (R, T) be a topological space in the classical sense and (S, [T.sub.1]) be a neutrosophic topological space. F : (R, T) [right arrow] (S, [T.sub.1]) is called a neutrosophic multifunction if and only if for each r [member of] R, F(r) is a neutrosophic set in S.

Definition 2.11. [11] For a neutrosophic multifunction F : (R, T) [right arrow] (S, [T.sub.1]), the upper inverse [F.sup.+]([lambda]) and lower inverse [F.sup.-] ([lambda]) of a neutrosophic set [lambda] in S are dened as follows:

[F.sup.+]([lambda]) = {r [member of] R|F(r) [less than or equal to] [lambda]} (2.1)

and

[F.sup.-]([lambda]) = {r [member of] R|F(r)q[lambda]} (2.2)

Lemma 2.12. [11] In a neutrosophic multifunction F : (R, T) [right arrow] (S, [T.sub.1]), we have [F.sup.-](1 - [lambda]) = R - [F.sup.+]([lambda]), for any neutrosophic set [lambda] in S.

A neutrosophic set S in S is said to be q-coincident with a neutrosophic set [psi], denoted by Sq[psi], if and only if there exists p [member of] S such that S(p) + [psi](p) > 1. A neutrosophic set S of S is called a neutrosophic neighbourhood of a fuzzy point [p.sub.[epsilon]] in S if there exists a neutrosophic open set [psi] in S such that [p.sub.[epsilon]] [member of] [psi] [less than or equal to] S.

3 Neutrosophic Pre-continuous multifunctions

Definition 3.1. In a neutrosophic multifunction F : (R, T) [right arrow] (L, [S.sub.1]) is said to be

(i) neutrosophic lower pre-continuous at a point [p.sub.0] [member of] R, if for any neutrosophic open set W [less than or equal to] S such that F([p.sub.0])qW there exists R [member of] NPO(R) containing [p.sub.0] such that F(R)qW, [[for all].sub.r] [member of] R.

(ii) neutrosophic upper pre-continuous at a point [p.sub.0] [member of] R, if for any neutrosophic open set W [less than or equal to] S such that F([p.sub.0]) [less than or equal to] W there exists R [member of] NPO(R) containing [p.sub.0] such that F(R) [less than or equal to] W.

(iii) neutrosophic upper pre-continuous (resp. neutrosophic lower pre-continuous) if it is neutrosophic upper pre-continuous (resp. neutrosophic lower pre-continuous) at every point of R.

A subset R of a neutrosophic topological space (R, T) is said to be neutrosophic neighbourhood (resp. neutrosophic-preneighbourhood) of a point r [member of] R if there exists a neutrosophic-open (resp. neutrosophicpreopen) set S such that r [member of] S [??] R, neutrosophic neighbourhood (resp. neutrosophic pre-neighbourhood) write briefly neutrosophic nbh (resp. neutrosophic pre-nbh).

Theorem 3.2. A neutrosophic multifunction F : (R, T) [right arrow] (S, [T.sub.1]), the the following statements are equivalent:

(i) F is neutrosophic upper pre-continuous at [p.sub.0];

(ii) [F.sub.+](S) [member of] NPO(X) for any neutrosophic open set S of S,

(iii) [F.sup.-] (T) is neutrosophic pre-closed in R for any neutrosophic closed set T of S,

(iv) pNCl([F.sup.-](W)) [??] [F.sup.-](NCl(W)) for each neutrosophic set W of S.

(v) for each point [p.sub.0] [member of] R and each neutrosophic nbh S of F(r), [F.sup.+](S) is a neutrosophic pre-nbh of [p.sub.0],

(vi) for each point [p.sub.0] [member of] R and each neutrosophic nbh S of F(r), there exists a neutrosophic pre-nbh of [p.sub.0] such that F(R) [less than or equal to] S,

(vii) [F.sup.+](NInt(W)) [??] pNInt([F.sup.+] (W)) for every neutrosophic subset W of [I.sup.S],

(viii) [F.sup.+](S) [??] NInt(NCl([F.sup.+](S))) for every neutrosophic open subset S of [I.sup.S],

(ix) for each point [p.sub.0] [member of] R and each neutrosophic nbh S of F(r), Cl([F.sup.+] (S)) is a neighbourhood of r.

Proof. (i) [??] (ii): Let S be any arbitrary NOS of S and [p.sub.0] [member of] [F.sup.+](S). Then F([p.sub.0]) [member of] S. There exists an NPO set R of R containing [p.sub.0] such that F(R) [??] S. Since

[p.sub.0] [member of] R [??] NInt(NCl(R)) [??] NInt(NCl([F.sup.+](S))) (3.1)

and so we have

[F.sup.+] (S) [??] NInt(NCl([F.sup.+](S))). (3.2)

Hence [F.sup.+] (S) is an NPO in R.

(ii) [??] (iii): It follows from the fact that [F.sup.+](S - B)= R - [F.sup.-](W) for any subset W of S.

(iii) [??] (iv): For any subset W of S, NCl(W) is an NCS in S and then [F.sup.-](NCl(W)) is neutrosophic pre-closed in R. Hence,

pNCl([F.sup.-](W)) [??] pNCl([F.sup.-](NCl(R))) [??] [F.sup.-](NCl(R)). (3.3)

(iv) [??] (iii): Let [beta] be any arbitrary NCS of S. Then

pNCl([F.sup.-](M)) [??] [F.sup.-](NCl([beta])) = [F.sup.-]([beta]), (3.4)

and hence [F.sup.-]([beta]) is NPC in R.

(ii) [??] (v): Let [p.sub.0] [member of] R and S be a nbh of F([p.sub.0]). There exists an NOS B of S such that

F([p.sub.0]) [??] B [??] S. (3.5)

Then we have [p.sub.0] [member of] [F.sup.+](B) [??] [F.sup.+](S) and since [F.sup.+](S) is neutrosophic pre-open in R, [F.sup.+](S) is a neutrosophic pre-nbh of [p.sub.0].

(v) [??] (vi): Let [p.sub.0] [member of] R and S be any neutrosophic nbh of F([p.sub.0]). Put R = [F.sup.+](S). By (v) R is a neutrosophic pre-nbh of [p.sub.0] and F(R) [??] S.

(vi) [??] (i): Let [p.sub.0] [member of] R and S be any neutrosophic open set of S such that F([p.sub.0]) [??] S. Then S is a neutrosophic nbh of F([p.sub.0]) and there exists a neutrosophic pre-nbh R of [p.sub.0] such that F(R) [??] S. Therefore, there exists an NPO B in R such that [p.sub.0] [member of] B [??] R and so F(B) [??] S.

(ii) [??] (vii): Let W be an NOs set of S, NInt(W) is an NO in S and then [F.sup.+](NInt(W)) is NPO in R. Hence,

[F.sup.+](NInt(W)) [??] pNInt([F.sup.+](W)). (3.6)

(vii) [??] (ii): Let S be any neutrosophic open set of S. By (vii) [F.sup.+](S) = [F.sup.+](Int(S)) [??] pNInt([F.sup.+] (S)) and hence [F.sup.+](S) is an NPO in R.

(viii) [??] (ix): Let [p.sub.0] [member of] R and S be any neutrosophic nbh of F(r). Then

[p.sub.0] [member of] [F.sup.+](S) [??] NInt(NCl([F.sup.+](S))) [??] NCl([F.sup.+] (S)), (3.7)

and hence NCl([F.sup.+](S)) is a neutrosophic nbh of [p.sub.0].

(viii) [??] (ix): Let S be any open set of S and

[p.sub.0] [member of] [F.sup.+] (S). (3.8)

Then

NCl([F.sup.+](S)) (3.9)

is a neutrosophic nbh of [p.sub.0] and thus

NInt(NCl([F.sup.+](S))). (3.10)

Hence,

[F.sup.+](S) [??] NInt(NCl(F + (S))). (3.11)

Theorem 3.3. For a neutrosophic multifunction F : (R, T) [right arrow] (S, [T.sub.1]), the following statements are equivalent:

(i) F is neutrosophic lower pre-continuous at [p.sub.0];

(ii) [F.sup.+](S) [member of] NPO(X) for any NOs S of S,

(iii) [F.sup.+](T) [member of] NPC(X) for any neutrosophic closed set T of S,

(iv) for each [p.sub.0] [member of] R and each neutrosophic nbh S which intersects F(r), [F.sup.-](S) is a neutrosophic pre-nbh of [p.sub.0],

(v) for each [p.sub.0] [member of] R and each neutrosophic nbh S which intersects F(r), there exists a neutrosophic preneighbourhood R of [p.sub.0] such that F(u) [intersection] S [not equal to] [phi] or any u [member of] R,

(vi) pNCl([F.sup.+](W)) [??] [F.sup.+](NCl(W)) for any neutrosophic set W of S.

(vii) [F.sup.-](NInt(W)) [??] pNInt([F.sup.-](W)) for every neutrosophic subset W of [I.sup.S],

(viii) [F.sup.-](S) [??] NInt(NCl([F.sup.-](S))) for any NOs subset S of [I.sup.S],

(ix) for each point [p.sub.0] [member of] R and each neutrosophic nbh S of F (r), Cl([F.sup.-](S)) is a neighbourhood of r.

Proof. (i) [??] (ii): Let S be any arbitrary NOS of S and [p.sub.0] [member of] [F.sup.+](S). Then by (a), there exists an NPO set R of R containing [p.sub.0] such that F(R) [??] S. Since

[p.sub.0] [member of] R [??] NInt(NCl(R)) [??] NInt(NCl([F.sup.-](S))) (3.12)

and so we have

[F.sup.-](S) [??] NInt(NCl([F.sup.-](S))), (3.13)

and hence

[F.sup.-] (S) [member of] NPO(R). (3.14)

(ii) = (iii) : It follows from the fact that

[F.sup.+](S \ B)= R \ [F.sup.-](W) (3.15)

for any W [member of] S.

(iii) [??] (vi): Let W in S, NCl(W) is an NCS in S. By (iii) [F.sup.+](NCl(W)) is neutrosophic pre-closed in R. Hence,

pNCl([F.sup.+](W)) [??] pNCl([F.sup.+](NCl(R))) [??] [F.sup.+] (NCl(R)). (3.16)

(iv) [??] (iii): Let [beta] be NCs of S. Then

pNCl([F.sup.+](M)) [??] pNCl([F.sup.+] ([F.sup.+] (NCl([beta]))) [??] [F.sup.+](NCl([beta])) = [F.sup.+]([beta]) [??] [F.sup.-]([beta]) (3.17) is NPC in R.

(ii) [??] (v): Let [p.sub.0] [member of] R and S be a neutrosophic nbh of F([p.sub.0]). There exists an NOS B of S such that

F([p.sub.0]) [??] B [??] S. (3.18)

Then we have

[p.sub.0] [member of] [F.sup.-](B) [??] [F.sup.-](S), (3.19)

and since [F.sup.-] (V) is neutrosophic pre-open in R, by (ii) [F.sup.-] (S) is a neutrosophic pre-nbh of [p.sub.0].

(v) [??] (vi): Let [p.sub.0] [member of] R and S be any neutrosophic nbh of F([p.sub.0]). Put R = [F.sup.-](S). By (v) R is a neutrosophic pre-nbh of [p.sub.0] and F(R) [??] S.

(vi) [??] (i): Let [p.sub.0] [member of] R and S be any NOs of S such that F([p.sub.0]) [??] S. Then S is a neutrosophic nbh of F([p.sub.0]) by (vi) there exists a neutrosophic pre-nbh R of [p.sub.0] such that F(R) [??] S. Therefore, there exists an NPO B in R such that

[p.sub.0] [member of] B (S) C R (S) (3.20)

and so

S [??] [F.sup.-](B). (3.21)

(ii) [??] (vii): Let W be an NOs set of S, NInt(W) is an NO in S and then F +(NInt(W)) is NPO in

R. Hence, [F.sup.+](NInt(W)) C pNInt([F.sup.+](W)).

(vii) [??] (ii): Let S be any NOs of S. By (vii)

[F.sup.+](S (S)) = [F.sup.-](Int(S (S))) C pNInt([F.sup.-](S (S))), (3.22)

and hence [F.sup.-] (S) is an NPO in R.

(vi) [??] (vii): Let W be any neutrosophic open set of S, then

[[F.sup.-](NInt(W))} (c) = F+(NCl(W (c))) D pNCl(F+(NInt(NCl(W (c))))) (3.23)

= pNCl(F+(NCl(NInt(W))) (c)) = pNCl([F.sup.-](NCl(NInt(W)))) (c) (3.24)

= [pNInt([F.sup.-](NCl(NInt(W))))} (c). (3.25)

Thus we obtained

[F.sup.-](NInt(W)) D pNInt([F.sup.-](NCl(NInt(W)))). (3.26)

(vi) [??] (vii): Obvious.

We now show by means of the following examples that lower neutrosophic pre-continuous [??] upper neutrosophic pre-continuous.

Example 3.1. Let R = {u,v,w} and S = [0, 1]. Let T and [T.sub.1], be respectively the topology on R and neutrosophic topology on S, given by T{[R.sub.N], [[phi].sub.N], {u, w}}, [T.sub.1] = {[C.sub.o], C, [[mu].sub.S], [[sigma].sub.S], [[gamma].sub.S], ([[mu].sub.S] [union] [[sigma].sub.S]), ([[mu].sub.S] [??] [[sigma].sub.S])}. Where [[mu].sub.S] (r) = r, [[sigma].sub.S] (r) = I - r, for r [member of] S, and

[mathematical expression not reproducible] (3.27)

We definitionne a neutrosophic multifunction F : (R, T) [right arrow] (S, [T.sub.1]) be letting F(u) = ([[mu].sub.S] [intersection] [[sigma].sub.S]), F(v) = [[sigma].sub.S] and F(w) = [[gamma].sub.S].

{u, w} is neutrosophic open set in R and therefore {u, w} is neutrosophic pre-open set. The other neutrosophic pre-open set in (R, T) are {u, v}, {v, w}, {u} and {w}. Then {u} is not neutrosophic pre-open set in (R, T). From definitionnition of [[mu].sub.S] and [[sigma].sub.S] we find that,

[mathematical expression not reproducible] (3.28)

[mathematical expression not reproducible] (3.29)

Now [[sigma].sub.S] [member of] [T.sub.1] but [F.sup.+]([[sigma].sup.S]) = {v} which is not neutrosophic pre-open set in (R, T). Hence F is not upper neutrosophic pre-continuous. Then [F.sup.-]([[sigma].sub.S]) = {v} which is not neutrosophic pre-open set in (R, T). Therefore F is not lower neutrosophic pre-continuous

Remark 3.4. [11] A subset [micro] of a topological space (R, T) can be considered as a neutrosophic set with characteristic function definitionned by

[mathematical expression not reproducible] (3.30)

Let (S, [T.sub.1]) be a neutrosophic topological space. The neutrosophic sets of the form [micro] x v with [micro] [member of] T and v [member of] [T.sub.1] make a basis for the product neutrosophic topology T x [T.sub.1] on R x S, where for any (u,v) [member of] R x S, ([micro] x v)(u, v) = min{[micro](u), v(v)}.

Definition 3.5. [11] For a neutrosophic multifunction F : (R, T) [right arrow] (S, [T.sub.1]), the neutrosophic graph multifunction [F.sub.G] : R x R [right arrow] S of F is definitionned by [F.sub.G](r) = [r.sub.1] x F(r) for every r [member of] R.

Lemma 3.6. In a neutrosophic multifunction F : (R, T") [right arrow] (S, [T.sub.1]), the following hold: a) [F.sup.+.sub.G](R x S) = R [intersection] [F.sup.+](S)

b) [F.sup.-.sub.G](R x S) = R [intersection] [F.sup.+](S) for all subsets R [member of] R and S [member of] S.

Theorem 3.7. If the neutrosophic graph multifunction [F.sub.G] of a neutrosophic multifunction F : (R, T) [right arrow] (S, [T.sub.1]) is neutrosophic lower precontinuous, then F is neutrosophic lower precontinuous.

Proof. Suppose that [F.sub.G] is neutrosophic lower precontinuous and s [member of] R. Let B be an NOs [member of] S such that F(r)qB. Then there exists r [member of] S such that (F(r))(r) + A(r) > 1. Then

([F.sub.G](r))(r, r) + (R x B)(r, r) = (F(r))(r) + B(r) > 1. (3.31)

Hence, [F.sub.G](r)q(R x B). Since [F.sub.G] is neutrosophic lower precontinuous, there exists an open set A [member of] R such that r [member of] A and [F.sub.G](b)q(R x B) [for all]a [member of] A. Let there exists [a.sub.0] [member of] A such that F([a.sub.0])qB. Then [for all]r [member of] S, (F([a.sub.0]))(r) + B(r) < 1. For any (b, c) [member of] R x S, we have

([F.sub.G]([a.sub.0]))(b,c) [??] (F([a.sub.0]))(c), (3.32)

and

(R x B)(b,c) [??] B(c). (3.33)

Since [for all]r [member of] S, (F([a.sub.0]))(r) + B(r) < 1,

([F.sub.G]([a.sub.0]))(b,c) + (R x B)(b,c) < 1. (3.34)

Thus, [F.sub.G]([a.sub.0])q(R x B), where [a.sub.0] [member of] A. This is a contradiction since

[F.sub.G](a)q(R x B), [for all]a [member of] A, (3.35)

Therefore, F is neutrosophic lower precontinuous.

Definition 3.8. A neutrosophic space (R, T) is said to be neutrosophic pre-regular (NP-regular) if for every NCs F and a point u [member of] F, there exist disjoint neutrosophic-preopen sets R and S such that F [??] R and u [member of] S.

Theorem 3.9. Let F : (R, T) [right arrow] (S, [T.sub.1]) be a neutrosophic multifunction and [F.sub.G] : R [right arrow] R x S the graph multifunction of F. If [F.sub.G] is neutrosophic upper pre-continuous (neutrosophic lower pre-continuous, then F is neutrosophic upper pre-continuous. (neutrosophic lower pre-continuous.) and R is NP-regular.

Proof. Let [F.sub.G] be a neutrosophic upper pre-continuous multifunction and S be a neutrosophic open set containing F(r) such that r [member of] [F.sup.+](S). Then R x S is a neutrosophic open set of R x S containing [F.sub.G](r). Since [F.sub.G] is neutrosophic upper pre-continuous, there exists an NPOs R of R containing r such that [R.sup.-.sub.p] [??] [F.sup.+.sub.G](R x S). Therefore we obtain

[R.sup.-.sub.p] [??] [F.sup.+](S). (3.36)

Now we show that R is NP-regular. Let R be any NPOs of R containing r. Since

[F.sub.G](r) [member of] R x S, (3.37)

and R x S is neutrosophic open in R x S, there exists an NPOs set U of R such that

[U.sup.-.sub.p] [??] [F.sup.+.sub.G](R x S). (3.38)

Therefore we have

r [member of] U [??] [U.sup.-.sub.p] [??] R. (3.39)

This shows that R is NP-regular.

The proof for neutrosophic upper lower-continuous is similar.

Theorem 3.10. Let R is NP-regular. A neutrosophic multifunction F : (R, T) [right arrow] (S, [T.sub.1]) is neutrosophic lower pre-continuous iff [F.sub.G](r) is neutrosophic lower pre-continuous.

Proof. [??] Let r [member of] R and A be any NPOs of R x S such that r [member of] [F.sub.G](A). Since

A [intersection] ({r}x F(r)) [not equal to] [phi], (3.40)

there exists s [member of] F(r) such that (r, s) [member of] A. Hence

(r, s) [member of] R x S [??] A (3.41)

for some NOs R [??] R and S [??] S. Since R is NP-regular, there exists B [member of] NPO(R, r) such that

r [member of] B [??] [B.sup.-.sub.p] [??] R. (3.42) Since F is neutrosophic lower pre-continuous, there exists W [member of] NPO(R, r) such that

[W.sup.-.sub.p] [??] [F.sup.-](S). (3.43)

By Lemma 3.6, we have

[W.sup.-.sub.p] [intersection] [B.sup.-.sub.p] [??] R [intersection] [F.sup.-] (S) = [F.sup.-.sub.G] (R x S) [??] [F.sup.-.sub.G] (B). (3.44)

Moreover, we have B [intersection] W [member of] NPO(R, r) and hence [F.sub.G](r) is neutrosophic lower pre-continuous. [??] Let r [member of] R and S be any NOs [member of] S such that r [member of] [F.sup.-] (S). Then R x S is

NO [member of] R x S. (3.45)

Since [F.sub.G] is neutrosophic lower pre-continuous and lemma,

[F.sup.-.sub.G] (R x S) = R [intersection] [F.sup.-] (S) = [F.sup.-] (S) (3.46)

is NPOs [member of] R. This shows that F is neutrosophic lower pre-continuous.

Definition 3.11. [11] A neutrosophic set [DELTA] of a neutrosophic topological space S is said to be neutrosophic compact relative to S if every cover [{[[DELTA].sub.[lambda]]}.sub.[lambda][member of][and]] of [DELTA] by neutrosophic open sets of S has a finite subcover [{[[DELTA].sub.i]}.sup.n.sub.i=1] of [DELTA].

Theorem 3.12. Let F : (R, T") [right arrow] (S, [T.sub.1]) be a neutrosophic multifunction such that F (r) is compact for each r [member of] R. And R is a NP-regular space. If F is neutrosophic upper pre-continuous then [F.sub.G] is neutrosophic upper pre-continuous.

Proof. Let r [member of] R and A be any NOs [member of] R x S containing [F.sub.G] (r). For each s [member of] F(r), there exist open sets R(s) [??] R and S(s) [??] S such that

(r, s) [member of] R(s) x S(s) [??] A. (3.47)

The family {S(s) : s [member of] F(r)} is a neutrosophic open cover of F(r). Since F(r) is compact, there exists a finite number of points [{[s.sub.j]}.sup.n.sub.j=1] in F(r) such that

F(r) [??] [union] {S([s.sub.j]) : j = 1,... n}. (3.48)

Use R = [intersection]{R([s.sub.j]) : j = 1,...,n} and S = {S([s.sub.j]) : j = 1,...,n}. Then R and S are NOs [member of] R and S, respectively and

{r} x F(r) [??] R x S [??] A. (3.49)

Since F is neutrosophic upper pre-continuous, there exists S [member of] NPO(R, r) such that

[S.sup.-.sub.p] [??] [F.sup.+] (S). (3.50)

Since R is NP-regular, there exists G [member of] NPO(R, r) such that

r [member of] G [??] [G.sup.-.sub.p] [??] R. (3.51)

Hence, we have

{r} x F(r) [??] [G.sup.-.sub.p] x S [??] R x S [??] A. (3.52)

Then we have

[(S [intersection] G).sup.-.sub.p] [??] S[G.sup.-.sub.p] [intersection] [G.sup.-.sub.p] [??] [F.sup.+](S) [intersection] [G.sup.-.sub.p] = [F.sup.+.sub.G] ([G.sup.-.sub.p] x S) [??] [F.sup.+.sub.G] (A). (3.53)

Moreover, we obtain S [intersection] G [member of] NPO(R, r) and hence [F.sub.G] is neutrosophic upper pre-continuous.

Proposition 3.2. Let B and [R.sub.o] be subsets of neutrosophic topological space (R, T).

(i) If B [member of] NPO(R) and [R.sub.o] is NSO in R, then B [intersection] [R.sub.o] [member of] NPO([R.sub.o]).

(ii) If B [member of] NPO([R.sub.o]) and [R.sub.o] [member of] NPO(R), then B [member of] NPO(R).

Proposition 3.3. Let B and R be subsets of neutrosophic topological space (R, T), B [??] R [??] R. Let the neutrosophic pre-closure [([B.sup.-.sub.p]).sub.R] of B in the neutrosophic subspace [R.sub.o]:

(i) If R is NSO in R, then [mathematical expression not reproducible] [??] ([B.sup.-.sub.p])R.

(ii) If B in NPO([R.sub.o]) and [R.sub.o] in NPO(R), then [B.sup.-.sub.p] [??] [mathematical expression not reproducible].

Theorem 3.13. A neutrosophic multifunction F : (R, T) [right arrow] (S, [T.sub.1]) is upper almost neutrosophic pre-continuous. (lower almost neutrosophic pre-continuous) if [for all] r [member of] R there exists an NPOs [R.sub.o] containing r such that F|[R.sub.o] : (R, T) [right arrow] (S, [T.sub.1]) is upper almost neutrosophic pre-continuous. (lower almost neutrosophic pre-continuous).

Proof. Let r [member of] R and S be an neutrosophic open set of S containing F(r) such that r [member of] [F.sup.+] (S) and there exists [R.sub.o] [member of] NPO(R, r) such that

F|[R.sub.o] :(R, T) [right arrow] (S, [T.sub.1]), (3.54)

is upper almost neutrosophic pre-continuous. Therefore, there exists R in NPO([R.sub.o], r) such that

[mathematical expression not reproducible]. (3.55)

By Proposition 3.2 and Proposition 3.3, R in NPO(X, r) and

[mathematical expression not reproducible]. (3.56)

Therefore

[mathematical expression not reproducible]. (3.57)

This shows that F is upper almost neutrosophic pre-continuous.

4 Almost Neutrosophic pre-continuous multifunctions

Definition 4.1. Let (R, T) be a neutrosophic topological space and (S, [T.sub.1]) a topological space. A neutrosophic multifunction F : (R, T) [right arrow] (S, [T.sub.1]) is said to be:

(i) upper almost neutrosophic pre-continuous at a point r [member of] R if for each open set S of [I.sup.S] such that r [member of] [F.sup.+](S), there exists a neutrosophic pre-open set R of R containing r such that R [??] [F.sup.+] (NInt(NCl(S)));

(ii) lower almost neutrosophic pre-continuous at a point r [member of] R if for each neutrosophic open set S [member of] S such that r [member of] F(S), there exists a neutrosophic pre-open R of R containing r such that R [??] [F.sup.-] (NInt(NCl(S)));

(iii) upper (resp. lower) almost neutrosophic pre-continuous if F has this property at each point of R.

Theorem 4.2. Let (R, T) be a neutrosophic topological space and F : (R, T) [right arrow] (S, [T.sub.1]) a neutrosophic multifunction from a neutrosophic topological space (R, T) to a topological space (S, [T.sub.1]). Then the following properties are equivalent:

(i) F is upper almost neutrosophic-pre-continuous;

(ii) for any r [member of] R and for all NOs S of S such that F (r) [??] S, there exists a neutrosophic pre-open R of R containing r such that if z [member of] R, then F(z) [??] N Int(NCl(S));

(iii) for any r [member of] R and for all NROs G of S such that F (r) [??] G, there exists a neutrosophic per-open R of R containing r such that F(R) [??] G;

(iv) for any r [member of] R and for all closed set [F.sup.+] (S - M), there exists a neutrosophic per-closed N of R such that r [member of] R - N and [F.sup.-] (NInt(NCl(M))) [??] N;

(v) [F.sup.+](NInt(NCl(S))) is neutrosophic pre-open in R for any NOs S of S;

(vi) [F.sup.-](NCl(NInt(F))) is neutrosophic pre-closed in R for each closed set F of S;

(vii) [F.sup.+](G) is neutrosophic pre-open in R for each regular open G of S;

(viii) [F.sup.+](H) is neutrosophic pre-closed in R for each regular closed H of S;

(ix) for any point r [member of] R and each neutrosophic-nbh S of F(r), [F.sup.+](NInt(NCl(S))) is a neutrosophic pre-nbh of r;

(x) for any point r [member of] R and each neutrosophic-nbh S of F(r), there exists a neutrosophic pre-nbh R of r such that F(R) [??] NInt(NCl(S));

(xi) pNCl([F.sup.-](NCl(NInt(A)))) [??] [F.sup.-](NCl(NInt(NCl(A)))) for any subset A of S;

(xii) [F.sup.+](NInt(NCl(NInt(A)))) [??] pNInt([F.sup.+](NInt(NCl(A)))) for any subset A of S.

Proof. (i) [??] (ii): Obvious.

(ii) [??] (iii): Let r [member of] R and G be a regular open set of S such that F(r) [??] G. By (ii), there exists an NPOs R containing r such that if z [member of] R, then F(z) [??] NInt(NCl(G)) = G. We obtain F(R) [??] G.

(iii) [??] (ii): Let r [member of] R and S be an NOs set of S such that F(r) [??] S. Then, NInt(NCl(S)) is NROs in S. By (iii), there exists an NPOS of R containing r such that

F(R) [??] NInt(NCl(S)). (4.1)

(ii) [??] (iv): Let r [member of] R and M be an NCs of S such that r [member of] [F.sup.+](S - M). By (ii), there exists an NPOs R of R containing r such that

F(R) [??] NInt(NCl(SM)). (4.2)

We have

NInt(NCl(S - M)) = S - NCl(NInt(M)) (4.3)

and

R [??] [F.sup.+](S - NCl(NInt(M))) = R - [F.sup.-](NCl(NInt(M))). (4.4)

We get

[F.sup.-](NCl(NInt(M))) [??] R - R. (4.5)

Let N = R - R. Then, r [member of] R - N and N is an NPCs.

(iv) [??] (ii): The proof is similar to (ii) [??] (iv).

(i) [??] (v): Let S be any neutrosophic open set of S and r [member of] [F.sup.+](NInt(NCl(S))). By (i), there exists an NPOs [R.sub.r] of R containing r such that

[R.sub.r] [??] [F.sup.+](NInt(NCl(S))). (4.6)

Hence, we obtain

[F.sup.+](NInt(NCl(S))) = [[union].sub.r] [member of] [F.sup.+](NInt(NCl(S)))[R.sub.r]. (4.7)

Therefore, [F.sup.+](NInt(NCl(S))) is an NPOs of R.

(v) [??] (i): Let S be any neutrosophic open set of S and r [member of] [F.sup.+](S). By (v), [F.sup.+](NInt(NCl(S))) is NPOs in R. Let R = [F.sup.+](NInt(NCl(S))). Then,

F(R) [??] NInt(NCl(S)). (4.8)

Therfore, F is upper neutrosophic pre-continuous.

(v) [??] (vi) : Let F be any neutrosophic closed set of S. Then, S - F is an NOs of S. By (v), [F.sup.+](NInt(NCl(S - F))) is NPOs [member of] R. Since NInt(NCl(S - F)) = S - NCl(NInt(F)), it follows that [F.sup.+](NInt(NCl(S - F))) = [F.sup.+](S - NCl(NInt(F))) = R - [F.sup.-](NCl(NInt(F))). We obtain that [F.sup.-](NCl(NInt(F))) is NPCs [member of] R.

(vi) [??] (v): The proof is similar to (v) [??] (vi).

(v) [??] (vii): Let G be any NROs of S. By (v),

[F.sup.+](NInt(NCl(G))) = [F.sup.+](G) (4.9) is NPOs [member of] R.

(vii) [??] (v): Let S be any neutrosophic-open set of S. Then, NInt(NCl(S)) is NRO sin S. By (vii),

[F.sup.+](NInt(NCl(S))) (4.10)

is NPOs [member of] R.

(vi) [??] (viii): The proof is similar to (v) [??] (vii).

(viii) [??] (vi): The proof is similar to (vii) [??] (v).

(v) [??] (ix): Let r [member of] R and S be a neutrosophic nbh of F(r). Then, there exists an open set G of S such that

F(r) [??] G [??] S. (4.11)

Hence, we obtain r [member of] [F.sup.+](G) [??] [F.sup.+](S). Since [F.sup.+](NInt(NCl(G))) is NPOs [member of] R, [F.sup.+](NInt(NCl(S))) is a neutrosophic pre-nbh of r.

(ix) [??] (x): Let r [member of] R and S be a neutrosophic nbh of F(r). By (ix), [F.sup.+](NInt(NCl(S))) is a neutrosophic-nbh of r. Let R = [F.sup.+](NInt(NCl(S))). Then,

F(R) [??] NInt(NCl(S)). (4.12)

(x) [??] (i): Let r [member of] R and S be any NOs of S such that F(r) [??] S. Then, S is a neutrosophic-nbh of F(r). By (r), there exists a neutrosophic pre-nbh R of r such that

F(R) [??] NInt(NCl(S)). (4.13)

Hence, there exists an NPOs G of R such that

r [member of] G [??] R, (4.14)

and hence

F(G) [??] F(R) [??] NInt(NCl(S)). (4.15)

We obtain that F is upper almost neutrosophic pre-continuous.

(vi) [??] (xi): For every subset A of S, NCl(A) is NCs in S. By (vi), [F.sup.-](NCl(NInt(NCl(A)))) is NPCs [member of] R. Hence, we obtain

pNCl([F.sup.-](NCl(NInt(A)))) [??] [F.sup.-] (NCl(NInt(NCl(A)))). (4.16)

(xi) [??] (vi) : For any NCs F of S. Then we have

pNCl([F.sup.-]F)))) [??] [F.sup.-] (NCl(NInt(NCl(F)))) = [F.sup.-] (NCl(NInt(F))). (4.17)

Thus, [F.sup.-](NCl(NInt(F))) is NPCs [??] R.

(v) [??] (xii): For every subset A of S, NInt(A) is NO [member of] S. By (v),

[F.sup.+] (NInt(NCl(NInt(A)))) (4.18) is NPOs in R. Therefore, we obtain

[F.sup.+](NInt(NCl(NInt(A)))) [??] pNInt([F.sup.+] (NInt(NCl(A)))). (4.19)

(xii) [??] (v): Let S be any subset of S. Then

[F.sup.+] (NInt(NCl(S))) [??] pNInt([F.sup.+](NInt(NCl(S)))). (4.20)

Therefore, [F.sup.+] (NInt(NCl(S))) is NPOs [member of] R.

Remark 4.3. If F : (R, T) [right arrow] (S, [T.sub.1]) are neutrosophic upper pre-continuous multifunctions, then F is a neutrosophic upper almost pre-continuous multifunction.

The implication is not reversible.

Example 4.1. Let R = {[micro],v, [omega]} and S = {u,v, w, t, h}. Let (R, T) be a neutrosophic topology on R and [[sigma].sub.S] a topology on S given by T = {[[phi].sub.N], {v}, {w}, {v,w}, [R.sub.N]} and [[sigma].sub.S] = {[[phi].sub.N], {u,v,w,t}, [S.sub.N]}. Definitionne the multifunction F : (R, T) [right arrow] (S, [T.sub.1]) by F([micro]) = {w}, F(v) = {v,t} and F([omega]) = {u,h}. Then F is upper almost neutrosophic precontinuous but not upper neutrosophic precontinuous, since {u,v,w,t,h} [member of] [[sigma].sub.S] and [F.sup.+]({u, v, w, t, h}) = {[micro], v} is not neutrosophic pre-open in R.

Theorem 4.4. Let F : (R, T) [right arrow] (S, [T.sub.1]) be a multifunction from a neutrosophic topological space (R, T) to a topological space (S, [T.sub.1]). Then the following properties are equivalent:

(i) F is lower almost neutrosophic-precontinuous;

(ii) for each r [member of] R and for each open set S of S such that F(r)[intersection]S [not equal to] [phi], there exists a neutrosophic-preopen R of R containing r such that if z [member of] R, then F(z) [intersection] NInt(NCl(S)) [not equal to] [phi];

(iii) for each r [member of] R and for each regular open set G of S such that F (r)[intersection]G [not equal to] [phi], there exists a neutrosophic-peropen R of R containing r such that if z [member of] R, then F(z) [intersection] G [not equal to] [phi];

(iv) for each r [member of] R and for each closed set M of S such that r [member of] [F.sup.+] (S - M), there exists a neutrosophic-perclosed N of R such that r [member of] R - N and [F.sup.+](NCl(NInt(M))) [??] N;

(v) [F.sup.-] (NInt(NCl(S))) is neutrosophic-pre-open in R for any NOs S of S;

(vi) [F.sup.+] (NCl(NInt(F))) is neutrosophic-pre-closed in R for any NCs F of S;

(vii) [F.sup.-] (G) is neutrosophic-per-open in R for any NROs G of S;

(viii) [F.sup.+] (H) is neutrosophic-perclosed in R for any NRCs H of S;

(ix) pNCl([F.sup.+](NCl(NInt(B)))) [??] [F.sup.+](NCl(NInt(NCl(B)))) for every subset B of S;

(x) [F.sup.-](NInt(NCl(NInt(B)))) [??] pNInt([F.sup.-](NInt(NCl(B)))) for every subset B of S.

Proof. It is similar to that of Theoremark 4.2.

5 Conclusions and/or Discussions

Topology on lattice is a type of theory developed on lattice which involves many problems on ordered structure. For instance, complete distributivity of lattices is a pure algebraic problem that establishes a connection between algebra and analysis. neutrosophic topology is a generalization of fuzzy topology in classical mathematics, but it also has its own marked characteristics. Some scholars used tools for examining neutrosophic topological spaces and establishing new types from existing ones. Attention has been paid to define and characterize new weak forms of continuity.

We have introduced neutrosophic upper and neutrosophic lower almost pre-continuous-multifunctions as a generalization of neutrosophic multifunctions over neutrosophic topology space. Many results have been established to show how far topological structures are preserved by these neutrosophic upper pre-continuous (resp. neutrosophic lower pre-continuous). We also have provided examples where such properties fail to be preserved. In this paper we have introduced the concept of upper and lower pre-continuous multifunction and study some properties of these functions together with the graph of upper and lower pre-continuous as well as upper and lower weakly pre-continuous multifunction.

6 Acknowledgments

We are thankful to the referees for their valuable suggestions to improve the paper.

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Received: December 26, 2018.

Accepted: April 30, 2019.

Wadei F. Al-Omeri (1), Saeid Jafari (2)

(1) W. F. Al-Omeri, Department of Mathematics, Al-Balqa Applied University, Salt 19117, Jordan. E-mail: wadeialomeri@bau.edu.jo or wadeimoon1@hotmail.com

(2) Department of Mathematics, College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark. E-mail: jafaripersia@gmail.com

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