Journal of Indian Acad. Math.  
Vol. 47, No. 1 (2025) pp. 17-29  
ISSN: 0970-5120  
Mahima Thakur 1  
NEUTROSOPHIC SEMI  
IRRESOLUTE MAPPINGS  
-PRE  
Jyoti Pandey Bajpai2  
Anita Singh Banafar3  
and  
S.S.Thakur4  
Abstract. This paper introduces a new class of mappings called  
neutrosophic semi δ-pre irresolute mappings in neutrosophic topological  
spaces and discusses some of its properties and characterizations.  
Keywords: Neutrosophic Set, Neutrosophic Topology, Neutrosophic Semi  
δ-Pre Open Sets, Neutrosophic Semi δ-Pre Continuous and  
Neutrosophic Semi δ-Pre Irresolute mappings.  
Mathematics Subject Classification No.: 54A.  
1. Introduction  
After the introduction of fuzzy sets [15] and intuitionistic fuzzy sets [4],  
Smarandache [10] created a neutrosophic set on a nonempty set by considering three  
components, namely membership, Indeterminacy,and non-membership whose sums  
lie between 0 and 3. In 2008, Lupiáñez [8] introduced the neutrosophic topology as  
an extension of intuitionistic fuzzy topology. Since, 2008 many authors such as  
Lupiáñez [8], Salama et.al. [10, 11], Acikgoz and his coworkers [1], Dhavaseelan  
et.al. [5], Al-Musaw [2], and others have contributed to neutrosophic topological  
spaces. Recently many weak and strong forms of neutrosophic open sets and  
neutrosophic continuity have been investigated by various authors [1, 2, 5, 6, 7, 12–  
14]. In this paper, we introduce a new class of mappings called neutrosophic fuzzy  
semi δ-pre irresolute mappings and obtain some of their characterizations and  
properties.  
18  
M. THAKUR, J. P. BAJPAI, A.S. BANAFAR AND S.S. THAKUR  
2. Preliminaries  
This section contains some basic definitions and preliminary results which  
will be needed in the sequel.  
Definition 2.1 [12]: A Neutrosophic set (NS) in  
X
is a structure  
A {x, µA(x),A(x),A(x) : x X}  
where µA : X ]0,1[,A : X ]0,1[, and  
A  
: X ]0,1[ denote the  
membership, indeterminacy, and non-membership of A, satisfying the condition that  
0 µA(x) A(x) A(x) 3,x X  
.
In real-life applications in scientific and engineering problems, using a  
neutrosophic set with values from a real standard or a non-standard subset of  
0, 1is difficult. Hence, we consider the neutrosophic set which takes the value  
from the closed interval [0,1] and the sum of membership, indeterminacy, and  
non-membership degrees of each element of the universe of discourse lies between 0  
and 3.  
Definition 2.2 [10]: Let  
X
be a nonempty set and let the neutrosophic sets  
A
and neutrosophic set be in the form A {x, µA(x),A(x),A(x) : x X}  
B
,
B {x, µB(x),B(x),B(x) : x X} and let {A : i J} be an arbitrary  
i
family of neutrosophic sets in  
X
. Then:  
A B if µA(x) µB(x),A(x) B(x), and A(x) B(x)  
A B if A B and B A  
(a)  
(b)  
.
.
(c)  
(d)  
Ac {x,A(x),A(x), µA(x) : x X}.  
∩ A {x, µAi(x), Ai(x), Ai(x) : x X}  
.
.
i
(e)  
(f)  
A {x, µAi(x), Ai(x), Ai(x) : x X}  
i
0 {x, 0, 0,1 : x X} and 1 {x,1,1, 0 : x X}  
NEUTROSOPHIC SEMI  
-PRE IRRESOLUTE MAPPINGS  
19  
Definition 2.3 [8, 9]: A neutrosophic topology on a nonempty set  
family τ of neutrosophic sets in that satisfies the following axioms.  
X
is a  
X
(NT1)  
0
and 1  
(NT2) Finite intersection of members of τ is a member of τ  
(NT3) Anyunion of members of τ is a member of τ  
In this case, the pair (X,) is called a neutrosophic topological space and  
each neutrosophic set in τ is known as a neutrosophic open set in . The  
X
complement Ac of a neutrosophic open set A is called a neutrosophic closed set in  
X
.
Definition 2.4 [5]: Let ,, [0,1] and 0 3 . A neutron-  
sophic point x(,,) of is a neutrosophic set in X defined by  
X
(, , ) if y x  
x
(,)(y)   
(0, 0, 1)  
if y x  
Definition 2.5 [1]: Let x(,,) be a neutrosophic point in  
A {x, µA, A, A : x X} is neutrosophic set in  
x (,,) A if and only if µA(x), A, and A(x)  
X
and  
a
X
.
Then  
.
Definition 2.6 [1]: A neutrosophic point x(,,) is said to be quasi-  
coincident ( -coincident, for short) with , denoted by x(,,)qA iff  
q
A
x(,,) Ac . If x(,,) is not quasi-coincident with  
A
, we denote by  
(x(,,)qA)  
.
Definition 2.7 [1]: Two neutrosophic set  
A
and  
B
of  
X
are said to be  
q
-coincident (denoted byA B ) if A Bc  
.
q
20  
M. THAKUR, J. P. BAJPAI, A.S. BANAFAR AND S.S. THAKUR  
Lemma 2.8 [1]: For any two neutrosophic sets  
A
and  
B
of  
X, (A B) A Bc where (A B) is not q-coincident with  
B
.
q
q
Definition 2.9 [9]: Let (X, ) be a NTS and F N(X). Then the  
neutrosophic interior and neutrosophic closure of are defined by:  
A
cl(F) {H : H NC(X) and F H}  
int(F) {K : K and K F}  
Definition 2.10 [3]: A neutrosophic set  
neutrosophic regular open (resp.neutrosophic regular closed) if A int(cl(A))  
(resp.A cl(int(A)))  
A
of a NTS (X, ) is called  
.
Definition 2.11 [1]: The δ-interior (denoted byint ) (resp.δ-closure  
(denoted bycl )) of a neutrosophicset of a NTS(X, ) is the union of all  
neutrosophic regular open sets contained in (resp.intersection of all neutrosophic  
regular closed sets containing)  
A
A
.
Definition 2.12 [3, 6, 13]: A neutrosophic set  
A
of a NTS (X, ) is called  
neutrosophic semi open (resp.neutrosophic pre open, neutrosophic α-open,  
neutrosophic semi preopen, neutrosophic δ-open, neutrosophic δ-preopen,  
neutrosophic δ-semi open, neutrosophic b-open) if  
A cl(int(A))(resp.A int(cl(A))  
,
A int(cl(int(A)))  
,
A cl(int(cl(A)))  
,
A int(A) A int(cl(A))  
,
,
A cl(int(A))  
,
A cl(int(A)) int(cl(A))  
.
Definition 2.13 [11]: A neutrosophic set  
space (X, ) is called neutrosophic semi δ-preopen if there exists an eutrosophic  
δ-pre open set in such that O A cl(O)  
A
of a neutrosophic topological  
O
X
.
The family of all neutrosophic semi δ-pre open set so fan neutrosophic  
topological space (X, ) is denoted byNSPO(X)  
.
Definition 2.14 [11]: A neutrosophic set  
A
in a neutrosophic topological  
space (X, ) is called neutrosophic semi δ-preclosed) ifAc NSPO(X). The  
NEUTROSOPHIC SEMI  
-PRE IRRESOLUTE MAPPINGS  
21  
family of all neutrosophic semi δ-preclosed) sets of an neutrosophictopological space  
(X, ) is denoted byNSPC(X)  
.
Remark 2.15 [11]: Every neutrosophic semi preopen (resp.neutrosophic  
δ-preopen) set is neutrosophic semi δ-preopen. But the separate converse may not be  
true.  
Definition 2.16 [11]: Let (X, ) be an neutrosophic topological space and  
A
be an neutrosophic set of . Then the neutrosophic semi δ-preinterior (denoted by  
X
sp int) and neutrosophic semi δ-preclosure (denoted byspcl ) of  
A
respectively  
defined as follows:  
sp int(A) {O : O A;O NSPO(X)}  
spcl(A) {O : O A;O NSPC(X)}  
,
.
Definition 2.17 [11]: Let  
A
be an neutrosophic set  
A
of an neutrosophic  
is called:  
topological space (X, ) and x(,,) be an neutrosophic point of  
X
.
A
(a) Neutrosophic semi δ-pre neighborhood of x(,,) if there exists an  
neutrosophic set O NSPO(X) such that x(,,) O A  
.
(b) Neutrosophic semi δ-pre  
Q -neighborhood of x(,,) if there exists an  
neutrosophic set O NSPO(X) such that x(, , ) O A  
.
Definition 2.19 [9,11]: A mapping f : (X, ) (Y, ) is called:  
(a) Neutrosophic continuous if f 1(A) is a neutrosophic open set in  
X
for each  
neutrosophic open set  
A
of  
Y
.
(b) Neutrosophic semi δ-pre continuous if  
neutrosophic open set of  
f 1(A) NSPO(X) for every  
A
Y
.
3. Neutrosophic Semi δ-preir Resolute Mappings  
In this section, we introduce the concept of neutrosophic semi δ-pre  
irresolute mappings and study some of their properties in neutrosophic topological  
spaces.  
22  
M. THAKUR, J. P. BAJPAI, A.S. BANAFAR AND S.S. THAKUR  
Definition 3.1: A mapping  
f
from aneutrosophic topological space (X, )  
to another neutrosophic topological space (Y, ) is said to be neutrosophic semi  
δ-pre irresolute if  
NSPO(Y)  
f 1(A) NSPO(X)  
for every neutrosophic set  
.
Remark 3.2: Every neutrosophic semi δ-pre irresolute mapping is  
neutrosophic semi δ-pre continuous but the converse may not be true.  
Example 3.3: Let X {a, b},Y {p, q}, and neutrosophic sets  
as follows:  
U
defined  
U {a, 0.5, 0.4, 0.5 , b, 0.4, 0.4, 0.6 }  
let {0,U, 1} and {0, 1} be neutrosophic topologies on  
X
and  
Y
respectively. Then the mapping f : (X, ) (Y, ) defined by f(a) p and  
f(b) q is neutrosophic semi δ-pre continuous and hence neutrosophic continuous  
but not neutrosophic semi δ-pre irresolute.  
Consider the following example:  
Example 3.4: Example 3.4. LetX {a, b} Y {p, q}, and neutrosophic  
,
sets  
V
defined as follows:  
V {a, 0.4, 0.3, 0.6 , b, 0.5, 0.3, 0.5 }  
let {0, 1} and {0,V, 1} be neutrosophic topologies on  
X
and  
Y
respectively. Then the mapping g : (X, ) (Y, ) defined by g(a) p and  
g(b) q is neutrosophic semi δ-pre irresolute but not neutrosophic continuous.  
Remark 3.5: Example (3.3) and Example (3.4) show that the concepts of  
neutrosophic semi δ-pre irresolute and neutrosophic continuous mappings are  
independent.  
Theorem 3.6: Let f : (X, ) (Y, ) be a mapping then the following  
statements are equivalent:  
 
NEUTROSOPHIC SEMI  
-PRE IRRESOLUTE MAPPINGS  
23  
(a) F is neutrosophic semi δ-pre irresolute  
(b) If f 1(A) NSPC(X) for every neutrosophic set A NSPC(Y)  
.
(c) For every neutrosophic point x(,,) in  
A NSPO(Y) such that f(x(,,)) A there is an neutrosophic  
set O NSPO(X) such that x(,,) O and f(O) A  
X
and every neutrosophic set  
.
(d) For every neutrosophic point x(,,) of  
X
and every neutrosophic  
semi δ-pre neighborhood  
A
of f(x(,,)), f 1(A) is an neutrosophic  
semi δ-pre neighborhood of x(,,)  
.
(e) For every neutrosophic point x(,,) of  
X
and every neutrosophic  
semi δ-pre neighbor hood A of f(x(,,)  
)
, there is an neutrosophic  
semi δ-pre neighborhood  
U
of x(,,) such that f(U) A  
.
(f) For every neutrosophic point x(,,) of  
X
and every neutrosophic set  
A NSPO(Y) such that f(x(,,))qA, there is an neutrosophic set  
O NSPO(X) such that x(, )qO and f(O) A  
.
(g) for every neutrosophic point x(,,) of  
X
and every neutrosophic semi  
δ-pre  
Q
Q
-neighborhood A of f(x(α,η,β)), f-1(A) is an neutrosophic semi  
δ-pre  
-neighborhood of x(,,)  
.
(h) for every neutrosophic point x(,,) of  
semi δ-pre -neighborhood of f(x(,,)  
semi pre -neighborhood  
(j) f(spcl(A)) spcl(f(A)), for every neutrosophicset  
X
and every neutrosophic  
Q
A
)
, there is an neutrosophic  
Q
U
of x(,,) such that f(U) A.  
A
of  
X
.
(j) spcl(f 1(O)) f 1(spcl(O)), for every neutrosophic set  
O
of  
Y
.
24  
M. THAKUR, J. P. BAJPAI, A.S. BANAFAR AND S.S. THAKUR  
(k) f1(sp int(O)) sp int(f 1(O)), for every neutrosophic set  
O
of  
Y
.
Proof: (a)  
(b) Obvious.  
(a) ⇒ (c) Let x(,,) be an eutrosophic point of  
X
and A NSPO(Y)  
such that f(x(,,)) A. Put O f 1(A), then by (a),O NSPO(X) such  
that x(,,) O and f(O) A  
.
(c) ⇒ (a) Let A NSPO(Y)and x(,,) f 1(A) . Then f(x(,,)) A  
.
Now by (c) there is an eutrosophic set O NSPO(X) such that x(,, ) O  
and f(O) A. Then x(,,) O f 1(A). Hence, f 1(A) NSPO(X)  
.
(a) ⇒ (d) Let x(,,) be a neutrosophic point of  
neighborhood of f(x(,,) .Then there is  
X
, and let  
A
be a semi  
set  
δ-pre  
)
an eutrosophic  
U NSPO(X) such that f(x(,,)) U A. Now f 1(U) NSPO(X)  
and f 1(U) f 1(A) . Thus, f 1(A) is an eutrosophic semi δ-pre neighborhood of  
x(,,) in  
X
.
(d) ⇒ (e) Let x(,,) be a neutrosophic point of  
X
, and let  
A
be a semi  
δ-pre neighborhood of f(x(,,)  
)
. Then U f 1(A) is an eutrosophic semi δ-pre  
neighborhood of x(,,) and f(U) f(f 1(A)) A  
.
(e) ⇒ (c) Let x(,,) be an neutrosophic point of  
such that f(x(,,)) A. So there is neutrosophic semi δ-pre neighborhood  
x(,,) in  
set O NSPO(X) such that x(, , ) O U and so f(O) f(U) A  
X
and A NSPO(Y)  
U
of  
X
such that x(,,) U and f(U) A. Hence there is an eutrosophic  
.
(a) ⇒ (f) Let x(,,) be an neutrosophic point of  
X
and A NSPO(Y)  
such that f(x(,, ))q A. Let O f 1(A). Then O NSPO(X)  
,
x(,,)qO  
and f(O) f(f 1(A)) A  
.
NEUTROSOPHIC SEMI  
-PRE IRRESOLUTE MAPPINGS  
25  
(f) ⇒ (a) Let A NSPO(Y)and x(,,) f1(A) clearly f(x(,,)) A  
choose the Neutrosophic point xc defined as  
(, , ) if z x  
xc(,,)(z)   
(1, 1, 0)  
if z x  
Then f(xc  
)
A
and so by(f) there exists an neutrosophic set  
(,,) q  
O NSPO(X)  
such that  
xc  
O
and f(O) A. Now xc(,,)q O  
(,,)q  
implies x(,,) O . Thus, x(,,) f 1A. Hence, f 1(A) NSPO(X)  
.
(f) ⇒ (g) Let x(,,) be an neutrosophic point of  
X
and  
A
be semi  
δ-Q-neighborhood of f(x(, , )). Then there is a neutrosophic open set  
A NSPO(Y) such that f(x(,, ))q A A. By hypothesis, there is a  
1
1
neutrosophic set O NSPO(X) such that x(,, )qO and f(O) A1 . Thus,  
x
(,,)qO f 1(A ) f 1(A). Hence, f 1(A) is an neutrosophic semi δ-pre  
1
Q-neighborhood of x(,,)  
.
(f) ⇒ (h) Let x(,,) be an eutrosophic point of  
X
and A be a semi δ-pre-Q-  
neighborhood of f(x(, , )). Then U f 1(A) is aneutrosophic semi δ-pre-Q-  
neighborhood of x(,,) and f(U) f(f 1(A)) A  
.
(h) ⇒ (f) Let x(,,) be an eutrosophic point of  
such that f(x(,,))qA. Then is neutrosophic semi δ-pre-Q-neighborhood of  
f(x(, , )). So there is an eutrosophic semi δ-pre Q-neighborhood of x(,,)  
such that f(U) A. Now being an eutrosophic semi δ-pre Q-neighborhood of  
x(,,) . Then there exists an eutrosophicset O NSPO(X) such that  
X
and A NSPO(Y)  
A
U
U
x
(,,)qO U . Hence,  
x
(,)qO and f(O) f(U) A  
.
(b) ⇒ (i) Let  
A
be an eutrosophic set of  
X
. Since,A f 1(f(A)), we have  
26  
M. THAKUR, J. P. BAJPAI, A.S. BANAFAR AND S.S. THAKUR  
A f 1(spcl(f 1(A)))  
.
Now  
spcl(f(A)) NSPC(Y)  
and  
hence,  
f 1(spcl(f(A))) NSPC(X) . There for espcl(A) f 1(spcl(f(A))) and  
f(spcl(A)) f(f 1(spcl(A))) spcl(f(A))  
.
(i) ⇒ (b) Let A NSPC(Y) then f(spcl(f 1(A))) spcl(f(f 1(A)))  
spcl(A) A. Hence, spcl(f 1(A)) f 1(A) andsof 1(A) NSPC(X)  
.
(i) ⇒ (j) Let  
O
be any neutrosophic set of  
Y
, then f 1(O) is an  
neutrosophic set of  
X
. Therefore by hypothesis (i),  
f(spcl(f 1(O)))  
spcl(f(f 1(O))) spcl(O) . Hence,spcl(f 1(O)) f 1(spcl(O))  
.
(j) ⇒ (i) Let  
neutrosophic set of  
A
be any neutrosophic set of  
X
, then  
f 1(A) is an  
Y
,and by (j),spcl(f 1(f(A))) f 1(spcl(f(A))). Hence,  
f(spcl(A)) spcl(f(A))  
.
(a) ⇒ (k) Let be any neutrosophic set of  
O
Y
, then spint(O) NSPO(Y)  
and f 1(sp int(O)) NSPO(X)  
.
Since,  
f 1(sp int(O)) f 1(O)  
,
then  
f 1(sp int(O)) sp int(f 1(O))  
.
(i)  
⇒
(a)  
LetO NSPO(Y)  
Thus,  
,
then  
sp int(O) O  
and  
and  
f 1(O) sp int(f 1(O))  
.
f 1(O) sp int(f 1(O))  
f 1(O) NSPO(X). Hence,  
f
is neutrosophic semi δ-pre irresolute  
Definition 3.7: A mapping f : (X, ) (Y, ) is called neutrosophic  
R-open if the image of every neutrosophic open set of is neutrosophic δ-open  
in  
X
Y
.
Theorem  
3.8:  
Iff : (X, ) (Y, )  
is neutrosophic  
δ-almost  
continuous and neutrosophic R-open mapping, then  
f
is neutrosophic semi  
δ-pre irresolute.  
NEUTROSOPHIC SEMI  
-PRE IRRESOLUTE MAPPINGS  
27  
Proof:  
LetA NSPO(Y)  
.
Then there  
exist  
aneutrosophic  
set  
O IFPO(X)  
such thatO A cl(O)  
,
therefore  
f 1(O) f 1(A)  
f 1(cl(O)) cl(f 1(O))because  
f
is neutrosophic R-open. Since  
f
is  
neutrosophic δ-almost continuous and neutrosophic R-open, f 1(O) IFPO(X)  
.
Hence, f 1(A) NSPO(X)  
.
Theorem 3.9: Let f : (X, ) (Y, ) and g : (Y, ) (Z,) be  
neutrosophic semi δ-pre irresolute mappings then gof is neutrosophic semi  
δ-pre irresolute.  
Proof: Let A NSPO(Z). Since  
g
is neutrosophic semi δ-preirresolute,  
g1(A) NSPO(Y)  
.
Therefore,  
(gof)1(A) f 1(g1(A)) NSPO(X)  
,
because  
f
is neutrosophic semi δ-pre irresolute. Hence, gof is neutrosophic semi  
δ-pre irresolute.  
Theorem 3.10: Let f : (X, ) (Y, ) is neutrosophic semi δ-pre  
irresolute and g : (Y, ) (Z,) is neutrosophic semi δ-pre continuous  
mapping, then gof is neutrosophic semi δ-pre continuous.  
Proof: Let  
O
be any neutrosophic open set of  
Z
. Since  
g
is neutrosophic  
semi δ-precontinuousg1(O) NSPO(Y) . Therefore, (gof)1(O) f 1(g1(O))  
NSPO(X) because  
f
is neutrosophic semi δ-pre irresolute. Hence, gof is  
neutrosophic semi δ-precontinuous.  
4. Conclusion  
In this paper, a new class of mappings called neutrosophic fuzzy semi δ-pre  
irresolute mappings have been introduced, it is shown by examples that the concepts  
of neutrosophic fuzzy semi δ-pre irresolute mappings are stronger than the  
neutrosophic fuzzy semi δ-pre continuous mappings and independent of the  
neutrosophic fuzzy continuous mappings. Several characterizations and properties of  
this class of neutrosophic fuzzy mappings have been studied. In the future, we study  
the images and inverse images of neutrosophic compact, and neutrosophic connected  
spaces under these classes of mappings.  
28  
M. THAKUR, J. P. BAJPAI, A.S. BANAFAR AND S.S. THAKUR  
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1,2,3,4. Department of Applied Mathematics,  
Jabalpur Engineering College,  
Jabalpur, 482011, India  
(Received, November 8, 2024)  
(Revised, January 29, 2025)  
1. E-mail-mahimavthakur@gmail.com  
2. E-mail: jyotipbajpai@gmail.com  
3. E-mail-anita.banafar1@gmail.com  
4. E-mail: ssthakur@jecjabalpur.ac.in