Johann Listing was first introducing the term topology in the 19^th century. In 1955, Kelly [27] studied different basic concepts of topological spaces namely neighborhood, open set, closed set, interior, closure, compactness, continuity. Afterwards, Levine [28] introduced the concept of semi-open set and semi-continuous functions via general topological space. Masshour et al. [31] grounded the notion of supra topological space in 1983. In 2002, Csaszar [4] presented the notion of generalized topology and generalized continuous functions on it. Csaszar [5] also studied the separation axioms for generalized topologies. Later on, the generalized open sets via generalized topological spaces was presented by Csaszar [6] in 2005. In 2009, Csaszar [7] also introduced the notion of products of generalized topologies. The concept of infi-topological spaces was grounded by Das et al. [16]. In 1965, Zadeh [49] grounded the notion of Fuzzy Set (FS) theory to deal with the situation involving the uncertainty. Afterwards, Chang [3] presented the notion of fuzzy topological spaces in 1968. Hutton [25] studied the normality via fuzzy topological spaces. In 1975, Gantner et al. [24] presented the idea of compactness via fuzzy topological spaces in 1978. Later on, Saha and Bhattacharya [44] introduced and studied the concept of countable fuzzy topological spaces and countable fuzzy vector spaces. In 2021, Das et al. [8] introduced the notion of fuzzy infi topological spaces by extending the notion of FS and infi-topological space. Later on, Atanassov [2] introduced the notion of Intuitionistic Fuzzy Set (IFS) theory in 1986. Afterwards, Smarandache [38] grounded the notion of Neutrosophic Set (in short NS) theory in 1998 by generalizing the idea of FS and IFS theory. In an NS, every element has three independent memberships namely truth membership, indeterminacy membership and false membership. Later on, Wang et al. [40] grounded the concept of single-valued NS. Till now, many researchers around the globe used NS, SVNS and their extensions in the theoretical [10, 15, 20, 22, 33,34, 35, 36, 37, 38, 39, 40, 41,  47] area as well as practical area of research [11, 17-18, 29]. In 2020, Das and Tripathy [19] introduced the neutrosophic multiset topological spaces. Thereafter, the idea of Neutrosophic Topological Space ( NTS) was grounded by Salama and Alblowi [48] in 2012. Later on, Arokiarani et al. [1] defined the notion of semi-open functions via NTS. In 2016, Iswaraya and Bageerathi [26] presented the concept of semi-open set and semi-closed set in NTSs. Afterwards, Rao and Srinivasa [43] introduced the idea of neutrosophic pre-open set and neutrosophic pre-closed set via NTSs. The notion of neutrosophic generalized closed sets via NTSs was studied by Pushpalatha and Nandhini [42]. The idea of neutrosophic b-open sets via NTSs was introduced by Ebenanjar et al. [23]. Maheswari et al. [30] studied the idea of neutrosophic generalized b-closed sets in NTSs. In the year 2019, the concept of generalized neutrosophic b-open set via NTSs was studied by Das and Pramanik [12]. Later on, Das and Pramanik [13] also grounded the notion of neutrosophic F-open sets and neutrosophic F-continuous mappings via NTSs. Afterwards, Das and Pramanik [14] presented the notions of neutrosophic simply soft open set via neutrosophic soft topological spaces. Mohammed Ali Jaffer and Ramesh [32] presented the notion of neutrosophic generalized pre regular closed set via NTSs. In 2021, Das [9] introduced the notion of neutrosophic supra simply open set and neutrosophic supra simply compactness via neutrosophic supra topological spaces. Recently, Das and Tripathy [21] presented the idea of neutrosophic simply b-open set via NTSs.

In this article, we procure the notion of neutrosophic infi-topological space, and studied some basic properties of neutrosophic infi-topological spaces like neutrosophic infi-interior, neutrosophic infi-closure, neutrosophic infi-continuous mapping and neutrosophic infi-open mapping, etc. via neutrosophic infi-topological spaces.

Research Gap: No investigation on neutrosophic infi-topological space has been reported in the literature.

Motivation: To fill the research gap, we procure the notion of neutrosophic infi-topological space, and study its different properties.

The rest of this paper has been designed as follows:

In section-2, we recall some relevant definitions and results on FS, NS, NTS, Infi-Topological Space, etc. In section-3, we procure the notion of neutrosophic infi-topological space, and formulate several interesting results on it. Finally, section-4 represents the concluding remarks of the work done in this article.

2. Some Relevant Results   

In this section, we provide some definitions and results those are very relevant and useful for the preparation of the main results of this article.

Definition 2.1.[16] Assume that W is a fixed set. Then, a collection Á of subsets of W is said to be an infi-topology on W if the following two conditions hold:

(i) Æ, WÎÁ;

(ii) Y₁, Y₂ ÎÁ Þ Y₁ÇY₂ ÎÁ;

Then, the structure (W, Á) is called an infi-topological space. Every member of Á is called an infi-open set. If YÎÁ, then Y^c is said to be an infi-closed set.

Definition 2.2.[49] Suppose that W is a fixed set. Then P, a FS over W is defined by:

P={(q, T_P(q)): qÎW},

where T_P(q) (Î[0, 1]) denotes the truth membership values of qÎW.

Definition 2.3.[49] The absolute FS (1_F) and null FS (0_F) over W are defined as follows:

(i) 1_N ={(q, 1): qÎW};

(ii) 0_N ={(q, 0): qÎW}.

Definition 2.4.[8] Suppose that W is a fixed set. Then, a collection Á of FSs over W is called a fuzzy infi-topology on W if the following two conditions hold:

(i) 0_N, 1_N ÎÁ;

(ii) Y₁, Y₂ ÎÁ Þ Y₁ÇY₂ ÎÁ.

Then, the structure (W, Á) is said to be a fuzzy infi-topological space. Every member of Á is called a fuzzy infi-open set. If YÎÁ, then Y^c is said to be a fuzzy infi-closed set in (W, Á).

Definition 2.5.[46] Suppose that W is a universe of discourse. Then P, an NS over W is defined by:

P={(q, T_P(q), I_P(q), F_P(q)): qÎW},

where T_P(q), I_P(q), F_P(q) (Î[0, 1]) are the truth membership, indeterminacy membership and falsity membership values of qÎW. So, 0 £ T_P(q) + I_P(q) + F_P(q) £ 3, for all qÎW.

Definition 2.6.[46] The absolute NS (1_N) and null NS (0_N) over a fixed set W are defined as follows:

(i) 1_N ={(q,1,0,0): qÎW};

(ii) 0_N ={(q,0,1,1): qÎW}.

Definition 2.3.[46] Assume that X={(q,T_X(q),I_X(q),F_X(q)): qÎW} and Y = {(q,T_Y(q),I_Y(q), F_Y(q)): qÎW} are any two NSs over a fixed set W. Then, XÍY if and only if T_X(q) £ T_Y(q), I_X(q) ³ I_Y(q), F_X(q) ³ F_Y(q), for all qÎW.

Definition 2.7.[46] Suppose that X={(q,T_X(q),I_X(q),F_X(q)): qÎW} and Y = {(q,T_Y(q),I_Y(q), F_Y(q)): qÎW} are any two NSs over a fixed set W. Then, the intersection of X and Y is defined by

XÇY={(q, min{T_X(q),T_Y(q)}, max{I_X(q),I_X(q)}, max{F_X(q),F_X(q)}): qÎW}.

Definition 2.8.[46] Assume that X={(q,T_X(q),I_X(q),F_X(q)): qÎW} and Y = {(q,T_Y(q),I_Y(q), F_Y(q)): qÎW} are any two NSs over a fixed set W. Then, the union of X and Y is defined by

XÈY={(q, max{T_X(q),T_Y(q)}, min{I_X(q),I_X(q)}, min{F_X(q),F_X(q)}): qÎW}.

Definition 2.9.[46] Suppose that X={(q,T_X(q),I_X(q),F_X(q)): qÎW} is an NS over a fixed set W. Then, the complement of X is defined by X^c={(q,1-T_X(q),1-I_X(q),1-F_X(q)): qÎW}.

3. Neutrosophic Infi-Topological space

In this section, an attempt is made to introduce the notion of neutrosophic infi-topological space (NITS) as an extension of infi-topological space and fuzzy infi-topological space, and study some of its basic properties. Further, we formulate several interesting results on NITSs.

Definition 3.1. Suppose that W is a fixed set. Then, a family Á of NSs over W is said to be an neutrosophic Infi-Topology (NIT) on W if the following two conditions holds:

(i) 0_N, 1_N ÎÁ;

(ii) Y₁, Y₂ ÎÁ Þ Y₁ÇY₂ ÎÁ.

Then, the structure (W, Á) is called an NITS. Every member of Á is called an neutrosophic infi-open set (NIOS). If YÎÁ, then Y^c is said to be an neutrosophic infi-closed set (NICS).

Clearly, every NTS is an NITS.

Example 3.1. Suppose that W={a, b} is a fixed set. Let Á={0_N, 1_N, P, Q} be a collection of NSs such that P={(a,0.5,0.3,0.6), (b,0.9,0.8,0.2)} and Q={(a,0.8,0.2,0.5), (b,1.0,0.5,0.2)}. Then, Á is an NIT on W. Therefore, (W, Á) is an NITS.

Remark 3.1. In an NITS (W, Á), the null NS (0_N) and the absolute NS (1_N) are both NIOS and NICS.

The notion of neutrosophic infi-interior i.e., N_i-int and neutrosophic infi-closure i.e., N_i-cl of an NS are defined as follows:

Definition 3.2. Assume that (W, Á) is an NITS. Suppose that X is an NS over W. Then, the neutrosophic infi-interior of X i.e., N_i-int(X) is the union of all NIOSs contained in X and the neutrosophic infi-closure of X i.e., N_i-cl(X)  is the intersection of all NICSs containing X.

Therefore, N_i-int(X) = È{Y: YÍX and Y is an NIOS in (W, Á)} and N_i-cl(X) = Ç{Z: XÍZ and Z is an NICS in (W,Á)}.

Remark 3.2. Clearly, N_i-int(X) is the largest NIOS in (W, Á) which is contained in X and N_i-cl(X) is the smallest NICS in (W, Á) that contains X.

Theorem 3.1. Suppose that (W, Á) is an NITS. Assume that Q and R are any two NSs over W. Then, the following properties hold:

(i) N_i-int(Q) Í Q Í P-N_i-cl(Q);

(ii) Q Í R Þ P-N_i-cl(Q) Í P-N_i-cl(R);

(iii) Q Í R Þ P-N_i-int(Q) Í P-N_i-int(R);

(iv) P-N_i-cl(QÈR) = P-N_i-cl(Q) È P-N_i-cl(R);

(v) P-N_i-cl(QÇR) Í P-N_i-cl(Q) Ç P-N_i-cl(R);

(vi) P-N_i-int(QÈR) Ê P-N_i-int(Q) È P-N_i-int(R);

(vii) P-N_i-int(QÇR) Í P-N_i-int(Q) Ç P-N_i-int(R).

Proof. (i) It is known that N_i-int(Q) = È{R: R is an NIOS in (W, Á) and RÍQ}. Since, each RÍQ, so È{R: R is an NIOS in (W, Á) and RÍQ} Í Q, i.e., N_i-int(Q)ÍQ.

Again, N_i-cl(Q) = Ç{Z: Z is an NICS in (W, Á) and QÍZ}. Since, each ZÊQ, so Ç{Z: Z is an NICS in (W, Á) and QÍZ} Ê Q, i.e., N_i-cl(Q)ÊQ.

Therefore, N_i-int(Q)ÍQÍ N_i-cl(Q).

(ii) Suppose that (W, Á) be an NITS. Assume that Q and R are two NSs over W such that QÍR.

Now, N_cl(Q) = Ç{Z: Z is an NICS in (W, Á) and QÍZ}

                       Í Ç{Z: Z is an NICS in (W, Á) and RÍZ}           [Since QÍR]

                       = N_i-cl(R)

Þ N_i-cl(Q) Í N_i-cl(R).

Therefore, QÍR Þ N_i-cl(Q)ÍN_i-cl(R).

(iii) Assume that (W, Á) is an NITS. Suppose that Q and R are two NSs over W such that QÍR.

Now, N_i-int(Q) = È{Z: Z is an NIOS in (W, Á) and ZÍQ}

                         Í È{Z: Z is an NIOS in (W, Á) and ZÍR}           [Since QÍR]

                          = N_i-int(R)

Þ N_i-int(Q) Í N_i-int(R).

Therefore, QÍR Þ N_i-int(Q)Í N_i-int(R).

(iv) Suppose that Q and R are any two neutrosophic subsets of an NITS (W, Á). It is known that QÍQÈR and RÍQÈR.

Now, Q Í QÈR

Þ N_i-cl(Q) Í N_i-cl(QÈR);

and R Í QÈR

Þ N_i-cl(R) Í N_i-cl(QÈR).

Therefore, N_i-cl(Q)ÈN_i-cl(R) Í N_i-cl(QÈR)                                                                                                                   (1)

We have, QÍN_i-cl(Q), RÍN_i-cl(R). Therefore, QÈR Í N_i-cl(Q)ÈN_i-cl(R).

Further, it is known that N_i-cl(Q)ÈN_i-cl(R) is an NICS in (W, Á). It is clear that, N_i-cl(Q)ÈN_i-cl(R) is an NICS in (W, Á), which contains QÈR. But it is known that N_i-cl(QÈR) is the smallest NICS in (W, Á), which contains QÈR. Therefore, N_i-cl(QÈR) Í N_i-cl(Q)ÈN_i-cl(R)                                                                                                                   (2)

From eq. (1) and eq. (2), we have N_i-cl(QÈR) = N_i-cl(Q)ÈN_i-cl(R).

(v) Suppose that Q and R are any two neutrosophic subsets of an NITS (W, Á). It is known that QÇRÍQ, QÇRÍR.

Now, QÇRÍQ

Þ N_i-cl(QÇR) Í N_i-cl(Q);

and QÇRÍR

Þ N_i-cl(QÇR) Í N_i-cl(R).

Therefore, N_i-cl(QÇR) Í N_i-cl(Q)ÇN_i-cl(R).

(vi) Assume that Q and R are two neutrosophic subsets of an NITS (W, Á). It is known that QÍQÈR and RÍQÈR.

Thus, we get

QÍQÈR

Þ N_i-int(Q) Í N_i-int(QÈR);

and RÍQÈR

Þ N_i-int(R) Í N_i-int(QÈR).

Therefore, N_i-int(Q)ÈN_i-int(R) ÍN_i-int(QÈR).

(vii) Suppose that Q and R are two neutrosophic subsets of an NITS (W, Á). It is known that QÇRÍQ, QÇRÍR.

Now, QÇRÍQ

Þ N_i-int(QÇR) Í N_i-int(Q);

and QÇRÍR

Þ N_i-int(QÇR) Í N_i-int(R).

Therefore, N_i-int(QÇR) Í N_i-int(Q)ÇN_i-int(R).

Theorem 3.2. Let Q be an neutrosophic subset of an NITS (W, Á). Then, the following properties hold:

(i) (N_i-int(Q))^c = N_i-cl(Q^c);

(ii) (N_i-cl(Q))^c = N_i-int(Q^c).

Proof. (i) Suppose that (W, Á) is an NITS, and Q={(w, T_Q(w), I_Q(w), F_Q(w)): wÎW} is an neutrosophic subset of W.

Therefore, P-N_int(Q)

= È{Z_i : iÎD and Z_i is an NIOS in (W, Á) such that Z_i Í Q}

= {(w, Ú (w), Ù (w), Ù (w)) : w ÎW}, where for all iÎD and Z_i is an NIOS in (W, Á) such that Z_iÍQ.

This implies, (N_i-int(Q))^c = {(w, Ù (w), Ú (w), Ú (w)): wÎW}.

Since, Ù (w) £ (w), Ú (w) ³ (w), Ú (w) ³ (w), for each iÎD and wÎW, so N_i-cl(Q^c) = {(w, Ù (w), Ú (w), Ú (w)): wÎW}} = Ç{Z_i: iÎD and Z_i is an NICS in (W, Á) such that Q^cÍZ_i}. Therefore, (N_i-int(Q))^c = N_i-cl(Q^c).

(ii) Let (W, Á) be an NITS, and Q = {(w, T_­Q(w), I_Q(w), F_Q(w)): wÎW} be an neutrosophic subset of W.

Therefore, P-N_cl(Q)

= Ç{Z_i: iÎD and Z_i is an NICS in (W, Á) such that Z_i ÊQ}

= {(w, Ù (w), Ú (w), Ú (w)): wÎW}, where Z_i is an NICS in (W, Á) such that Z_i  ÊQ, for all iÎD.

This implies, (N_i-cl(Q))^c = {(w, Ú (w), Ù (w), Ù (w)): wÎW}.

Since Ú (w) ³ (w),  £ (w), Ù (w) £ (w), for each iÎD and wÎW, so N_i-int(Q^c) = {(w, Ú (w), Ù (w), Ù (w)): wÎW} = È{Z_i: iÎD and Z_i is an NIOS in (W, Á) such that Z_i Í Q^c}. Therefore, (N_i-cl(Q))^c = N_i-int(Q^c).

Theorem 3.3. Let X be an neutrosophic subset of an NITS (W, Á). Then, the following properties hold:

(i) Q is an NIOS if and only if N_i-int(Q) = Q;

(ii)Q is a NIOS if and only if N_i-cl(Q) = Q.

Proof. (i) Let Q be an NIOS in an NITS (W, Á). Now, N_i-int(Q) = È{Z: Z is an NIOS in (W, Á) and Z ÍQ}. Since, Q is an NIOS in (W, Á), so Q is the largest NIOS, which is contained in Q. This implies, È{Z: Z is an NIOS in (W, Á) and Z ÍQ} = Q. Therefore, N_i-int(Q) = Q.

(ii) Let Q be an NICS in an NITS (W, Á). Now, N_i-cl(Q) = Ç{Z : Z is an NICS in (W, Á) and QÍZ}. Since, Q is an NICS in (W, Á), so Q is the smallest NICS, which contains Q. This implies, Ç{Z: Z is an NICS in (W, Á) and QÍZ} = Q. Therefore, N_i-cl(Q) = Q.

Definition 3.3. Let (W, Á) be an NITS. Then X, an NS over W is called as

(i) neutrosophic infi-semi-open (in short NISO) set if and only if X Í N_i-cl(N_i-int(X));

(ii) neutrosophic infi-pre-open (in short NIPO) set if and only if X Í N_i-int(N_i-cl(X).

Remark 3.3. The complement of NISO set and NIPO set in an NITS (W, Á) are called neutrosophic infi-semi-closed (in short NISC) set and neutrosophic infi-pre-closed (in short NIPC) set respectively.

Theorem 3.4. Suppose that (W, Á) is an NITS. Then,

(i) every NIOS is an NISO set.

(ii) every NIOS is an NIPO set.

Proof. (i) Let (W, Á) be an NITS. Let X be an NIOS. Therefore, X=N_i-int(X). It is known that XÍN_i-cl(X). This implies, XÍN_i-cl(N_i-int(X)). Therefore, X is an NISO set in (W, Á).

(ii) Let (W, Á) be an NITS. Let X be an NIOS. Therefore, X=N_i-int(X). It is known that, XÍN_i-cl(X). This implies, N_i-int(X)ÍN_i-int(N_i-cl(X)) i.e. X=N_i-int(X) ÍN_i-int(N_i-cl(X)). Therefore, XÍN_i-int(N_i-cl(X)). Hence, X is an NIPO set in (W, Á).

Remark 3.4. The converse of the theorem 3.4 may not be true in general, which follows from the following example.

Example 3.2. Assume that (W, Á) is an NITS, where Á={0_N, 1_N, {(a,0.4,0.4,0.3), (b,0.3,0.3,0.4)}, {(a,0.6,0.4,0.1), (b,0.4,0.1,0.3)}}. Then,

(i) Q={(a,0.6,0.4,0.1), (b,0.8,0.1,0.2)} is an NISO set but it is not an NIOS in (W, Á).

(ii) P={(a,0.7,0.9,0.2), (b,0.7,0.4,0.3)} is an NIPO set but it is not an NIOS in (W, Á).

Theorem 3.5. In an NITS (W, Á), the union of any two NISO sets is also an NISO set.

Proof. Suppose that X and Y be any two NISO sets in an NITS (W, Á). Therefore,

XÍN_i-cl(N_i-int(X))                                                                                                                                                           (3)

and YÍN_i-cl(N_i-int(Y))                                                                                                                                                    (4)   

From eq. (3) and eq. (4), we have

XÈY Í N_i-cl(N_i-int(X))È N_i-cl(N_i-int(Y))

        = N_i-cl(N_i-int(X)È N_i-int(Y))

          Í N_i-cl(N_i-int(XÈY)).

Therefore, XÈY Í N_i-cl(N_i-int(XÈY)). Hence, XÈY is an NISO set in (W, Á).

Theorem 3.6. In an NITS (W, Á), the union of any two NIPO sets is also an NIPO set.

Proof. Suppose that X and Y are any two NIPO sets in an NITS (W, Á). Therefore,

XÍ N_i-int(N_i-cl(X))                                                                                                                                                          (5)

and YÍ N_i-int(N_i-cl(Y))                                                                                                                                                   (6)

From eq. (5) and eq. (6), we have,

XÈYÍ N_i-int(N_i-cl(X))ÈN_i-int(N_i-cl(Y))

      Í N_i-int(N_i-cl(X)ÈN_i-cl(Y))

        = N_i-int(N_i-cl(XÈY)).

Therefore, XÈY Í N_i-int(N_i-cl(XÈY)). Hence, XÈY is an NIPO set in (W, Á).

Definition 3.4. Let (W, Á) is an NITS. Then, an NS X over W is called a neutrosophic infi-a-open (in short NI-a-O) set if and only if XÍN_i-int(N_i-cl(N_i-int(X))). The complement of an NI-a-O set is called an neutrosophic infi-a-closed (in short NI-a-C) set.

Corollaries 3.1. In an NITS (W, Á), every NIOS is an NI-a-O set.

Remark 3.5. The converse of the above proposition may not be true in general, which follows from the following example.

Example 3.3. Let us consider an NITS (W, Á) as shown in Example 3.2. Clearly, the NS Q = {(a,0.6,0.4,0.1), (b,0.8,0.1,0.2)} is an NI-a-O set but it is not an NIOS in (W, Á).

Theorem 3.7. In an NITS (W, Á), every NI-a-O set is an NISO set.

Proof. Assume that X be an NI-a-O set in (W, Á). Therefore, XÍN_i-int(N_i-cl(N_i-int (X))). It is known that N_i-int(N_i-cl(N_i-int(X))) ÍN_i-cl(N_i-int(X)). Thus we have, XÍ N_i-cl(N_i-int(X)). Hence, X is a NISO set. Therefore, every NI-a-O set is an NISO set.

Remark 3.6. The converse of the above example may not be true in general, which follows from the following example.

Example 3.4. Suppose that (W, Á) be an NITS, where Á ={0_N, 1_N, {(a,0.6,0.7,0.8), (b,0.5,0.5, 0.6)}, {(a,0.4,0.8,0.8), (b,0.5,0.8,0.8)}}. Then, it can be easily verified that A={(a,0.6,0.3,0.3), (b,0.5,0.4,0.4)} is an NISO set in (W, Á), but it is not an NI-a-O set in (W, Á).

Theorem 3.8. In an NITS (W, Á), every NI-a-O set is an NIPO set.

Proof. Let (W, Á) is an NITS. Assume that X be an NI-a-O set in (W, Á). Therefore, XÍN_i-int(N_i-cl(N_i-int(X))). It is known that, N_i-int(X)ÍX. This implies, N_i-cl(N_i-int(X))Í N_i-cl(X). Which implies N_i-int(N_i-cl(N_i-int(X))) Í N_i-int(N_i-cl(X). Therefore, XÍN_i-int(N_i-cl(X). Hence, X is an NIPO set. Therefore, every NI-a-O set is an NIPO set in (W, Á).

Remark 3.7. The converse of the above example may not be true in general, which follows from the following example.

Example 3.5. Suppose that (W, Á) is an NITS as shown in Example 3.2. Then, the NS P={(a, 0.7,0.9,0.2), (b,0.7,0.4,0.3)} is an NIPO set in (W, Á) but it is not an NI-a-O set in (W, Á).

Definition 3.5. Assume that (W, Á) is an NITS. Then, an NS X over W is called an neutrosophic infi-b-open (in short NI-b-O) set if and only if X Í N_i-int(N_i-cl(X)) È N_i-cl(N_i-int(X)).

Remark 3.8. An NS X is called an neutrosophic infi-b-closed (in short NI-b-C) set iff X^c is an NI-b-O set i.e., if N_i-int(N_i-cl(X))ÇN_i-cl(N_i-int(X)) Í X.

Theorem 3.9. In an NITS (W, Á), every NIPO (NISO) set is an NI-b-O set.

Proof. Let X be an NIPO set in an NITS (W, Á). Therefore, X Í N_i-int(N_i-cl(X)). This implies, X Í N_i-int(N_i-cl(X)) È N_i-cl(N_i-int(X)). Hence, X is an NI-b-O set. Therefore, every NIPO set is an NI-b-O set.

Similarly, it can be easily shown that every NISO set is an NI-b-O set.

Theorem 3.10. The union of any  two NI-b-O sets in an NITS (W, Á) is also an NI-b-O set.

Proof. Suppose that X and Y be any two NI-b-O sets in an NITS (W, Á).

Therefore, X Í N_i-int(N_i-cl(X)) È N_i-cl(N_i-int(X))                                                                                                            (7)

and Y Í N_i-int(N_i-cl(Y)) È N_i-cl(N_i-int(Y))                                                                                                                       (8)

It is known that, X Í XÈY and Y Í XÈY.

Now, X Í XÈY

Þ N_i-int(X) Í N_i-int(A B)

ÞN_i-cl(N_i-int(X)) Í N_i-cl(N_i-int(XÈY))                                                                                                                            (9)

and X Í XÈY

Þ N_i-cl(X) Í N_i-cl(A B)

ÞN_i-int(N_i-cl(X)) Í N_i-int(N_i-cl(XÈY))                                                                                                                          (10)

Similarly, it can be shown that

N_i-cl(N_i-int(Y)) Í N_i-cl(N_i-int(XÈY))                                                                                                                              (11)

N_i-int(N_i-cl(Y)) Í N_i-int(N_i-cl(XÈY))                                                                                                                              (12)

From eq. (7) and eq. (8) we have,

XÈY Í N_i-cl(N_i-int(X)) È N_i-int(N_i-cl(X)) È N_i-cl(N_i-int(Y)) È N_i-int(N_i-cl(Y))

  Í N_i-cl(N_i-int(XÈY)) È N_i-int(N_i-cl(XÈY)) È N_i-cl(N_i-int(XÈY)) È N_i-int(N_i-cl(XÈY))

[ By eqs. (9), (10), (11), & (12)]

        = N_i-cl(N_i-int(XÈY)) È N_i-int(N_i-cl(XÈY))

Þ XÈY Í N_i-cl(N_i-int(XÈY)) È N_i-int(N_i-cl(XÈY)).

Therefore, XÈY is an NI-b-O set.

Hence, the union of two NI-b-O sets is also an NI-b-O set.

Theorem 3.11. In an NITS (W, Á), the intersection of any two NI-b-C sets is also an NI-b-C set.

Proof. Assume that (W, Á) is an NITS. Suppose that X and Y are any two NI-b-C sets in (W, Á). Therefore,

N_i-int(N_i-cl(X)) Ç N_i-cl(N_i-int(X)) Í X                                                                                                                            (13)

and N_i-int(N_i-cl(Y)) Ç N_i-cl(N_i-int(Y)) Í Y                                                                                                                     (14)

Since, XÇY X and XÇY Y, so we get

N_i-int(XÇY) N_i-int(X) Þ N_i-cl(N_i-int(XÇY)) N_i-cl(N_i-int(X))                                                                                    (15)

N_i-cl(XÇY)  N_i-cl(X) Þ N_i-int(N_i-cl(XÇY)) N_i-int(P-N_cl(X))                                                                                   (16)

N_i-int(XÇY)  N_i-int(Y) Þ N_i-cl(N_i-int(XÇY)) N_i-cl(N_i-int(Y))                                                                                   (17)

and N_i-cl(XÇY)  N_i-cl(Y) Þ N_i-int(N_i-cl(XÇY)) N_i-int(N_i-cl(Y))                                                                              (18)

From eq. (13) & (14) we get,

XÇY Ê N_i-int(N_i-cl(X)) Ç N_i-cl(N_i-int(X)) Ç N_i-int(N_i-cl(Y)) Ç N_i-cl(N_i-int(Y))

       Ê N_i-int(N_i-cl(XÇY)) Ç N_i-cl(N_i-int(XÇY)) Ç N_i-int(N_i-cl(XÇY)) Ç N_i-cl(N_i-int(XÇY))

[By eqs. (15), (16), (17) & (18)]

         = N_i-int(N_i-cl(X Y)) Ç N_i-cl(N_i-int(XÇY))

Þ XÇY  N_i-cl(N_i-int(XÇY)) Ç N_i-int(N_i-cl(XÇY)).

Hence, XÇY is an NI-b-C set in (W, Á).

Therefore, the intersection of any two NI-b-C sets is also an NI-b-C set.

Definition 3.6. Suppose that (W, Á₁) and (M, Á₂) are any two NITSs. Then, a one to one and onto mapping x:(W, Á₁)® (M, Á₂) is called as

(i) neutrosophic infi-continuous mapping (NI-C-mapping) if and only if x^-1(L) is an NIOS in W, whenever L is an NIOS in M;

(ii) neutrosophic infi-semi-continuous mapping (NIS-C-mapping) if and only if x^-1(L) is an NISO set in W, whenever L is an NIOS in M;

(iii) neutrosophic infi-pre-continuous mapping (NIP-C-mapping) if and only if x^-1(L) is an NIPO set in W, whenever L is an NIOS in M;

(iv) neutrosophic infi-b-continuous mapping (NIP-C-mapping) if and only if x^-1(L) is an NI-b-O set in W, whenever L is an NIOS in M;

(v) neutrosophic infi-a-continuous mapping (NIP-C-mapping) if and only if x^-1(L) is an NI-a-O set in W, whenever L is an NIOS in M;

Theorem 3.12. Suppose that x:(W, Á₁)® (M, Á₂) and z:(M, Á₂)®(V, Á₃) are any two NI-C-mappings. Then, the composition mapping z x:(W, Á₁)→(V, Á₃) is also an NI-C-mapping.

Proof. Let x:(W, Á₁)® (M, Á₂) and z:(M, Á₂)®(V, Á₃) be any two NI-C-mappings. Assume that Q is an NIOS in (V, Á₃). Since, z:(M, Á₂)®(V, Á₃) is an NI-C-mapping, so z^-1(Q) is an NIOS in (M, Á₂). Further, since x:(W, Á₁)® (M, Á₂) is an NI-C-mapping, so x^-1(z^-1(Q))=(z x)^-1(Q) is an NIOS in (Y, t₁, t₂). Hence, (z x)^-1(Q) is an NIOS in (W, Á₁) whenever Q is an NIOS in (V, Á₃). Therefore, the composition mapping z x:(W, Á₁)→(V, Á₃) is also an NI-C-mapping.

Theorem 3.13. Let x:(W, Á₁)® (M, Á₂) be an NIS-C-mapping, and z:(M, Á₂)®(V, Á₃) be an NI-C-mapping. Then, the composition mapping z x:(W, Á₁)→(V, Á₃) is an NIS-C-mapping.

Proof. Let x:(W, Á₁)® (M, Á₂) be an NIS-C-mapping, and z:(M, Á₂)®(V, Á₃) be an NI-C-mapping. Assume that Q is an NIOS in (V, Á₃). Since, z:(M, Á₂)®(V, Á₃) is an NI-C-mapping, so z^-1(Q) is an NIOS in (M, Á₂). Further, since x:(W, Á₁)® (M, Á₂) is an NIS-C-mapping, so x^-1(z^-1(Q))=(z x)^-1(Q) is an NISO set in (Y, t₁, t₂). Hence, (z x)^-1(Q) is an NISO set in (W, Á₁) whenever Q is an NIOS in (V, Á₃). Therefore, the composition mapping z x:(W, Á₁)→(V, Á₃) is an NIS-C-mapping.

Theorem 3.14. Let x:(W, Á₁)® (M, Á₂) be an NIP-C-mapping, and z:(M, Á₂)®(V, Á₃) be an NI-C-mapping. Then, the composition mapping z x:(W, Á₁)→(V, Á₃) is an NIP-C-mapping.

Proof. Let x:(W, Á₁)® (M, Á₂) be an NIP-C-mapping, and z:(M, Á₂)®(V, Á₃) be an NI-C-mapping. Assume that Q is an NIOS in (V, Á₃). Since, z:(M, Á₂)®(V, Á₃) is an NI-C-mapping, so z^-1(Q) is an NIOS in (M, Á₂). Further, since x:(W, Á₁)® (M, Á₂) is an NIP-C-mapping, so x^-1(z^-1(Q))=(z x)^-1(Q) is an NIPO set in (Y, t₁, t₂). Hence, (z x)^-1(Q) is an NIPO set in (W, Á₁) whenever Q is an NIOS in (V, Á₃). Therefore, the composition mapping z x:(W, Á₁)→(V, Á₃) is an NIP-C-mapping.

Theorem 3.15. Let x:(W, Á₁)® (M, Á₂) be an NI-b-C-mapping, and z:(M, Á₂)®(V, Á₃) be an NI-C-mapping. Then, the composition mapping z x:(W, Á₁)→(V, Á₃) is an NI-b-C-mapping.

Proof. Let x:(W, Á₁)® (M, Á₂) be an NI-b-C-mapping, and z:(M, Á₂)®(V, Á₃) be an NI-C-mapping. Assume that Q is an NIOS in (V, Á₃). Since, z:(M, Á₂)®(V, Á₃) is an NI-C-mapping, so z^-1(Q) is an NIOS in (M, Á₂). Further, since x:(W, Á₁)® (M, Á₂) is an NI-b-C-mapping, so x^-1(z^-1(Q))=(z x)^-1(Q) is an NI-b-O set in (Y, t₁, t₂). Hence, (z x)^-1(Q) is an NI-b-O set in (W, Á₁) whenever Q is an NIOS in (V, Á₃). Therefore, the composition mapping z x:(W, Á₁)→(V, Á₃) is an NI-b-C-mapping.

Theorem 3.16. Let x:(W, Á₁)® (M, Á₂) be an NI-a-C-mapping, and z:(M, Á₂)®(V, Á₃) be an NI-C-mapping. Then, the composition mapping z x:(W, Á₁)→(V, Á₃) is an NI-a-C-mapping.

Proof. Let x:(W, Á₁)® (M, Á₂) be an NI-a-C-mapping, and z:(M, Á₂)®(V, Á₃) be an NI-C-mapping. Assume that Q is an NIOS in (V, Á₃). Since, z:(M, Á₂)®(V, Á₃) is an NI-C-mapping, so z^-1(Q) is an NIOS in (M, Á₂). Further, since x:(W, Á₁)® (M, Á₂) is an NI-a-C-mapping, so x^-1(z^-1(Q))=(z x)^-1(Q) is an NI-a-O set in (Y, t₁, t₂). Hence, (z x)^-1(Q) is an NI-a-O set in (W, Á₁) whenever Q is an NIOS in (V, Á₃). Therefore, the composition mapping z x:(W, Á₁)→(V, Á₃) is an NI-a-C-mapping.

Theorem 3.17.

(i) Every NI-C-mapping is an NIP-C-mapping;

(ii) Every NI-C-mapping is an NIS-C-mapping;

(iii) Every NIP-C-mapping is an NI-b-C-mapping;

(iv Every NIS-C-mapping is an NI-b-C-mapping;

(v) Every NI-C-mapping is an NI-b-C-mapping.

Proof. (i) Let x:(W, Á₁)® (M, Á₂) be an NI-C-mapping, and L be an NIOS in M. Since x is an NI-C-mapping, so x^-1(L) is an NIOS in W. Further, since every NIOS is again an NIPO set, so x^-1(L) is an NIPO set in (W, Á₁). Therefore, x:(W, Á₁)® (M, Á₂) is an NIP-C-mapping.

(ii) Suppose that x:(W, Á₁)® (M, Á₂) is an NI-C-mapping, and L be an NIOS in M. Since x is an NI-C-mapping, so x^-1(L) is an NIOS in W. Further, since every NIOS is again an NISO set, so x^-1(L) is an NISO set in (W, Á₁). Therefore, x:(W, Á₁)® (M, Á₂) is an NIS-C-mapping.

(iii) Suppose that x:(W, Á₁)® (M, Á₂) is an NIP-C-mapping, and L is an NIOS in M. Since x is an NIP-C-mapping, so x^-1(L) is an NIPO set in W. Further, since every NIPO set is an NI-b-O set, so x^-1(L) is an NI-b-O set in W. Therefore, x^-1(L) is an NI-b-O set in W whenever L is an NIOS in M. Hence, x:(W, Á₁)® (M, Á₂)  is an NI-b-C-mapping.

(iv) Suppose that x:(W, Á₁)® (M, Á₂) is an NIS-C-mapping, and L is an NIOS in M. Since x is an NIS-C-mapping, so x^-1(L) is an NISO set in W. Further, since every NISO set is an NI-b-O set, so x^-1(L) is an NI-b-O set in W. Therefore, x^-1(L) is an NI-b-O set in W whenever L is an NIOS in M. Hence, x:(W, Á₁)® (M, Á₂)  is an NI-b-C-mapping.

(v) Suppose that x:(W, Á₁)® (M, Á₂) is an NI-C-mapping, and L is an NIOS in M. Since x is an NI-C-mapping, so x^-1(L) is an NIOS in W. Further, since every NIOS is again an NI-b-O set, so x^-1(L) is an NI-b-O set in (W, Á₁). Therefore, x:(W, Á₁)® (M, Á₂) is an NI-b-C-mapping.

5. Conclusions 

In this article, we introduce the notion of neutrosophic infi-topological space, and study different types of open sets, namely, NIOS, NIPO set, NISO set, NI-b­-O set and NI-a-O set. By defining NIOS, NIPO set, NISO set, NI-b­-O set and NI-a-O set, we formulate some interesting results on NITSs in the form of theorems, propositions, etc. It is hoped that, in the future, based on these notions and various open sets on NITS, many new investigations can be easily done.

Funding: “This research received no external funding.”

Conflicts of Interest: “The authors declare that they have no conflict of interest.”

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