On fuzzy semi δ – V continuity in fuzzy δ – V topological space
New concepts of fuzzy semi δ – V and fuzzy semi δ – Λ sets were introduced in the work „On fuzzy semi δ – Λ sets and fuzzy semi δ – V sets V – 6” by the authors (J. Trip. Math. Soc., 6, 81 – 88 (2004)). It was shown that the family of all fuzzy semi δ – V sets forms a fuzzy supra topological space...
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Mukherjee, A. Halder, S. 2020-02-10T13:46:11Z 2020-02-10T13:46:11Z 2008 On fuzzy semi δ – V continuity in fuzzy δ – V topological space / A. Mukherjee, S. Halder // Український математичний журнал. — 2008. — Т. 60, № 5. — С. 712–717. — Бібліогр.: 4 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164674 517.5 New concepts of fuzzy semi δ – V and fuzzy semi δ – Λ sets were introduced in the work „On fuzzy semi δ – Λ sets and fuzzy semi δ – V sets V – 6” by the authors (J. Trip. Math. Soc., 6, 81 – 88 (2004)). It was shown that the family of all fuzzy semi δ – V sets forms a fuzzy supra topological space on X denoted by ( X, FS δV ). The aim of this paper is to introduce the concept of fuzzy semi δ – V continuity in a fuzzy δ – V topological space. Finally, some properties, preservation theorems, etc., are studied. Нові поняття нечітких напів δ - V та нечітких напів δ - Λ множин введено у роботі авторів „On fuzzy semi δ - Λ sets and fuzzy semi δ - V sets V - 6" (J. Trip. Math. Soc. - 2004. - 6. - C. 81 - 88). Було показано, що сім'я усіх нечітких напів δ - V множин формує нечіткий супра-топологічний простір в X, що позначається як ( X, FS δ V ). Метою даної статті є введення поняття нестійкої напів δ - V неперервності у нестійкому δ - V топологічному просторі. Також досліджено деякі її властивості, наведено теорему про збереження та інші питання. en Інститут математики НАН України Український математичний журнал Короткі повідомлення On fuzzy semi δ – V continuity in fuzzy δ – V topological space Про нечітку напів δ - V неперервність в нечіткому δ - V топологічному просторі Article published earlier |
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On fuzzy semi δ – V continuity in fuzzy δ – V topological space |
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On fuzzy semi δ – V continuity in fuzzy δ – V topological space Mukherjee, A. Halder, S. Короткі повідомлення |
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On fuzzy semi δ – V continuity in fuzzy δ – V topological space |
| title_full |
On fuzzy semi δ – V continuity in fuzzy δ – V topological space |
| title_fullStr |
On fuzzy semi δ – V continuity in fuzzy δ – V topological space |
| title_full_unstemmed |
On fuzzy semi δ – V continuity in fuzzy δ – V topological space |
| title_sort |
on fuzzy semi δ – v continuity in fuzzy δ – v topological space |
| author |
Mukherjee, A. Halder, S. |
| author_facet |
Mukherjee, A. Halder, S. |
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Короткі повідомлення |
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Короткі повідомлення |
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2008 |
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English |
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Український математичний журнал |
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Інститут математики НАН України |
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Article |
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Про нечітку напів δ - V неперервність в нечіткому δ - V топологічному просторі |
| description |
New concepts of fuzzy semi δ – V and fuzzy semi δ – Λ sets were introduced in the work „On fuzzy
semi δ – Λ sets and fuzzy semi δ – V sets V – 6” by the authors (J. Trip. Math. Soc., 6, 81 – 88
(2004)). It was shown that the family of all fuzzy semi δ – V sets forms a fuzzy supra topological space
on X denoted by ( X, FS δV ). The aim of this paper is to introduce the concept of fuzzy semi δ – V
continuity in a fuzzy δ – V topological space. Finally, some properties, preservation theorems, etc., are
studied.
Нові поняття нечітких напів δ - V та нечітких напів δ - Λ множин введено у роботі авторів „On fuzzy semi δ - Λ sets and fuzzy semi δ - V sets V - 6" (J. Trip. Math. Soc. - 2004. - 6. - C. 81 - 88). Було показано, що сім'я усіх нечітких напів δ - V множин формує нечіткий супра-топологічний простір в X, що позначається як ( X, FS δ V ). Метою даної статті є введення поняття нестійкої напів δ - V неперервності у нестійкому δ - V топологічному просторі. Також досліджено деякі її властивості, наведено теорему про збереження та інші питання.
|
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/164674 |
| citation_txt |
On fuzzy semi δ – V continuity in fuzzy δ – V topological space / A. Mukherjee, S. Halder // Український математичний журнал. — 2008. — Т. 60, № 5. — С. 712–717. — Бібліогр.: 4 назв. — англ. |
| work_keys_str_mv |
AT mukherjeea onfuzzysemiδvcontinuityinfuzzyδvtopologicalspace AT halders onfuzzysemiδvcontinuityinfuzzyδvtopologicalspace AT mukherjeea pronečítkunapívδvneperervnístʹvnečítkomuδvtopologíčnomuprostorí AT halders pronečítkunapívδvneperervnístʹvnečítkomuδvtopologíčnomuprostorí |
| first_indexed |
2025-11-24T16:10:03Z |
| last_indexed |
2025-11-24T16:10:03Z |
| _version_ |
1850483909304778752 |
| fulltext |
UDC 517.5
A. Mukherjee, S. Halder (Tripura Univ., India)
ON FUZZY SEMI δδδδ – V CONTINUITY
IN FUZZY δδδδ – V TOPOLOGICAL SPACE
PRO NEÇITKU NAPIV δδδδ – V NEPERERVNIST|
V NEÇITKOMU δδδδ – V TOPOLOHIÇNOMU PROSTORI
New concepts of fuzzy semi δ – V and fuzzy semi δ – Λ sets were introduced in the work „On fuzzy
semi δ – Λ sets and fuzzy semi δ – V sets V – 6” by the authors (J. Trip. Math. Soc., 6, 81 – 88
(2004)). It was shown that the family of all fuzzy semi δ – V sets forms a fuzzy supra topological space
on X denoted by ( X, FS
δV
). The aim of this paper is to introduce the concept of fuzzy semi δ – V
continuity in a fuzzy δ – V topological space. Finally, some properties, preservation theorems, etc., are
studied.
Novi ponqttq neçitkyx napiv δ – V ta neçitkyx napiv δ – Λ mnoΩyn vvedeno u roboti avtoriv
„On fuzzy semi δ – Λ sets and fuzzy semi δ – V sets V – 6” (J. Trip. Math. Soc. – 2004. – 6. – C. 81
– 88). Bulo pokazano, wo sim’q usix neçitkyx napiv δ – V mnoΩyn formu[ neçitkyj supra-topo-
lohiçnyj prostir v X, wo poznaça[t\sq qk ( X, FS
δV
). Metog dano] statti [ vvedennq ponqttq
nestijko] napiv δ – V neperervnosti u nestijkomu δ – V topolohiçnomu prostori. TakoΩ do-
slidΩeno deqki ]] vlastyvosti, navedeno teoremu pro zbereΩennq ta inßi pytannq.
1. Introduction. The notion of fuzzy semi δ – V and fuzzy semi δ – Λ set has been
introduced by the authors in [1]. It was shown that the set of all fuzzy semi δ – V sets
forms a fuzzy supra topological space denoted by ( )X FS V, δ .
In Section 2, we study some properties on fuzzy semi δ – V continuity. Some
theorems are also studied. Finally, in Section 3, we study some relationships between
the above two continuities. Here we denote the complement of λ, i.e., 1 – λ , by λc
,
and the fuzzy topological space is denoted by ( X, F ).
Below, we present several definitions and results for reader’s convenience.
1.1 [2]. Let S be a fuzzy subset of a fuzzy space ( X, F ). We define Λδ ( S ) and
Vδ ( S ) as follows
Λδ ( S ) = inf { G : S ≤ G, G ∈ Fδ O ( X, F ) }
and
Vδ ( S ) = sup { D : D ≤ S, D ∈ Fδ C ( X, F ) }.
1.2 [2]. Fuzzy subsets S, Q, and { Sj , j ∈ J } of a fuzzy space ( X, F ) possess the
following properties:
(1) S ≤ Λδ ( S ),
(2) Q ≤ S implies that Λδ ( Q ) ≤ Λδ ( S ),
(3) Λδ Λδ ( S ) = Λδ ( S ),
(4) if S ∈ Fδ O ( X, F ), then S = Λδ ( S ),
(5) Λδ { ∪ ( Sj , j ∈ J ) } = ∪ { Λδ ( Sj ) , j ∈ J },
(6) Λδ { ∩ ( Sj , j ∈ J ) } ≤ ∩ { Λδ ( Sj ) , j ∈ J },
(7) Λδ ( 1 – S ) = 1 – Vδ ( S ),
(8) Λδ ( 0 ) = 0 and Λδ ( 1 ) = 1,
(9) V q A iff V q Λδ ( A ), where V is a δ-closed subset of X.
1.3 [2]. Fuzzy subsets S, Q, and { Sj , j ∈ J } of a space ( X, F ) possess the fol-
© A. MUKHERJEE, S. HALDER, 2008
712 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ON FUZZY SEMI δ – V CONTINUITY IN FUZZY δ – V TOPOLOGICAL SPACE 713
lowing properties:
(1) Vδ ( 0 ) = 0 and Vδ ( 1 ) = 1,
(2) Vδ ( S ) ≤ S,
(3) Q ≤ S implies that Vδ ( Q ) ≤ Vδ ( S ),
(4) Vδ Vδ ( S ) = Vδ ( S ),
(5) if S ∈ Fδ C ( X, F ), then S = Vδ ( S ),
(6) Vδ { ∩ ( Sj , j ∈ J ) } = ∩ { Vδ ( Sj ) , j ∈ J },
(7) ∪ { Vδ ( Sj ) , j ∈ J } ≤ Vδ { ∪ ( Sj , j ∈ J ) },
(8) M q A iff Vδ ( A ) q M, where M is a fuzzy δ-open subset of X.
1.4 [2]. A fuzzy subset S of a fuzzy space ( X, F ) is called a fuzzy δ – Λ set
(resp., a fuzzy δ – V set) if S = Λδ ( S ) (resp., Vδ ( S ) = S ).
1.5 [2]. Let S be a fuzzy subset of a fuzzy space ( X, F ). We define η δ ( S ) and
γ δ ( S ) as follows:
η δ ( S ) = Vδ Λδ ( S )
and
γ δ ( S ) = Λδ Vδ ( S ).
1.6 [2]. Fuzzy subsets S, Q, and { Sj , j ∈ J } of a space ( X, F ) possess the fol-
lowing properties:
(1) Q ≤ S implies that η δ ( Q ) ≤ η δ ( S ),
(2) Vδ ( S ) ≤ η δ ( S ) ≤ Λδ ( S ),
(3) η δ ( 0 ) = 0, η δ ( 1 ) = 1,
(4) if S ∈ Fδ O ( X, F ) , then η δ ( S ) = Vδ ( S ),
(5) η δ ( 1 – S ) = 1 – γ δ ( S ),
(6) η δ ( S ) is bounded,
(7) η δ { ∪ ( Sj , j ∈ J ) } ≥ ∪ { η δ ( Sj ) , j ∈ J },
(8) ∩ { η δ ( Sj ) , j ∈ J } ≥ η δ { ∩ ( Sj , j ∈ J ) },
(9) V q A iff η δ ( A ) q Λδ ( V ), where V is a δ-closed subset of X.
1.7 [2]. Fuzzy subsets S, Q, and { Sj , j ∈ J } of a fuzzy space ( X, F ) possess the
following properties:
(1) Q ≤ S implies that γ δ ( Q ) ≤ γ δ ( S ),
(2) Vδ ( S ) ≤ γ δ ( S ) ≤ Λδ ( S ),
(3) γ δ ( 0 ) = 0, γ δ ( 1 ) = 1,
(4) if S ∈ Fδ C ( X, F ) then γ δ ( S ) = Λδ ( S ),
(5) γ δ ( S ) is bounded,
(6) γ δ { ∪ ( Sj , j ∈ J ) } ≥ ∪ { γ δ ( Sj ) , j ∈ J },
(7) γ δ { ∩ ( Sj , j ∈ J ) } ≤ ∩ γ δ { ( Sj ) , j ∈ J },
(8) V q A iff γ δ ( A ) q Vδ ( V ), where V is a δ-open subset of X.
1.8 [2]. A fuzzy subset S of a fuzzy space ( X, F ) is called a fuzzy δ – η set
(resp., a fuzzy δ – γ set) iff S = η δ ( S ) (resp., S = γ δ ( S ) ).
1.9 [2]. A fuzzy set S of a fuzzy space X is a δ – η set iff Sc is a fuzzy δ – γ
set.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
714 A. MUKHERJEE, S. HALDER
1.10 [2]. The families of all fuzzy δ – Λ sets and all fuzzy δ – V sets form fuzzy
Alexandroff spaces. We denote them by ( X, FΛδ
) and ( X, FVδ
), respectively, and call
them the fuzzy δ – Λ topological space and the fuzzy δ – V topological space,
respectively.
1.11 [1]. A fuzzy subset S of a fuzzy space ( X, F ) is called a fuzzy semi δ – V
set if there exists a fuzzy δ – V set H such that
H ≤ S ≤ Λδ ( H ) = γ S ( H ),
and a fuzzy subset G is said to be a fuzzy semi δ – Λ set if there exists a fuzzy δ – Λ
set K such that Vδ ( K ) ≤ G ≤ K.
1.12 [1]. If S is a fuzzy δ – V set, then it is obviously a fuzzy semi δ – V set, but
the converse statement is not necessarily true. Similarly, if G is a fuzzy δ – Λ set,
then it is obviously a fuzzy semi δ – Λ set.
1.13 [1]. For any fuzzy subset λ of a fuzzy space X, the following statements are
equivalent:
(a) λ is a fuzzy semi δ – V set,
(b) λc is a fuzzy semi δ – Λ set,
(c) η δ ( λc
) ≤ λc,
(d) γ δ ( λ ) ≥ λ.
1.14 [1]. Let S be a fuzzy subset of a fuzzy space ( X, F ). We define ωδ ( S ) and
mδ ( S ) as follows:
ωδ ( S ) = ∩ { G : G ≥ S, G is a fuzzy semi δ – Λ set }
and
mδ ( S ) = ∪ { G : G ≤ S, G is a fuzzy semi δ – V set }.
1.15 [1]. A fuzzy subset S of a fuzzy space ( X, F ) is a fuzzy semi δ – V set
(resp., a fuzzy semi δ – Λ set) iff mδ ( A ) = A [resp., ωδ ( A ) = A ].
1.16 [1]. Fuzzy subsets S, Q, and { Sj , j ∈ J } of a fuzzy space ( X, F ) possess
the following properties:
(1) S ≥ mδ ( S ),
(2) Q ≤ S implies that mδ ( Q ) ≤ mδ ( S ),
(3) mδ mδ ( S ) = mδ ( S ),
(4) mδ { ∪ ( Sj , j ∈ J ) } ≥ ∪ { mδ ( Sj ) , j ∈ J },
(5) mδ { ∩ ( Sj , j ∈ J ) } ≤ ∩ { mδ ( Sj ) , j ∈ J },
(6) mδ ( 1 – S ) = 1 – ωδ ( S ),
(7) mδ ( 0 ) = 0 and mδ ( 1 ) = 1,
(8) mδ ( S ) is a fuzzy semi δ – V set,
(9) mδ ( S ) ≤ Λδ ( S ).
1.17 [1]. Fuzzy subsets S, Q, and { Sj , j ∈ J } of a fuzzy space ( X, F ) possess
the following properties:
(1) S ≤ ωδ ( S ),
(2) Q ≤ S implies that ωδ ( Q ) ≤ ωδ ( S ),
(3) ωδ ωδ ( S ) = ωδ ( S ),
(4) ωδ { ∪ ( Sj , j ∈ J ) } ≥ ∪ { ωδ ( Sj ) , j ∈ J },
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ON FUZZY SEMI δ – V CONTINUITY IN FUZZY δ – V TOPOLOGICAL SPACE 715
(5) ωδ { ∩ ( Sj , j ∈ J ) } ≤ ∩ { ωδ ( Sj ) , j ∈ J },
(6) ωδ ( 0 ) = 0 and ωδ ( 1 ) = 1,
(7) ωδ ( S ) is a fuzzy semi δ – Λ set,
(9) Vδ ( S ) ≤ ωδ ( S ).
1.18 [1]. The family of all fuzzy semi δ – V sets forms a fuzzy supra topological
space and is denoted by FS
δV ; the supra topological space may be denoted by
( X, FS
δV
) and can be written as a fuzzy supra semi δ – V topological space.
1.19 [3]. A function f : ( )X FS V, 1
δ → ( )Y FS V, 2
δ is said to be fuzzy supra semi δ –
– V continuous if the inverse image of every fuzzy semi δ – V set in Y is a fuzzy
semi δ – V set in X.
1.20 [2]. A function f : ( )X F, 1
Λδ → ( )Y F, 2
Λδ is called fuzzy Λδ
- continuous iff
the inverse image of a fuzzy Λδ
- open set in ( )Y F, 2
Λδ is a fuzzy Λδ
-open set in
( )X F, 1
Λδ .
1.21 [4]. A function f : ( X, F1 ) → ( Y, F2 ) is called fuzzy δ – Λ continuous iff the
inverse image of a δ – Λ set in ( Y, F2 ) is a fuzzy δ-open set in ( X, F1 ).
2. Fuzzy semi δδδδ – V continuity in a fuzzy δδδδ – V topological space. In this sec-
tion, we introduce fuzzy semi δ – V continuity in the fuzzy δ – V topological space.
Some related properties are to be studied. Some important theorems are also intro-
duced.
Definition 2.1. A function f : ( )X FV, 1
δ → ( )Y FV, 2
δ from a fuzzy topological
space X into a fuzzy topological space Y is called fuzzy semi δ – V continuous if
f –
1
( B ) is a fuzzy semi δ – V set of X for each B ∈ FV
2
δ .
Example 2.1. Let X = { a, b, c } and let A and B be fuzzy sets of X defined as
follows:
A ( a ) = 0.3, A ( b ) = 0.2, A ( c ) = 0.4,
B ( a ) = 0.15, B ( b ) = 0.1, B ( c ) = 0.2.
We set F1 = { 0, A, 1 } and F2 = { 0, B, 1 } and define f : ( )X FV, 1
δ → ( )Y FV, 2
δ as
f ( x ) = x. It is clear that FV
1
δ = { 0, Ac, 1 } and FV
2
δ = { 0, Bc, 1 }. Here, f –
1
( 0 ) = 0,
f –
1
( 1 ) = 1, and f –
1
( Bc
) = Bc ∉ FV
1
δ . Then Vδ ( A ) = 0. Hence, 0 = Vδ ( A ) ≤ B ≤ A. If
A is a fuzzy δ – Λ set in X, then B is a fuzzy semi δ – Λ set in X. Hence, Bc is a
fuzzy semi δ – V set in X. Therefore, the inverse image of a fuzzy δ – V set in Y is
a fuzzy semi δ – V set in X. Hence, f is fuzzy semi δ – V continuous but not fuzzy
Vδ
-continuous. Again, f is also fuzzy supra semi δ – V continuous. Taking into
account that Bc is a fuzzy δ – V set and using 1.12, we conclude that Bc is a fuzzy
semi δ – V set.
Theorem 2.1. Let f : ( )X FV, 1
δ → ( )Y FV, 2
δ be a mapping from a fuzzy topologi-
cal space ( )X FV, 1
δ into a fuzzy topological space ( )X FV, 2
δ . Then the following
statements are equivalent:
(i) f is fuzzy semi δ – V continuous;
(ii) f –
1
( B ) is a fuzzy semi δ – Λ set in X for every fuzzy δ – Λ set B in Y ;
(iii) f [ ωδ ( A ) ] ≤ Λδ [ f ( A ) ] for every fuzzy set A of X;
(iv) ωδ [ f –
1
( B ) ] ≤ f –
1
[ Λδ ( B ) ] for every fuzzy set B of Y ;
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
716 A. MUKHERJEE, S. HALDER
(v) f –
1
[ Vδ ( Bc
) ] ≤ mδ [ f –
1
( Bc
) ], where Bc is the complement of B.
Proof. (i) ⇒ (ii). f is fuzzy semi δ – V continuous iff f –
1
( Bc
) is a fuzzy semi
δ – V set in X for every Bc ∈ FV
2
δ iff 1 – f –
1
( Bc
) is fuzzy semi δ – Λ in X for
every 1 – Bc being a fuzzy δ – Λ set in Y iff f –
1
( B ) is a fuzzy semi δ – Λ set in X
for every B being a fuzzy δ – Λ set in Y [from 1.13, 1.2 (7), and 1.4]. The required
implication is proved.
(ii) ⇒ (iii). Let A be a fuzzy set of X and let f ( A ) be a fuzzy set of Y. Then
Λδ [ f ( A ) ] [from 1.2 (3) and 1.4] is a fuzzy δ – Λ set in Y. Hence, it follows from (ii)
that f –
1
Λδ [ f ( A ) ] is a fuzzy semi δ – Λ set in X and
ωδ ( A ) ≤ ωδ [ f –
1 f ( A ) ] ≤ ωδ f –
1
Λδ [ f ( A ) ] [from 1.2 (1)] =
= f –
1
Λδ [ f ( A ) ] [from 1.15].
Therefore, f [ ωδ ( A ) ] ≤ f f –
1
Λδ [ f ( A ) ] ≤ Λδ [ f ( A ) ].
(iii) ⇒ (iv). Let A = f –
1
( B ). Then f ( A ) = f f –
1
( B ) ≤ B, and from 1.2 (2) we have
Λδ [ f ( A ) ] ≤ Λδ ( B ). Hence, it follows from (iii) that f [ ωδ [ f –
1
( B ) ] ] ≤ Λδ ( B ), i.e.,
f –
1 f [ ωδ [ f –
1
( B ) ] ] ≤ f –
1
Λδ ( B ), i.e., ωδ [ f –
1
( B ) ] ≤ f –
1 f [ ωδ [ f –
1
( B ) ] ] ≤ f –
1
Λδ ( B ).
(iv) ⇒ (v). 1 – ωδ [ f –
1
( B ) ] ≥ 1 – f –
1
Λδ ( B ), i.e., f –
1
[ Vδ ( Bc
) ] ≤ mδ [ f –
1
( Bc
) ]
[from 1.2 (7) and 1.16 (6)].
(v) ⇒ (i). Let Bc be a fuzzy δ – V set. Then it follows from 1.4 that Bc = Vδ ( Bc
),
i.e., f –
1
( Bc
) = f –
1
[ Vδ ( Bc
) ] ≤ mδ [ f –
1
( Bc
) ] ≤ f –
1
( Bc
) [from 1.16 (1)]. Hence,
mδ [ f –
1
( Bc
) ] = f –
1
( Bc
).
Thus, f is fuzzy semi δ – V continuous.
Theorem 2.2. Let f : ( )X FV, 1
δ → ( )Y FV, 2
δ be a bijective mapping from a fuzzy
topological space ( )X FV, 1
δ into a fuzzy topological space ( )X FV, 2
δ . Then f is
fuzzy semi δ – V continuous iff f [ mδ ( Ac
) ] ≥ Vδ [ f ( Ac
) ].
Proof. f [ mδ ( Ac
) ] ≥ Vδ [ f ( Ac
) ] iff 1 – f [ mδ ( Ac
) ] ≤ 1 – Vδ [ f ( Ac
) ] iff f [ ω δ ( A ) ] ≤
≤ Λδ [ f ( A ) ] [from 1.2 (7) and 1.16 (6)] iff f is fuzzy semi δ – V continuous [from
Theorem 2.1].
Theorem 2.3. Let f : ( )X FV, 1
δ → ( )Y FV, 2
δ be a mapping from a fuzzy topologi-
cal space ( )X FV, 1
δ into a fuzzy topological space ( )X FV, 2
δ . Then the following
statements are equivalent:
(i) f is fuzzy semi δ – V continuous;
(ii) η δ [ f –
1
( B ) ] ≤ f –
1
[ Λδ ( B ) ] for any fuzzy set B of Y ;
(iii) γδ [ f –
1
( Bc
) ] ≥ f –
1
[ Vδ ( Bc
) ], where Bc is the complement of B;
(iv) f [ η δ ( A ) ] ≤ Λδ f ( A ) for any fuzzy set Ac of X.
Proof. (i) ⇒ (iii). It follows from 1.3 (4) and 1.4 that Vδ ( Bc
) is a fuzzy δ – V
set in X . If f is fuzzy semi δ – V continuous, then f –
1
[ Vδ ( Bc
) ] is a fuzzy semi
δ – V set in X . Hence, γδ [ f –
1
( Vδ ( Bc
) ) ] ≥ f –
1
[ Vδ ( Bc
) ] [from 1.13], i.e.,
f –
1
[ Vδ ( Bc
) ] ≤ γδ [ f –
1
( Vδ ( Bc
) ) ] ≤ γδ [ f –
1
( Bc
) ] [from 1.3 (2)].
(iii) ⇔ (ii). 1 – f –
1
[ Vδ ( Bc
) ] ≥ 1 – γδ [ f –
1
( Bc
) ] iff η δ [ f –
1
( B ) ] ≤ f –
1
[ Λδ ( B ) ]
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ON FUZZY SEMI δ – V CONTINUITY IN FUZZY δ – V TOPOLOGICAL SPACE 717
[from 1.6 (5) and 1.2 (7)] for any fuzzy set B of Y.
(ii) ⇒ (iv). Let B = f ( A ). Then f –
1
( B ) = f –
1 f ( A ) ≥ A. Hence, it follows from (ii)
that η δ ( A ) ≤ η δ [ f –
1
( B ) ] [from 1.6 (1)] ≤ f –
1
[ Λδ ( B ) ] ≤ f –
1
[ Λδ f ( A ) ], i.e.,
f f –
1
[ Λδ f ( A ) ] ≥ f [ η δ ( A ) ], i.e., [ Λδ f ( A ) ] ≥ f f –
1
[ Λδ f ( A ) ] ≥ f [ η δ ( A ) ].
(iv) ⇒ (ii). Let A = f –
1
( B ). Then f ( A ) = f f –
1
( B ) ≤ B, i.e., Λδ ( B ) ≥ [ Λδ f ( A ) ]
[from 1.2 (2)] ≥ f [ η δ [ f –
1
( B ) ] ]. Consequently, f –
1
[ Λδ ( B ) ] ≥ f –
1 f [ η δ [ f –
1
( B ) ] ] ≥ ≥
η δ [ f –
1
( B ) ].
(iii) ⇒ (i). Let Bc be a fuzzy δ – V set in Y . Then Vδ ( Bc
) = Bc [from 1.4] and
f –
1
( Bc
) = f –
1
[ Vδ ( Bc
) ] ≤ γδ [ f –
1
( Bc
) ]. Hence, it follows from 1.13 that f –
1
( Bc
) is a
fuzzy semi δ – V set in X. Therefore, f is fuzzy semi δ – V continuous. Thus, we
have shown that (i) ⇒ (iii) ⇔ (ii) ⇔ (iv) and (iii) ⇒ (i).
Theorem 2.4. Let f : ( )X FV, 1
δ → ( )Y FV, 2
δ be a bijective mapping from a fuzzy
topological space ( )X FV, 1
δ into a fuzzy topological space ( )X FV, 2
δ . Then f is
fuzzy semi δ – V continuous iff f [ γδ ( Ac
) ] ≥ Vδ [ f ( Ac
) ].
Proof. f [ γδ ( Ac
) ] ≥ Vδ [ f ( Ac
) ] iff 1 – V δ [ f ( Ac
) ] ≥ 1 – f [ γδ ( Ac
) ] iff
[ Λδ f ( A ) ] ≥ f [ η δ ( A ) ] [from 1.2 (7) and 1.6 (5)] iff f is fuzzy δ – V continuous
[from Theorem 2.3].
3. Relationship between fuzzy supra semi δδδδ – V continuity and fuzzy semi δδδδ –
– V continuity. In this section, we introduce the relationship between fuzzy semi δ –
V continuity, fuzzy supra semi δ – V continuity, and fuzzy Λδ continuity.
Theorem 3.1. If a function f : X → Y is fuzzy supra semi δ – V continuous,
then f is fuzzy semi δ – V continuous.
Proof. Let A be a fuzzy δ – V set in Y. Then it follows from 1.12 that A is a
fuzzy semi δ – V set. Since f is fuzzy supra semi δ – V continuous, f –
1
( A ) is a
fuzzy semi δ – V set in X, i.e., the inverse image of a fuzzy δ – V set in Y is a fuzzy
semi δ – V set in X. Hence, f is fuzzy semi δ – V continuous.
Remark 3.1. The converse statement is not necessarily true because a fuzzy semi
δ – V set is not necessarily a fuzzy δ – V set.
Theorem 3.2. If a function f : X → Y is fuzzy Λδ continuous, then it is also
fuzzy semi δ – V continuous.
Proof. Let A be a fuzzy δ – Λ set in Y. Since f is fuzzy Λδ continuous, we
conclude that f –
1
( A ) is a fuzzy δ – Λ set in X , i.e., f –
1
( A ) is a fuzzy semi δ – Λ
set in X [from 1.12]. Hence, by virtue of Theorem 2.1, f is fuzzy semi δ – V contin-
uous.
Remark 3.2. The converse statement is not necessarily true because a fuzzy semi
δ – Λ set is not necessarily a fuzzy δ – Λ set. This follows from Example 2.1.
1. Mukherjee A., Halder S. On fuzzy semi δ – Λ sets and fuzzy semi δ – V sets V-6 // J. Trip.
Math. Soc. – 2004. – 6. – P. 81 – 88.
2. Mukherjee A., Halder S. Some properties of fuzzy δ – Λ sets and fuzzy δ – V sets V-6 // Ibid. –
P. 23 – 34.
3. Mukherjee A., Halder S. On fuzzy supra semi δ – V continuity in fuzzy supra semi δ – V topo-
logical space // Acta math. hung. (to appear).
4. Mukherjee A., Halder S. On fuzzy Λδ continuous functions // Indian J. Pure and Appl. Math. (to
appear).
Received 23.08.05
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
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