Uni e si y o New Mexico
On Fe ma ean Neu osophic e-con inuous and e-i esolu e Maps
Vadi el A1∗, Thamila asi P2, and P iya S3
1,2PG and Resea ch Depa men o Ma hema ics, A igna Anna Go e nmen A s College, Namakkal - 637
002, India; (Vadi el A) a[email p o ec ed], (Thamila asi P) [email p o ec ed]
3Depa men o Ma hema ics, M.Kuma asamy College o Enginee ing, Ka u - 639 113.;
(P iya S) [email p o ec ed]
1,2,3Depa men o Ma hema ics, Annamalai Uni e si y, Annamalai Naga - 608 002, India;
∗Co espondence: (P iya S) [email p o ec ed]; Tel.: (+91 )
Abs ac .The p ima y objec i e o his pape is o in oduce and de elop he concep o Fe ma ean neu o-
sophic e-con inuous maps in he amewo k o Fe ma ean neu osophic opological spaces. This new class o
mappings ex ends he idea o con inui y by inco po a ing he highe exp essi e powe o Fe ma ean neu osophic
se s, which a e capable o handling mo e complex deg ees o u h, inde e minacy, and alsi y han classical
uzzy o in ui ionis ic uzzy se ings. In addi ion o de ining and s udying he basic o mula ion o Fe ma ean
neu osophic e-con inui y, we in es iga e i s undamen al p ope ies, s uc u al beha io , and in e ela ions wi h
o he exis ing ypes o con inui y. Fu he mo e, we examine he no ion o Fe ma ean neu osophic e-i esolu e
maps, which play an impo an ole in he p ese a ion o Fe ma ean neu osophic opological s uc u es unde
mappings. Se e al cha ac e iza ions and p ope ies o hese maps a e p o ided, es ablishing hei signi icance
in ex ending he heo y o neu osophic opology. The esul s p esen ed no only gene alize exis ing concep s
om classical and uzzy opology bu also open new a enues o applica ions o Fe ma ean neu osophic heo y
in decision-making, in o ma ion sys ems, and unce ain da a analysis.
Keywo ds: Fe ma ean neu osophic e-closed se s, e ma ean neu osophic e-con inuous maps and Fe ma ean
neu osophic e-i esolu e maps.
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1. In oduc ion
In 1965, Zadeh [17] in oduced he concep o uzzy se s o model ambigui y and unce -
ain y in eal-wo ld scena ios. La e , in 1986, A anasso [1] ex ended his idea by p opos-
ing he concep o in ui ionis ic uzzy se s (i s’s), which inco po a ed bo h membe ship and
non-membe ship unc ions add essing a limi a ion in Zadehs model ha only conside ed he
membe ship deg ee. Howe e , in p ac ical applica ions, he sum o membe ship and non-
membe ship deg ees can exceed one, indica ing a need o mo e lexible models.
Vadi el A, Thamila asi P, and P iya S, On Fe ma ean Neu osophic e-con inuous and e-i esolu e Maps
Neu osophic Se s and Sys ems, Vol. 97, 2026
To add ess hese challenges, Yage [16] in oduced he Py hago ean uzzy se (p s), whe e
he sum o he squa es o he membe ship and non-membe ship deg ees is cons ained o be
less han o equal o one. Building on his, Senapa i and Yage [9] p oposed he Fe ma ean
uzzy se (F s), whe e he cube o he membe ship and non-membe ship deg ees mus sum
o less han o equal o one. This ex ension p o ides g ea e lexibili y and enhances he
capabili y o managing unce ain y, making Fe ma ean uzzy se s mo e efficien han i s’s and
p s’s in decision-making and o he eal-li e applica ions.
Sma andache [10] la e in oduced he concep o neu osophic se s (NS’s), ep esen ing a
majo ad ancemen in he ield o decision heo y and beyond. Neu osophic se s a e buil on
he idea ha any concep inhe en ly in ol es deg ees o u h (T), inde e minacy (I), and alsi y
(F). Unlike in ui ionis ic uzzy se s whe e membe ship and non-membe ship a e dependen
neu osophic se s allow hese h ee componen s o be mu ually independen , he eby offe ing
g ea e modeling lexibili y o incomple e, inconsis en , and inde e mina e in o ma ion.
This independence makes neu osophic se s a powe ul ool o ep esen ing unce ain y in a
wide ange o ields, including enginee ing, philosophy, economics, and in o ma ion science. In
pa icula , hey suppo complex analyses in ol ing bo h dependen and independen a iables,
con ibu ing signi ican ly o decision-making and s a egic planning.
To le e age he s eng hs o bo h Fe ma ean uzzy se s and neu osophic se s, he Fe ma ean
neu osophic se was de eloped. This hyb id amewo k effec i ely handles unce ain y, im-
p ecision, and inde e minacy in complex decision making con ex s. Fo example, in e alua ing
a local es au an , Fe ma ean uzzy heo y is applied o assess he impo ance o a ibu es
such as quali y, na u alness, eshness, as e, and p esen a ion which a e o en subjec o hu-
man hesi a ion and subjec i e judgmen . By inco po a ing neu osophic p inciples, he model
be e cap u es he agueness and unce ain y in cus ome p e e ences.
The domain o local ood se ice was chosen as a p ac ical applica ion due o he high deg ee
o unce ain y in consume decision making unce ain y ha neu osophic se s can ep esen
mo e accu a ely han adi ional me hods.
F om a opological pe spec i e, Vadi el e al. [12] in oduced he no ion o δ-open se s
wi hin neu osophic opological spaces. P io o his, in 2008, Ekici [4] in oduced e-open se s
in gene al opology. Building on his ounda ion, Seeni asan e al. [8] in 2014 de eloped he
concep o uzzy e-open se s and hei co esponding uzzy e-con inui y. La e , Vadi el e al. [3]
ex ended hese ideas in o he ealm o in ui ionis ic uzzy opological spaces. Mo e ecen ly,
Vadi el and collabo a o s [13–15] ha e made no able con ibu ions by explo ing a ious ypes o
open se s in Fe ma ean uzzy opological spaces, ad ancing he s udy o uzzy and neu osophic
opology.
Vadi el A, Thamila asi P, and P iya S, On Fe ma ean Neu osophic e-con inuous and
e-i esolu e Maps
Neu osophic Se s and Sys ems, Vol. 97, 2026 411
Resea ch Gap: To he bes o ou knowledge, no p io in es iga ion has been ca ied ou on
he concep s o con inui y and i esolu eness wi h espec o Fe ma ean neu osophic e-open
se s wi hin he amewo k o Fe ma ean uzzy opological spaces. While a ious no ions o
con inui y and i esolu e mappings ha e been s udied ex ensi ely in classical opology, uzzy
opology, and in ui ionis ic uzzy opology, he speci ic ea men o hese no ions unde he
iche s uc u e o Fe ma ean neu osophic se s emains unexplo ed in he exis ing li e a u e.
This absence highligh s a signi ican esea ch gap, he eby mo i a ing he p esen s udy o
ini ia e and ad ance he heo y o Fe ma ean neu osophic e-con inuous and e-i esolu e maps.
In his pape , we pu o wa d he no el concep o Fe ma ean neu osophic e-con inui y
wi hin he amewo k o Fe ma ean neu osophic opological spaces. The s udy begins wi h
a o mal de ini ion o his new ype o con inui y, ollowed by an in es iga ion o i s essen ial
p ope ies. To enhance cla i y and demons a e he p ac ical signi icance o he p oposed ideas,
se e al illus a i e examples a e p o ided. These examples no only alida e he heo e ical
amewo k bu also highligh how Fe ma ean neu osophic e-con inui y ex ends and gene al-
izes exis ing no ions o con inui y in uzzy and neu osophic opological se ings. Fu he mo e,
he pape del es in o he s udy o Fe ma ean neu osophic e-i esolu e maps, p esen ing hei
undamen al p ope ies and cha ac e iza ions in a comp ehensi e manne . Special a en ion is
gi en o hei s uc u al beha io and hei ole in p ese ing Fe ma ean neu osophic e-open
se s unde mappings. By es ablishing hese esul s, he pape con ibu es o he en ichmen
o Fe ma ean neu osophic opology and p o ides a ounda ion o u he heo e ical ad-
ancemen s and p ac ical applica ions in a eas in ol ing unce ain y, agueness, and complex
decision-making p ocesses.
2. P elimina ies
De ini ion 2.1. [9] Le Xbe a uni e se o discou se. A Fe ma ean uzzy se (FFs)Fin
Xis an objec ha ing he o m F={< x, αF(x), βF(x)>:x∈X}whe e αF(x) : X→
[0,1] and βF(x) : X→[0,1], including he condi ion 0 ≤(αF(x))3+ (βF(x))3≤1, o all
x∈X. The numbe s αF(x) and βF(x) deno e, espec i ely, he deg ee o membe ship and
he deg ee o non-membe ship o he elemen xin he se F. Fo any FFs F and x∈X,
πF(x) = 3
√1−[(αF(x))3−(βF(x))3] is iden i ied as he deg ee o inde e minacy o x o F.
In he in e es o simplici y, we shall men ion he symbol F= (αF, βF) o he FFs F ={<
x, αF(x), βF(x)>:x∈X}.
De ini ion 2.2. [7] Le Xbe a non-emp y se . A neu osophic se (b ie ly, Ns)Lis an
objec ha ing he o m L={⟨x, µL(x), νL(x), σL(x)⟩:x∈X}whe e µL→[0,1] deno e he
deg ee o membe ship unc ion, νL→[0,1] deno e he deg ee o inde e minacy unc ion and
Vadi el A, Thamila asi P, and P iya S, On Fe ma ean Neu osophic e-con inuous and
e-i esolu e Maps
Neu osophic Se s and Sys ems, Vol. 97, 2026 412
σL→[0,1] deno e he deg ee o non-membe ship unc ion espec i ely o each elemen x∈X
o he se Land 0 ≤µL(x) + νL(x) + σL(x)≤3 o each x∈X.
De ini ion 2.3. [6] A neu osophic opology (b ie ly, N ) on a non-emp y se Xis a amily
τNo neu osophic subse s o Xsa is ying
(i) 0N, 1N∈τN.
(ii) L1∩L2∈τN o any L1, L2∈τN.
(iii) ∪La∈τN,∀La:a∈A⊆τN.
Then (X, τN) is called a neu osophic opological space (b ie ly, N s) in X. The τNelemen s
a e called neu osophic open se s (b ie ly, Nos) in X. A Ns C is called a neu osophic closed
se s (b ie ly, Ncs) iff i s complemen Ccis Nos.
De ini ion 2.4. [11] Le Xbe a non-emp y se . A Fe ma ean neu osophic se (b ie ly, FNs)
Lis an objec ha ing he o m L={⟨x, µL(x), νL(x), σL(x)⟩:x∈X}whe e µL→[0,1]
deno e he deg ee o membe ship unc ion, νL→[0,1] deno e he deg ee o inde e minacy
unc ion and σL→[0,1] deno e he deg ee o non-membe ship unc ion espec i ely o each
elemen x∈X o he se Lsuch ha 0 ≤(µL(x))3+(σL(x))3≤1 and 0 ≤(νL(x))3≤1. Then
0≤(µL(x))3+ (νL(x))3+ (σL(x))3≤2 o all x∈X. He e µL(x) and σL(x) a e dependen
componen s and νL(x) is an independen componen .
The de ini ions o 1FN and 0FN ha will be needed be o e p oceeding o se ope a ions will
be gi en. In [6], possible de ini ions o 1FN and 0FN neu osophic se s a e gi en. In his pape ,
he heo y will be cons uc ed by de ining 0FN and 1FN Fe ma ean neu osophic se s in a single
way. 0FN and 1FN a e de ined as 0FN ={(x, 0,0,1) : x∈X}and 1FN ={(x, 1,1,0) : x∈X}
Now, he union, in e sec ion and complemen de ini ions necessa y o he de ini ion o he
opological space will be gi en. These de ini ions a e gi en in se e al diffe en ways in classical
neu osophic spaces in [2]; o a oid con usion he e, only one me hod will be gi en o se s wi h
Fe ma ean s uc u e, and his me hod is diffe en om he me hod chosen in [6].
De ini ion 2.5. [11] Le Xbe a non-emp y se & he FNs’s L&Min he o m L=
{⟨x, µL(x), νL(x), σL(x)⟩:x∈X},M={⟨x, µM(x), νM(x), σM(x)⟩:x∈X}, hen
(i) 0FN =⟨x, 0,0,1⟩and 1FN =⟨x, 1,1,0⟩,
(ii) L⊆Miff µL(x)≤µM(x), νL(x)≤νM(x) & σL(x)≥σM(x) : x∈X,
(iii) L=Miff L⊆Mand M⊆L,
(i ) 1FN −L={⟨x, σL(x),1FN −νL(x), µL(x)⟩:x∈X}=Lco C(L),
( ) L∪M={⟨x, max(µL(x), µM(x)),max(νL(x), νM(x)),min(σL(x), σM(x))⟩:x∈X},
( i) L∩M={⟨x, min(µL(x), µM(x)),min(νL(x), νM(x)),max(σL(x), σM(x))⟩:x∈X}.
Vadi el A, Thamila asi P, and P iya S, On Fe ma ean Neu osophic e-con inuous and
e-i esolu e Maps
Neu osophic Se s and Sys ems, Vol. 97, 2026 413
De ini ion 2.6. [5] A Fe ma ean neu osophic opology (b ie ly, FN ) on a non-emp y se X
is a amily τFN o Fe ma ean neu osophic subse s o Xsa is ying
(i) 0FN, 1FN ∈τFN,
(ii) L1∩L2∈τFN o any L1, L2∈τFN,
(iii) ∪La∈τFN,∀La:a∈A⊆τFN.
Then (X, τFN) is called a Fe ma ean neu osophic opological space (b ie ly, FN s) in X. The
τFN elemen s a e called Fe ma ean neu osophic open se s (b ie ly, FNos) in X. A FNs C is
called a Fe ma ean neu osophic closed se s (b ie ly, FNcs) iff i s complemen Ccis FNos.
De ini ion 2.7. [5] Le (X, τFN) be FN s on Xand Lbe an FNson X, hen he Fe ma ean
neu osophic in e io o L(b ie ly, FNin (L)) and he Fe ma ean neu osophic closu e o L
(b ie ly, FNcl(L)) a e de ined as
FNin (L) = ∪{I:I⊆L&Iis a FNos in X}
FNcl(L) = ∩{I:L⊆I&Iis a FNcs in X}.
Theo em 2.8. [5] Le Lbe an FNson X. In his case, he ollowing ou p ope ies hold:
(i) FNcl(L) is a closed Fe ma ean neu osophic se ,
(ii) FNcl(1FN) = 1FN,FNcl(0FN) = 0FN,
(iii) FNin (L) is an open Fe ma ean neu osophic se .
(i ) FNin (1FN) = 1FN,FNin (0FN) = 0FN.
Lemma 2.9. [5] Fo any Fe ma ean neu osophic se Ain (X, τFN), we ha e C(FNin (A)) =
FNcl(C(A)) and C(FNcl(A)) = FNin (C(A)).He e C(A) o Adeno es complemen o A.
3. Fe ma ean neu osophic e-con inuous maps
In his sec ion we in oduce Fe ma ean neu osophic e-con inuous maps and s udy some o
i s p ope ies.
De ini ion 3.1. Le (X, τFN) be an FN s and Abe an FNs. Then Ais said o be an Fe ma ean
neu osophic (i) egula open se (FN os in sho ) i A=FNin (FNcl(A)).(ii) egula closed
se (FN cs in sho ) i A=FNcl(FNin (A)).By Lemma 2.9, i ollows ha Ais an FN os iff
¯
Ais an FN cs.
De ini ion 3.2. Le (X, τFN) be an FN s and A={< a, µA(a), νA(a), σA(a)>|a∈X}be an
FNsin X. Then he δ-in e io and he δ-closu e o Aa e deno ed by FNδin (A) and FNδcl(A)
and a e de ined as ollows. FNδin (A) = ∪{G|Gis an FN os and G⊆A},FNδcl(A) = ∩{K|K
is an FN cs and A⊆K}.
Vadi el A, Thamila asi P, and P iya S, On Fe ma ean Neu osophic e-con inuous and
e-i esolu e Maps
Neu osophic Se s and Sys ems, Vol. 97, 2026 414
De ini ion 3.3. Le (X, τFN) be an FN s and A={< a, µA(a), νA(a), σA(a)>|a∈X}be an
FNsin X. A se Ais said o be FN
(i) δ-open se (b ie ly, FNδos) i A=FNδin (A),
(ii) δ-p e open se (b ie ly, FNδPos) i A⊆FNin (FNδcl(A)).
(iii) δ-semi open se (b ie ly, FNδSos) i A⊆FNcl(FNδin (A)).
(i ) δα open se o a-open se (b ie ly, FNδαos o FNaos) i A⊆FNin (FNcl(FNδin (A))).
( ) δβ open se o e∗-open se (b ie ly, FNδβos o FNe∗os) i A⊆FNcl(FNin (FNδcl(A))).
( i) eopen se (b ie ly, FNeos) i A⊆FNcl(FNδin (A)) ∪FNin (FNδcl(A)).
( ii) δ( esp. δ-p e, δ-semi, δα,eand δβ) dense i FNδcl(A) ( esp. FNδPcl(A),
FNδScl(A),FNδαcl(A),FNecl(A) and FNδβcl(A)) = 1FN.
The complemen o an FNδos ( esp. FNδPos, FNδSos, FNδαos,FNeos and FNδβos) is
called an FNδ( esp. FNδP,FNδS,FNδα,FNeand FNδβ) closed se (b ie ly, FNδcs ( esp.
FNδPcs, FNδScs, FNδαcs,FNecs and FNδβcs)) in X.
The amily o all FNδos ( esp. FNδcs, FNδPos, FNδPcs, FNδSos, FNδScs, FNδαos,
FNδαcs, FNeos, FNecs, FNδβos and FNδβcs) o Xis deno ed
by FNδOS(X),( esp. FNδCS(X),FNδPOS(X),FNδPCS(X),FNδSOS(X),FNδSCS(X),
FNδαOS(X),FNδαCS(X),FNeOS(X),FNeCS(X), FNδβOS(X) and FNδβCS(X)).
De ini ion 3.4. Le (X, τFN) be an FN s and A={< a, µA(a), νA(a), σA(a)>|a∈X}be
an FNsin X. Then he FNδ( esp. FNδ-p e, FNδ-semi, FNδα,FNeand FNδβ)-in e io
and he FNδ( esp. FNδ-p e, FNδ-semi, FNδα,FNeand FNδβ)-closu e o Aa e deno ed by
FNδin (A) ( esp. FNδPin (A), FNδSin (A), FNδαin (A), FNein (A) and FNδβin (A)) and
he FNδcl(A) ( esp. FNδPcl(A), FNδScl(A),FNδαcl(A), FNecl(A) and FNδβcl(A)) and a e
de ined as ollows:
FNδin (A) ( esp. FNδPin (A), FNδSin (A),FNδαin (A), FNein (A) and FNδβin (A) ) =
∪{G|Gin a FNδos ( esp. FNδPos, FNδSos, FNδαos, FNeos, and FNδβos) and G⊆A}and
FNδcl(A) ( esp. FNδPcl(A),FNδScl(A),FNδαcl(A),FNecl(A) and FNδβcl(A) ) = ∩{K|K
is an FNδcs ( esp. FNδPcs, FNδScs, FNδαcs, FNecs FNδβcs) and A⊆K}.
De ini ion 3.5. A map : (X, τFN)→(Y, σFN) is called Fe ma ean neu osophic
(i) con inuous (b ie ly, FNC s) i he in e se image o e e y FNos in (Y, σFN) is a FNos in
(X, τFN),
(ii) δ-con inuous (b ie ly, FNδC s) i he in e se image o e e y FNos in (Y, σFN) is a FNδos
in (X, τFN),
(iii) δS-con inuous (b ie ly, FNδSC s) i he in e se image o e e y FNos in (Y, σFN) is a
FNδSos in (X, τFN),
Vadi el A, Thamila asi P, and P iya S, On Fe ma ean Neu osophic e-con inuous and
e-i esolu e Maps
Neu osophic Se s and Sys ems, Vol. 97, 2026 415
(i ) δP-con inuous (b ie ly, FNδPC s) i he in e se image o e e y FNos in (Y, σFN) is a
FNδPos in (X, τFN),
( ) δα-con inuous (b ie ly, FNδαC s) i he in e se image o e e y FNos in (Y, σFN) is a
FNδαos in (X, τFN),
( i) e-con inuous (b ie ly, FNeC s) i he in e se image o e e y FNos in (Y, σFN) is a FNeos
in (X, τFN),
( ii) δβ-con inuous (b ie ly, FNδβC s) i he in e se image o e e y FNos in (Y, σFN) is a
FNδβos in (X, τFN).
P oposi ion 3.6. The s a emen s a e hold bu he con e se does no ue.
(i) E e y FNδC s is a FNC s.
(ii) E e y FNδC s is a FNδSC s.
(iii) E e y FNδC s is a FNδPC s.
(i ) E e y FNδPC s is a FNeC s.
( ) E e y FNδSC s is a FNeC s.
( i) E e y FNδPC s is a FNδβC s.
( ii) E e y FNδSC s is a FNδβC s.
( iii) E e y FNδαC s is a FNδSC s.
(ix) E e y FNδαC s is a FNδPC s.
P oo . We p o e only (i ) and ( ), he o he s a e simila .
(i ) Le λbe a FNos in Y. Since is FNδPC s, −1(λ) is a FNδPos in X. Since e e y
FNδPos is a FNeos, −1(λ) is a FNeos in X. Hence is a FNeC s.
( ) Le λbe a FNos in Y. Since is FNδSC s, −1(λ) is a FNδSos in X. Since e e y
FNδSos is a FNeos, −1(λ) is a FNeos in X. Hence is a FNeC s.
Example 3.7. Le X=Y={a, b}and he FNs’s A1,A2and A3a e de ined as
µA1(a) = 0.8, νA1(a) = 0.8, σA1(a) = 0.1,
µA1(b) = 0.9, νA1(b) = 0.8, σA1(b) = 0.2;
µA2(a) = 0.6, νA2(a) = 0.7, σA2(a) = 0.2,
µA2(b) = 0.5, νA2(b) = 0.7, σA2(b) = 0.6;
µA3(a) = 0.1, νA3(a) = 0.2, σA3(a) = 0.8,
µA3(b) = 0.2, νA3(b) = 0.2, σA3(b) = 0.9.
Le τFN =σFN ={0FN,1FN, A1, A2, A3}be a FN s on Xand le : (X, τFN)→(Y, σFN)
be an iden i y mapping, hen is FNC s ( esp. FNδPC s,FNδPC s,FNeC s and FNδβC s)
bu no FNδC s ( esp. FNδαC s,FNδC s,FNδSC s and FNδSC s), he se A2is a FNos
in Ybu −1(A2) = A2is no FNδos ( esp. FNδαos,FNδos,FNδSos and FNδSos) in X.
Vadi el A, Thamila asi P, and P iya S, On Fe ma ean Neu osophic e-con inuous and
e-i esolu e Maps
Neu osophic Se s and Sys ems, Vol. 97, 2026 416
Example 3.8. Le X=Y={a, b}and he FNs’s A1,A2and A3a e de ined as
µA1(a) = 0.2, νA1(a) = 0.5, σA1(a) = 0.8,
µA1(b) = 0.3, νA1(b) = 0.5, σA1(b) = 0.7;
µA2(a) = 0.1, νA2(a) = 0.5, σA2(a) = 0.9,
µA2(b) = 0.1, νA2(b) = 0.5, σA2(b) = 0.9;
µA3(a) = 0.2, νA3(a) = 0.5, σA3(a) = 0.8,
µA3(b) = 0.4, νA3(b) = 0.5, σA3(b) = 0.6.
Le τFN =σFN ={0FN,1FN, A1, A2, A3}be a FN s on Xand le : (X, τFN)→(Y, σFN)
be an iden i y mapping, hen is FNδSC s bu no FNδC s, he se A1is a FNos in Ybu
−1(A1) = A1is no FNδos in X.
Example 3.9. Le X=Y={a, b, c, d}and he FNs’s A1,A2,A3,A4and A5a e de ined as
µA1(a) = 1, νA1(a) = 0.5, σA1(a) = 0,
µA1(b) = 1, νA1(b) = 0.5, σA1(b) = 0,
µA1(c) = 1, νA1(c) = 0.5, σA1(c) = 0,
µA1(d) = 1, νA1(d) = 0.5, σA1(d) = 0;
µA2(a) = 1, νA2(a) = 0.5, σA2(a) = 0,
µA2(b) = 0, νA2(b) = 0.5, σA2(b) = 1,
µA2(c) = 0.2, νA2(c) = 0.5, σA2(c) = 0.7,
µA2(d) = 0, νA2(d) = 0.5, σA2(d) = 1;
µA3(a) = 0, νA3(a) = 0.5, σA3(a) = 1,
µA3(b) = 1, νA3(b) = 0.5, σA3(b) = 0,
µA3(c) = 0, νA3(c) = 0.5, σA3(c) = 0,
µA3(d) = 0, νA3(d) = 0.5, σA3(d) = 1;
µA4(a) = 1, νA4(a) = 0.5, σA4(a) = 0,
µA4(b) = 1, νA4(b) = 0.5, σA4(b) = 0,
µA4(c) = 0.2, νA4(c) = 0.5, σA4(c) = 0.7,
µA4(d) = 0, νA4(d) = 0.5, σA4(d) = 0.1;
µA5(a) = 0, νA5(a) = 0.5, σA5(a) = 1,
µA5(b) = 0, νA5(b) = 0.5, σA5(b) = 1,
µA5(c) = 0, νA5(c) = 0.5, σA5(c) = 1,
µA5(d) = 0, νA5(d) = 0.5, σA5(d) = 1.
Le τFN =σFN ={0FN,1FN, A1, A2, A3, A4, A5}be a FN s on Xand le : (X, τFN)→
(Y, σFN) be an iden i y mapping, hen is FNδSC s ( esp. FNeC s and FNδβC s) bu no
FNδαC s ( esp. FNδPC s and FNδPC s), he se A2is a FNos in Ybu −1(A2) = A2is
no FNδαos ( esp. FNδPos and FNδPos) in X.
Vadi el A, Thamila asi P, and P iya S, On Fe ma ean Neu osophic e-con inuous and
e-i esolu e Maps
Neu osophic Se s and Sys ems, Vol. 97, 2026 417
F om he abo e P oposi ion 3.6, and he ollowing Example, he implica ions a e hold.
FNC s
FNδC s
FNδαC s
FNδSC s FNeC s FNδPC s
FNδβC s
No e: A→Bdeno es Aimplies B, bu no con e sely.
Theo em 3.10. A map : (X, τFN)→(Y, σFN) is FNeC s iff he in e se image o each FNcs
in Yis FNecs in X.
P oo . Le λbe a FNcs in Y. This implies λcis FNos in Y. Since is FNeC s, −1(λc)
is FNeos in X. Since −1(λc) = ( −1(λ))c, −1(λ) is a FNecs in X.
Con e sely, le λbe a FNcs in Y. Then λcis a FNos in Y. By hypo hesis −1(λc) is FNeos
in X. Since −1(λc) = ( −1(λ))c,( −1(λ))cis a FNeos in X. The e o e −1(λ) is a FNecs in
X. Hence is FNeC s.
De ini ion 3.11. AFN (X, τFN)is said o be an Fe ma ean neu osophic eU1
2(in sho
FNeU1
2)-space, i e e y FNeos in Xis a FNos in X.
Theo em 3.12. Le : (X, τFN)→(Y, σFN) be a FNeC s, hen is a FNC s i Xis a
FNeU1
2-space.
P oo . Le λbe a FNos in Y. Then −1(λ) is a FNeos in X, by hypo hesis. Since Xis a
FNeU1
2-space, −1(λ) is a FNos in X. Hence is a FNeC s.
Theo em 3.13. Le : (X, τFN)→(Y, σFN) be a FNeC s map and g: (Y, σFN)→(Z, ρFN)
be an FNeC s, hen g◦ : (X, τFN)→(Z, ρFN) is a FNeC s.
P oo . Le λbe a FNeos in Z. Then g−1(λ) is a FNos in Y, by hypo hesis. Since is a
FNeC s map, −1(g−1(λ))is a FNeos in X. Hence g◦ is a FNeC s map.
Theo em 3.14. Le : (X, τFN)→(Y, σFN) be a FNeC s map. Then he ollowing condi ions
a e hold.
(i) (FNecl(λ)) ⊆FNcl( (λ)), o all FNcs λ in X.
Vadi el A, Thamila asi P, and P iya S, On Fe ma ean Neu osophic e-con inuous and
e-i esolu e Maps
Neu osophic Se s and Sys ems, Vol. 97, 2026 418
