International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August-2013 973

ISSN 2229-5518

𝑓𝑓

𝛼-Continuous Functions In Fuzzy Topological

Spaces

Anjana Bhattacharyya

Abstract— This paper deals with several types of fuzzy generalized closed sets and their interrelations. Also 𝑓𝑓𝛼 --continuous, 𝑓𝑓𝛼- open functions and 𝑓𝑓𝛼- closed functions are introduced and studied. Again, some important properties of such functions are studied in the newly defined spaces using 𝑓𝑓𝛼 –closed sets.

Index Terms𝑓𝑓𝛼 –open sets, 𝑓𝑓𝛼 –closed sets, 𝑓𝑓𝛼 –continuity, 𝑓𝑓𝛼 –open functions, 𝑓𝑓𝛼 –closed functions, 𝑓𝑓𝛼𝑇𝛼 –space,

𝑓𝑓𝛼𝑇𝑐 space.

1 INTRODUCTION

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HROUGHOUT the paper, by (𝑋, 𝜏), (𝑌, 𝜏1 ), (𝑍, 𝜏2 ) or simply by 𝑋, 𝑌, 𝑍 respectively we mean fuzzy topological
spaces (fts, for short) in the sense of Chang [3]. A fuzzy set
is a mapping from a nonempty set 𝑋 to the unit closed interval
𝐼 = [0, 1] [6]. 0𝑋 , 1𝑋 are the constant fuzzy sets taking values 0
and 1 respectively in 𝑋. The complement of a fuzzy set 𝐴 in 𝑋
will be denoted by 1𝑋 \𝐴. The two fuzzy sets 𝐴 and 𝐵in 𝑋, we
write 𝐴 ≤ 𝐵if and only if 𝐴(𝑥) ≤ 𝐵(𝑥), for all 𝑥 ∈ 𝑋.
𝑐𝑐𝐴and𝑖𝑖𝑖𝐴 of a fuzzy set 𝐴 in 𝑋 [6] respectively stand for the
fuzzy closure and fuzzy interior of 𝐴 in 𝑋.

2 𝒇𝒇𝜶-OPEN SETS AND ITS PROPERTIES

We now recall the following definitions, which are
useful in the sequel.

Definition 2.1. A fuzzy set 𝐴 in an fts (𝑋, 𝜏) is called fuzzy

(i) semiopen [1] if 𝐴 ≤ 𝑐𝑐 𝑖𝑖𝑖 𝐴
(ii) α-open [2] if 𝐴 ≤ 𝑖𝑖𝑖 𝑐𝑐 𝑖𝑖𝑖 𝐴
(iii) regular open [1] if 𝐴 = 𝑖𝑖𝑖 𝑐𝑐 𝐴
(iv) preopen [5] if 𝐴 ≤ 𝑖𝑖𝑖 𝑐𝑐 𝐴
The set of all fuzzy semiopen (resp. fuzzy α-open, fuzzy
regular open, fuzzy preopen) sets in 𝑋 is denoted by FSO(X)
(resp. FαO(X), FRO(X), FPO(X)).
The complements of the above mentioned sets are called fuzzy
semiclosed sets, fuzzy α-closed sets , fuzzy regular closed sets
and fuzzy preclosed sets respectively.
Fuzzy semiclosure [1] (resp., fuzzy α-closure [2], fuzzy
preclosure [5]) of a fuzzy set 𝐴in 𝑋, denoted by 𝑠𝑐𝑐 𝐴 (resp.
𝛼𝑐𝑐 𝐴, 𝑝 𝑐𝑐𝐴) is defined to be the intersection of all fuzzy
semiclosed (resp., fuzzy α-closed, fuzzy preclosed) sets
containing 𝐴. It is known that 𝑠𝑐𝑐 𝐴 (resp. 𝛼𝑐𝑐 𝐴, 𝑝𝑐𝑐𝐴) is a
fuzzy semiclosed (resp., fuzzy α-closed, fuzzy preclosed) set.

Definition 2.2. A fuzzy set 𝐴 in an fts (𝑋, 𝜏) is called fuzzy

(i) generalized closed (𝑓𝑓-closed, for short) if
𝑐𝑐 𝐴 ≤ 𝑈 whenever 𝐴 ≤ 𝑈 and 𝑈 ∈ 𝜏,
(ii) semi-generalized closed (𝑓𝑠𝑓-closed, for short) if
𝑠𝑐𝑐 𝐴 ≤ 𝑈 whenever 𝐴 ≤ 𝑈 and 𝑈 ∈ FSO(X),

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Assistant Professor, Department of Mathematics, Victoria Institution (College), A.P.C. Road, Kolkata – 700009, India. PH - +919883118254. E- mail: anjanabhattacharyya@hotmail.com

(iii) generalized semiclosed (𝑓𝑓𝑠-closed, for short) if
𝑠𝑐𝑐 𝐴 ≤ 𝑈 whenever 𝐴 ≤ 𝑈 and 𝑈 ∈ 𝜏,
(iv) generalized α-closed (𝑓𝑓𝛼-closed, for short) if
𝛼𝑐𝑐 𝐴 ≤ 𝑈 whenever 𝐴 ≤ 𝑈 and 𝑈 ∈ FαO(X),
(v) α-generalized closed (𝑓𝛼𝑓-closed, for short) if
𝛼𝑐𝑐 𝐴 ≤ 𝑈 whenever 𝐴 ≤ 𝑈 and 𝑈 ∈ 𝜏,
(vi) 𝑓#-closed (𝑓𝑓#-closed, for short) if 𝑐𝑐 𝐴 ≤ 𝑈
whenever 𝐴 ≤ 𝑈 and 𝑈 is 𝑓𝛼𝑓-open in (𝑋, 𝜏),
(vii) 𝑤𝑓𝛼-closed (𝑓𝑤𝑓𝛼-closed, for short) if
𝛼𝑐𝑐 (𝑖𝑖𝑖 𝐴) ≤ 𝑈 whenever 𝐴 ≤ 𝑈 and 𝑈 ∈
FαO(X),
(viii) 𝑤𝛼𝑓-closed (𝑓𝑤𝛼𝑓-closed, for short) if
𝛼𝑐𝑐 (𝑖𝑖𝑖 𝐴) ≤ 𝑈 whenever 𝐴 ≤ 𝑈 and 𝑈 ∈ 𝜏,
(ix) 𝑓𝛼-closed (𝑓𝑓𝛼-closed, for short) if 𝛼𝑐𝑐 𝐴 ≤ 𝑈
whenever 𝐴 ≤ 𝑈 and 𝑈 is 𝑓𝑓𝛼-open in (𝑋, 𝜏),
(x) 𝛼𝑓𝛼-closed (𝑓𝛼𝑓𝛼-closed, for short) if 𝛼𝑐𝑐 𝐴 ≤ 𝑈
whenever 𝐴 ≤ 𝑈 and 𝑈 ∈ FRO(X),
(xi) 𝑓𝑝𝛼-closed (𝑓𝑓𝑝𝛼-closed, for short) if 𝑝𝑐𝑐 𝐴 ≤ 𝑈
whenever 𝐴 ≤ 𝑈 and 𝑈 ∈FRO(X).
The complements of the above mentioned sets are called their
respective open sets.

Definition 2.3. An fts(𝑋, 𝜏) is called an

(i) 𝑓𝑇𝑏 -space if every 𝑓𝑓𝑠-closed set in (𝑋, 𝜏) is fuzzy
closed in (𝑋, 𝜏),
(ii) 𝑓𝛼𝑇𝑏-space if every 𝑓𝛼𝑓-closed set in (𝑋, 𝜏) is
fuzzy closed in (𝑋, 𝜏),
(iii) 𝑓𝑓𝛼𝑇𝑐 -space if every 𝑓𝑓𝛼-closed set in (𝑋, 𝜏) is
fuzzy closed in (𝑋, 𝜏),
(iv) 𝑓𝑓𝛼𝑇𝛼 -space if every 𝑓𝑓𝛼-closed set in (𝑋, 𝜏) is
fuzzy α-closed in (𝑋, 𝜏),

(v) 𝑓𝑤𝑓𝛼𝑇𝑔𝛼 -space if every 𝑓𝑤𝑓𝛼-closed set n (𝑋, 𝜏)

is 𝑓𝑓𝛼-closed in (𝑋, 𝜏).
Definition 2.4. A function 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) is called fuzzy
(i) α-continuous [4] ( 𝑓𝛼-continuous, for short) if
𝑓−1 (𝑉) ∈ FαO(X) for every 𝑉 ∈ 𝜏1 ,
(ii) semicontinuous [1] (𝑓𝑠-continuous, for short) if
𝑓−1 (𝑉) ∈ FSO(X) for every 𝑉 ∈ 𝜏1 ,
(iii) 𝑓-continuous (𝑓𝑓-continuous, for short) if 𝑓 −1 (𝑉)
is fuzzy 𝑓-open in (𝑋, 𝜏) for every 𝑉 ∈ 𝜏1 ,
(iv) 𝑠𝑓-continuous (𝑓𝑠𝑓-continuous, for short) if
𝑓−1 (𝑉) is 𝑓𝑠𝑓-open in (𝑋, 𝜏) for every 𝑉 ∈ 𝜏1 ,
(v) 𝑓𝑠-continuous (𝑓𝑓𝑠-continuous, for short) if

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International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August-2013 974

ISSN 2229-5518

𝑓−1 (𝑉) is 𝑓𝑓𝑠-open in (𝑋, 𝜏) for every 𝑉 ∈ 𝜏1 ,
(vi) 𝑓𝛼-continuous (𝑓𝑓𝛼-continuous, for short) if
𝑓−1 (𝑉) is 𝑓𝑓𝛼-open in (𝑋, 𝜏) for every 𝑉 ∈ 𝜏1 ,
(vii) 𝛼𝑓-continuous (𝑓𝛼𝑓-continuous, for short) if
𝑓−1 (𝑉) is 𝑓𝛼𝑓-open in (𝑋, 𝜏) for every 𝑉 ∈ 𝜏1 ,
(viii) Completely continuous if 𝑓 −1 (𝑉) ∈ FRO(X) for
every 𝑉 ∈ 𝜏1 ,
(ix) 𝛼-irresolute (𝑓𝛼-irresolute, for short) if 𝑓−1 (𝑉) ∈
FαO(X) for every 𝑉 ∈ FαO(Y),
(x) 𝑤𝑓𝛼-continuous (𝑓𝑤𝑓𝛼-continuous, for short) if
𝑓−1 (𝑉) is 𝑓𝑤𝑓𝛼-open in (𝑋, 𝜏) for every 𝑉 ∈ 𝜏1 ,
(xi) 𝑤𝛼𝑓-continuous (𝑓𝑤𝛼𝑓-continuous, for short) if
𝑓−1 (𝑉) is 𝑓𝑤𝛼𝑓-open in (𝑋, 𝜏) for every 𝑉 ∈ 𝜏1 ,
(xii) 𝑓#-continuous (𝑓𝑓#-continuous, for short) if
𝑓−1 (𝑉) is 𝑓𝑓#-open in (𝑋, 𝜏) for every 𝑉 ∈ 𝜏1 ,
(xiii) 𝑓𝑝𝛼-continuous (𝑓𝑓𝑝𝛼-continuous, for short) if
𝑓−1 (𝑉) is 𝑓𝑓𝑝𝛼-open in (𝑋, 𝜏) for every 𝑉 ∈ 𝜏1 ,
(xiv) 𝛼𝑓𝛼-continuous (𝑓𝛼𝑓𝛼-continuous, for short) if
𝑓−1 (𝑉) is 𝑓𝛼𝑓𝛼-open in (𝑋, 𝜏) for every 𝑉 ∈ 𝜏1 .
Proposition 2.5. Every fuzzy open set 𝑉 in an fts (𝑋, 𝜏) is 𝑓𝑓𝛼-

open in (𝑋, 𝜏).

Proof. Let 𝑉 ∈ 𝜏 be arbitrary. Then 1𝑋 \𝑉 ∈ 𝜏𝑐 . Let 1𝑋 \𝑉 ≤ 𝐺
where 𝐺 is 𝑓𝑓𝛼-open in (𝑋, 𝜏). Then 𝛼𝑐𝑐 (1𝑋 \𝑉) ≤ 𝑐𝑐 (1𝑋 \𝑉) =
1𝑋 \𝑉 ≤ 𝐺. Therefore, 1𝑋 \𝑉 is 𝑓𝑓𝛼-closed in (𝑋, 𝜏) and hence
𝑉is 𝑓𝑓𝛼-open in (𝑋, 𝜏).

Proposition 2.6.Every fuzzy regular open set in an fts(𝑋, 𝜏) is

𝑓𝑓𝛼-open in (𝑋, 𝜏).

Proof. Since every fuzzy regular open set is fuzzy open, the

proof follows from Proposition 2.5.
Proposition 2.7.Every 𝑓𝑓𝛼-open set in an fts(𝑋, 𝜏) is 𝑓𝑓𝛼-open in
(𝑋, 𝜏).
Proof. Let 𝑉 be 𝑓𝑓𝛼-open set in (𝑋, 𝜏). Then 1𝑋 \𝑉 is 𝑓𝑓𝛼-
closed in (𝑋, 𝜏). Let 𝑈 ∈ FαO(X) be such that 1𝑋 \𝑉 ≤ 𝑈. Then
𝑈 is fuzzy 𝑓𝑓𝛼-open in (𝑋, 𝜏). Indeed, 1𝑋 \𝑈 is fuzzy α-closed
in (𝑋, 𝜏) and let 1𝑋 \𝑈 ≤ 𝑊 where 𝑊 ∈ FαO(X). Then 𝛼𝑐𝑐 (1𝑋 \
𝑈) = 1𝑋 \𝑈 ≤ 𝑊 and so 1𝑋 \𝑈 is 𝑓𝑓𝛼-closed and hence 𝑈 is
𝑓𝑓𝛼-open in (𝑋, 𝜏). Since 1𝑋 \V is 𝑓𝑓𝛼-closed in (𝑋, 𝜏) and
1𝑋 \𝑉 ≤ 𝑈 where 𝑈 is 𝑓𝑓𝛼-open in (𝑋, 𝜏), 𝛼𝑐𝑐 (1𝑋 \𝑉) ≤ 𝑈
implies that 1𝑋 \𝑉 is 𝑓𝑓𝛼-closed and hence 𝑉 is 𝑓𝑓𝛼-open in
(𝑋, 𝜏).
Proposition 2.8.Every fuzzy α-open set is 𝑓𝑓𝛼-open set in (𝑋, 𝜏).
Proof. Let 𝑈 ∈ FαO(X). Then 1𝑋 \𝑈 is fuzzy α-closed in (𝑋, 𝜏).
Let 1𝑋 \𝑈 ≤ 𝐺 where 𝐺 is 𝑓𝑓𝛼-open in (𝑋, 𝜏). Then 𝛼𝑐𝑐 (1𝑋 \
𝑈) = 1𝑋 \𝑈 ≤ 𝐺 and so 1𝑋 \𝑈 is 𝑓𝑓𝛼-closed set and hence 𝑈 is
𝑓𝑓𝛼-open set in (𝑋, 𝜏).
Proposition 2.9.Every 𝑓𝑓𝛼-open set is 𝑓𝛼𝑓-open n (𝑋, 𝜏).
Proof. Let 𝑈 be an 𝑓𝑓𝛼-open set in (𝑋, 𝜏). Then 1𝑋 \𝑈 is 𝑓𝑓𝛼-
closed in (𝑋, 𝜏). Let 𝑉 ∈ 𝜏 be such that 1𝑋 \𝑈 ≤ 𝑉. Then 𝑉 ∈
FαO(X) and so by Proposition 2.8, 𝑉 is 𝑓𝑓𝛼-open and hence by Proposition 2.7, 𝑉 is 𝑓𝑓𝛼-open in (𝑋, 𝜏). Since 1𝑋 \𝑈 is 𝑓𝑓𝛼- closed, 𝛼𝑐𝑐 (1𝑋 \𝑈) ≤ 𝑉 and then 1𝑋 \𝑈 is 𝑓𝛼𝑓-closed and consequently, 𝑈 is 𝑓𝛼𝑓-open in (𝑋, 𝜏).
Proposition 2.10. Every 𝑓𝑓#-open set is 𝑓𝑓𝛼-open in (𝑋, 𝜏).
Proof. Let 𝐴be 𝑓𝑓#-open set in an fts (𝑋, 𝜏). Then 1𝑋 \𝐴 is 𝑓𝑓#-
closed in (𝑋, 𝜏). Let 𝐺 be any 𝑓𝑓𝛼-open set in 𝑋 such that
1𝑋 \𝐴 ≤ 𝐺. Then 𝐺 is 𝑓𝛼𝑓-open set in 𝑋. Indeed, 1𝑋 \𝐺 is 𝑓𝑓𝛼-
closed in 𝑋. Let 𝑊 ∈ 𝜏 be such that 1𝑋 \G ≤ 𝑊. Then 𝑊 ∈
FαO(X). Then 𝛼𝑐𝑐 (1𝑋 \G) ≤ 𝑊 and so 1𝑋 \𝐺 is 𝑓𝛼𝑓-closed and
hence 𝐺 is 𝑓𝛼𝑓-open in (𝑋, 𝜏). Therefore, 𝑐𝑐 (1𝑋 \𝐴) ≤ 𝐺 ⟹
𝛼𝑐𝑐 (1𝑋 \𝐴) ≤ 𝑐𝑐 (1𝑋 \𝐴) ≤ 𝐺 and so 1𝑋 \𝐴 is 𝑓𝑓𝛼—closed and
hence 𝐴 is 𝑓𝑓𝛼-open in (𝑋, 𝜏).
Proposition 2.11.Every 𝑓𝑓𝛼-open set is 𝑓𝑤𝑓𝛼-open in (𝑋, 𝜏).
Proof. Let 𝑈 be 𝑓𝑓𝛼-open in (𝑋, 𝜏). Then 1𝑋 \𝑈 is 𝑓𝑓𝛼-closed
in (𝑋, 𝜏). Let 𝐺 ∈ FαO(X) be such that 1𝑋 \𝑈 ≤ 𝐺. Then by
Proposition 2.7 and Proposition 2.8, 𝐺 is 𝑓𝑓𝛼-open in (𝑋, 𝜏).
Since 1𝑋 \𝑈 is 𝑓𝑓𝛼-closed, 𝛼𝑐𝑐 (1𝑋 \𝑈) ≤ 𝐺 ⟹ 𝛼𝑐𝑐 𝑖𝑖𝑖 (1𝑋 \
𝑈) ≤ 𝛼𝑐𝑐 (1𝑋 \𝑈) ≤ 𝐺 and so 1𝑋 \𝑈is 𝑓𝑤𝑓𝛼-closed and hence 𝑈
is 𝑓𝑤𝑓𝛼-open in (𝑋, 𝜏).
Proposition 2.12.Every 𝑓𝑓𝛼-open set is 𝑓𝑓𝑠-open set in (𝑋, 𝜏).
Proof. Let 𝑈 be 𝑓𝑓𝛼-open in (𝑋, 𝜏). Then 1𝑋 \𝑈 is 𝑓𝑓𝛼-closed
in 𝑋. Let 𝑉 ∈ 𝜏 be such that 1𝑋 \𝑈 ≤ 𝑉. Then 𝑉 ∈ FαO(X) and
then by Proposition 2.7 and Proposition 2.8, 𝑉 is 𝑓𝑓𝛼-open set
in 𝑋. As 1𝑋 \𝑈 is 𝑓𝑓𝛼-closed, 𝛼𝑐𝑐 (1𝑋 \𝑈) ≤ 𝑉 ⟹ 𝑠𝑐𝑐 (1𝑋 \𝑈) ≤
𝛼𝑐𝑐 (1𝑋 \𝑈) ≤ 𝑉 and so 1𝑋 \𝑈 is 𝑓𝑓𝑠-closed in 𝑋 and
consequently, 𝑈 is 𝑓𝑓𝑠-open in 𝑋.
Proposition 2.13.Every 𝑓𝑓𝛼-open set is 𝑓𝛼𝑓𝛼-open in (𝑋, 𝜏).
Proof.Let 𝐴 be 𝑓𝑓𝛼-open in 𝑋. Then 1𝑋 \𝐴 is 𝑓𝑓𝛼-closed in 𝑋.
Let 𝑈 ∈ FRO(X) be such that 1𝑋 \𝐴 ≤ 𝑈. Since 𝑈 ∈ FRO(X)
⇒𝑈 ∈ 𝜏 and hence𝑈 is 𝑓𝑓𝛼-open in 𝑋, as 1𝑋 \𝐴 is 𝑓𝑓𝛼-closed,
𝛼𝑐𝑐 (1𝑋 \𝐴) ≤ 𝑈 and hence 1𝑋 \𝐴 is 𝑓𝛼𝑓𝛼-closed in 𝑋 and
consequently, 𝐴 is 𝑓𝛼𝑓𝛼-open in (𝑋, 𝜏).

Proposition 2.14.Every 𝑓𝑓𝛼-open set is 𝑓𝛼𝑓-open set in (𝑋, 𝜏).

Proof. Let 𝐴 be 𝑓𝑓𝛼-open in 𝑋. Then 1𝑋 \𝐴 is 𝑓𝑓𝛼-closed in 𝑋.
Let 𝑈 ∈ 𝜏 be such that 1𝑋 \𝐴 ≤ 𝑈. Then 𝑈 ∈ FαO(X) and so
𝛼𝑐𝑐 (1𝑋 \𝐴) ≤ 𝑈 and so 1𝑋 \𝐴 is 𝑓𝛼𝑓-closed and hence 𝐴 is 𝑓𝛼𝑓-
open in 𝑋.
Proposition 2.15.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be a fuzzy function. Then

the following statements are true :

(i) 𝑓is fuzzy continuous [3] implies 𝑓 is 𝑓𝑓𝛼-continuous. (ii) 𝑓is𝑓𝑤𝑓𝛼-continuous implies 𝑓 is 𝑓𝑤𝛼𝑓-continuous. (iii) 𝑓is𝑓𝛼𝑓𝛼-continuous implies 𝑓 is 𝑓𝑓𝑝𝛼-continuous.

Proof. (i) Let 𝑓 be fuzzy continuous and 𝑉 ∈ 𝜏1 . Then
𝑓−1 (𝑉) ∈ 𝜏. Since every fuzzy open set is 𝑓𝑓𝛼-open in 𝑋 (by
Proposition 2.5 and Proposition 2.7), 𝑓 −1 (𝑉) is 𝑓𝑓𝛼-open in 𝑋

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International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August-2013 975

ISSN 2229-5518

and hence 𝑓 is 𝑓𝑓𝛼-continuous.
(ii) Let 𝑓be 𝑓𝑤𝑓𝛼-continuous and 𝑉 ∈ 𝜏1 . Then 𝑓−1 (𝑉) is
𝑓𝑤𝛼𝑓-open in𝑋. We claim that 𝑓−1 (𝑉) is 𝑓𝑤𝑓𝛼-open in 𝑋.
Indeed, let 𝑈 be any 𝑓𝑤𝑓𝛼-open in 𝑋. Then 1𝑋 \𝑈 is 𝑓𝑤𝑓𝛼-
closed in𝑋. Let 𝐺 ∈ 𝜏 be such that 1𝑋 \𝑈 ≤ 𝐺. Then 𝐺 ∈
FαO(X) and as 1𝑋 \𝑈 is 𝑓𝑤𝑓𝛼-closed, 𝛼𝑐𝑐 �𝑖𝑖𝑖 (1𝑋 \𝑈)� ≤ 𝐺
and so 1𝑋 \𝑈 is 𝑓𝑤𝛼𝑓-closed and hence 𝑈 is 𝑓𝑤𝛼𝑓-open in 𝑋.
Hence 𝑓 is 𝑓𝑤𝛼𝑓-continuous.
(iii) Let 𝑓 be 𝑓𝛼𝑓𝛼-continuous and𝑉 ∈ 𝜏1 . Then 𝑓 −1 (𝑉) is
𝑓𝛼𝑓𝛼-open in (𝑋, 𝜏). Since fuzzy α-open sets are fuzzy
preopen, it follows that for any 𝐴 ∈ 𝐼𝑋 , 𝑝𝑐𝑐 𝐴 ≤ 𝛼𝑐𝑐 𝐴 and
hence 𝑓 is 𝑓𝑓𝑝𝛼-continuous.
Definition 2.16. A fuzzy function 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) is called
fuzzy pre-α-closed if 𝑓(𝛼𝑐𝑐 𝐴) is fuzzy α-closed in (𝑌, 𝜏1 ), for
every fuzzy set 𝐴 in 𝑋.

3𝒇𝒇𝜶-CONTINUOUS FUNCTIONS

In this section the concept of 𝑓𝑓𝛼-continuous function in an
fts (𝑋, 𝜏) has been introduced and studied some of its
properties and found the relationship of this function with the
previously defined functions.
Definition 3.1. A fuzzy function 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) is said to be fuzzy generalized*α-continuous (𝑓𝑓𝛼 –continuous, for short) if𝑓−1 (𝑉)is 𝑓𝑓𝛼-open in 𝑋 for every 𝑉 ∈ 𝜏1 .
𝐵 ∈ 𝜏1 and 𝑖 −1 (𝐵) = 𝐵 ∉ FRO(𝑋, 𝜏). Hence 𝑖 is not fuzzy
completely continuous.
Theorem 3.8.Every 𝑓𝑓𝛼-continuous function is𝑓𝑓𝛼-continuous.
Proof. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be 𝑓𝑓𝛼-continuous and 𝑉 ∈ 𝜏1 .
Then 𝑓−1 (𝑉) is 𝑓𝑓𝛼-open in 𝑋. By Proposition 2.7, 𝑓−1 (𝑉) is
𝑓𝑓𝛼-open in 𝑋 and hence 𝑓 is 𝑓𝑓𝛼-continuous.

Remark 3.9. The converse of the above theorem need not be

true as seen from the following example.
Example 3.10.𝑓𝑓𝛼-continuity⇏ 𝑓𝑓𝛼-continuity
Let 𝑋 = {𝑎, 𝑏}, 𝜏 = { 0𝑋 , 1𝑋 ,𝐴} , 𝜏1 = {0𝑋 , 1𝑋 , 𝐵} where 𝐴(𝑎) =
0.5, 𝐴(𝑏) = 0.4, 𝐵(𝑎) = 0.6, 𝐵(𝑏) = 0.4. Then (𝑋, 𝜏) and (𝑋, 𝜏1 )
are fts’s. Consider the identity function 𝑖 ∶ (𝑋, 𝜏) → (𝑋, 𝜏1 ).
Fuzzy α-open sets in (𝑋, 𝜏)are 0𝑋 , 1𝑋 , 𝐴. Then fuzzy α-closed
sets in (𝑋, 𝜏)are 0𝑋 , 1𝑋 , 1𝑋 \𝐴. Now 𝑓𝑓𝛼-closed sets in (𝑋, 𝜏) are
0𝑋 , 1𝑋 , 𝑈 where 𝑈 ≰ 𝐴 and so 𝑓𝑓𝛼-open sets in (𝑋, 𝜏) are
0𝑋 , 1𝑋 , 1𝑋 \𝑈where 1𝑋 \𝑈 ≱ 1𝑋 \𝐴. Now 1𝑋 \𝐵 ∈ 𝜏𝑐 .𝑖 −1(1 \𝐵) =

1 𝑋

1𝑋 \𝐵which is 𝑓𝑓𝛼-closed in (𝑋, 𝜏). Therefore, 𝑖 is 𝑓𝑓𝛼-
continuous. But 1𝑋 \𝐵 is not 𝑓𝑓𝛼-closed as 1𝑋 \𝐵 is 𝑓𝑓𝛼-open
in (𝑋, 𝜏)and 𝛼𝑐𝑐 (1𝑋 \𝐵) = 1𝑋 \𝐴 ≰ 1𝑋 \𝐵. Hence 𝑖 is not 𝑓𝑓𝛼-
continuous.
Theorem 3.11. Every 𝑓𝛼-continuous function is 𝑓𝑓𝛼-continuous.
Proof. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be 𝑓𝛼-continuous and 𝑉 ∈ 𝜏1 .
Theorem 3.2.Every fuzzy continuous function 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 )
Then 𝑓−1
(𝑉) ∈ FαO(X). By Proposition 2.8, 𝑓−1
(𝑉)is 𝑓𝑓𝛼-open
is 𝑓𝑓𝛼-continuous.
Proof.Let 𝑉 ∈ 𝜏1 . Then 𝑓−1 (𝑉) ∈ 𝜏. By Proposition 2.5, 𝑓−1 (𝑉)
is 𝑓𝑓𝛼-open in 𝑋 and hence 𝑓 is 𝑓𝑓𝛼-continuous.

Remark 3.3. The converse of the above theorem need not be

true as seen from the following example.
Example 3.4. 𝑓𝑓𝛼-continuity fuzzy continuity
Let 𝑋 = {𝑎, 𝑏}, 𝜏 = { 0𝑋 , 1𝑋 ,𝐴}, 𝜏1 = {0𝑋 , 1𝑋 , 𝐵} where 𝐴(𝑎) =
0.5, 𝐴(𝑏) = 0.4, 𝐵(𝑎) = 0.4, 𝐵(𝑏) = 0.4. Then (𝑋, 𝜏) and (𝑋, 𝜏1 )
are fts’s. Consider the identity function 𝑖 ∶ (𝑋, 𝜏) → (𝑋, 𝜏1 ).
in (𝑋, 𝜏) and hence 𝑓 is 𝑓𝑓𝛼-continuous.

Remark 3.12. The converse of the above theorem need not be

true as seen from the following example.
Example 3.13.𝑓𝑓𝛼-continuity⇏ 𝑓𝛼-continuity
Consider Example 3.4. Here 𝑖 is 𝑓𝑓𝛼-continuous. Now 𝐵 ∈
𝜏1, 𝑖 −1 (𝐵) = 𝐵 ∉ FαO(𝑋, 𝜏). Hence 𝑖 is not 𝑓𝛼-continuous.
Theorem 3.14.Every 𝑓𝑓𝛼-continuous function is 𝑓𝛼𝑓-continuous.
Proof. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 )be 𝑓𝑓𝛼-continuous and 𝑉 ∈ 𝜏1 .
Now 𝑖 −1 (1𝑋 \𝐵) = 1𝑋 \𝐵 and 1𝑋 is the only 𝑓𝑓𝛼-open set in
Then 𝑓−1
(𝑉) is 𝑓𝑓𝛼-open in 𝑋. By Proposition 2.9, 𝑓−1
(𝑉) is
(𝑋, 𝜏) containing 1𝑋 \𝐵 and so 𝑖 is 𝑓𝑓𝛼-continuous. Again,
𝐵 ∈ 𝜏1 and 𝑖 −1(𝐵) = 𝐵 ∉ 𝜏1 . Hence 𝑖 is not fuzzy continuous.
Theorem 3.5.Every fuzzy completely continuous function is 𝑓𝑓𝛼-

continuous.

Proof. Let𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be fuzzy completely continuous
𝑓𝛼𝑓-open in 𝑋 and hence 𝑓 is 𝑓𝛼𝑓-continuous.

Remark 3.15. The converse of the above theorem need not be

true as seen from the following example.
Example 3.16.𝑓𝛼𝑓-continuity⇏ 𝑓𝑓𝛼-continuity
Let 𝑋 = {𝑎}, 𝜏 = {0𝑋 , 1𝑋 , 𝐵}, 𝜏1 = { 0𝑋 , 1𝑋 , 𝐴} where 𝐵(𝑎) = 0.6

1

function and 𝑉 ∈ 𝜏1 be arbitrary. Then 𝑓−1 (𝑉) ∈ FRO(X) and
and 𝐴(𝑎) =
. Then (𝑋, 𝜏) and (𝑋, 𝜏 ) are fts’s. Consider the

3

hence 𝑓 −1 (𝑉) ∈ 𝜏 and then by Proposition 2.5, 𝑓 −1 (𝑉) is 𝑓𝑓𝛼-
open in 𝑋. Consequently, 𝑓 is 𝑓𝑓𝛼-continuous.

Remark 3.6. The converse of the above theorem need not be

true in general as seen from the following example. Example 3.7.𝑓𝑓𝛼-continuity fuzzy completely continuity Consider Example 3.4. Here 𝑖 is 𝑓𝑓𝛼-continuous. Now
identity function 𝑖 ∶ (𝑋, 𝜏) → (𝑋, 𝜏1 ). We claim that 𝑖 is 𝑓𝛼𝑓-
continuous but not 𝑓𝑓𝛼-continuous.
Now fuzzy α-open sets in (𝑋, 𝜏) are 0𝑋 , 1𝑋 , 𝐵, 𝑈where 𝑈(𝑎) ≥
0.6. Then fuzzy α-closed sets in (𝑋, 𝜏) are 0𝑋 , 1𝑋 , 1𝑋 \𝐵, 1𝑋 \
𝑈where (1𝑋 \𝐵)(𝑎) = 0.4, (1𝑋 \𝑈)(𝑎) ≤ 0.4. Again 𝑓𝑓𝛼-closed
sets in (𝑋, 𝜏) are 0𝑋 , 1𝑋 , 𝑉 where 𝑉(𝑎) ≤ 0.4 [Indeed, 𝛼𝑐𝑐 𝑉 ≤
1𝑋 \𝐵whereas 𝑉 ≤ 𝑈]. And so 𝑓𝑓𝛼-open sets in (𝑋, 𝜏) are
0𝑋 , 1𝑋 , 1𝑋 \𝑉 where (1𝑋 \𝑉)(𝑎) ≥ 0.6.Now 1𝑋 \𝐴 ∈ 𝜏𝑐 .

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ISSN 2229-5518

Therefore, 𝑖 −1 (1𝑋\A) = 1𝑋 \𝐴 is 𝑓𝑓𝛼-open set in (𝑋, 𝜏). Therefore, 1𝑋 \𝐴 ≤ 1𝑋 \𝐴, but 𝛼𝑐𝑐 (1𝑋 \𝐴) = 1𝑋 ≰ 1𝑋 \𝐴. Therefore, 1𝑋 \𝐴 is not 𝑓𝑓𝛼-closed in (𝑋, 𝜏) and so 𝑖 is not
𝑓𝑓𝛼-continuous. Again, 1𝑋 is the only fuzzy open set in (𝑋, 𝜏)
such that 1𝑋 \𝐴 ≤ 1𝑋 .
Proposition 3.17.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be an 𝑓𝛼𝑓-continuous
function where (𝑋, 𝜏) is an 𝑓𝛼𝑇𝑏 -space. Then 𝑓 is 𝑓𝑓𝛼-continuous.
Proof. Let 𝑉 ∈ 𝜏1 . As 𝑓 is 𝑓𝛼𝑓-continuous, 𝑓−1 (𝑉) is 𝑓𝛼𝑓-open
in (𝑋, 𝜏). Then 1𝑋 \𝑓−1(𝑉) is 𝑓𝛼𝑓-closed in (𝑋, 𝜏). As (𝑋, 𝜏) is
𝑓𝛼𝑇𝑏 -space, 1𝑋 \𝑓−1(𝑉) is fuzzy closed in (𝑋, 𝜏) and hence
𝑓−1 (𝑉) is fuzzy open in (𝑋, 𝜏) . By Proposition 2.5, 𝑓 −1 (𝑉) is
𝑓𝑓𝛼-open in (𝑋, 𝜏) and hence 𝑓 is 𝑓𝑓𝛼-continuous.
Theorem 3.18.Every 𝑓𝑓#-continuous function is 𝑓𝑓𝛼-continuous.

continuous.

Proof.Let𝑉 ∈ 𝜏1 . As 𝑓 is 𝑓𝑤𝑓𝛼-continuous, 𝑓−1 (𝑉) is 𝑓𝑤𝑓𝛼- open in (𝑋, 𝜏). As (𝑋, 𝜏) is 𝑓𝑤𝑓𝛼𝑇𝑔𝛼 -space, 1𝑋 \𝑓−1 (𝑉) is 𝑓𝑓𝛼- closed in (𝑋, 𝜏)and hence 𝑓−1 (𝑉) is 𝑓𝑓𝛼-open in (𝑋, 𝜏). Consequently, 𝑓 is 𝑓𝑓𝛼-continuous.

Theorem 3.25.Every 𝑓𝑓𝛼-continuous function is 𝑓𝑤𝛼𝑓-

continuous.

Proof. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be 𝑓𝑓𝛼-continuous function. By
Theorem 3.21, 𝑓 is 𝑓𝑤𝑓𝛼-continuous. Then by Proposition
2.15(ii), 𝑓 is 𝑓𝑤𝛼𝑓-continuous.

Remark 3.26. The converse of the above theorem need not be

true as seen from the following example.
Example 3.27.𝑓𝑤𝛼𝑓-continuity⇏ 𝑓𝑓𝛼-continuity
Proof.Let 𝑉 ∈ 𝜏1 . Then 𝑓−1 (𝑉) is 𝑓𝑓#-open in (𝑋, 𝜏). By
Consider Example 3.16. Here 1𝑋 \𝐴 ∈ 𝜏𝑐 , 𝑖 −1
(1𝑋\A) = 1𝑋 \𝐴.
Proposition 2.10, 𝑓−1 (𝑉) is 𝑓𝑓𝛼-open in (𝑋, 𝜏) and hence 𝑓 is
𝑓𝑓𝛼-continuous.

Remark 3.19. The converse of the above theorem need not be

true as seen from the following example.
Example 3.20.𝑓𝑓𝛼-continuity⇏ 𝑓𝑓# - continuity
Let 𝑋 = {𝑎, 𝑏}, 𝜏 = { 0𝑋 , 1𝑋 ,𝐴} , 𝜏1 = {0𝑋 , 1𝑋 , 𝐵} where 𝐴(𝑎) =
0.4, 𝐴(𝑏) = 0.6, 𝐵(𝑎) = 0.5, 𝐵(𝑏) = 0.7. Then (𝑋, 𝜏) and (𝑋, 𝜏1 )
are fts’s. Consider the identity function 𝑖 ∶ (𝑋, 𝜏) → (𝑋, 𝜏1 ).
Now fuzzy α-open sets in (𝑋, 𝜏) are 0𝑋 , 1𝑋 , 𝐴, 𝑈 where 𝑈 ≥ 𝐴
and so fuzzy α-closed sets in (𝑋, 𝜏) are 0𝑋 , 1𝑋 , 1𝑋 \𝐴, 1𝑋 \𝑈where
1𝑋 \𝑈 ≤ 1𝑋 \𝐴. Now 𝑓𝑓𝛼-closed sets in (𝑋, 𝜏) are 0𝑋 , 1𝑋 , 1𝑋 \
𝐴, 1𝑋 \𝑈 and so 𝑓𝑓𝛼-open sets in (𝑋, 𝜏)are 0𝑋 , 1𝑋 , 𝐴, 𝑈. Again,
𝑓𝛼𝑓-closed sets in (𝑋, 𝜏) are 0𝑋 , 1𝑥 , 𝑉, 𝑊 where 𝑉(𝑎) ≤
0.4, 𝑉(𝑏) ≤ 0.4 and 𝑊 > 𝐴. Then 𝑓𝛼𝑓-open sets in (𝑋, 𝜏) are
0𝑋 , 1𝑥 , 1𝑋 \ 𝑉, 1𝑋 \𝑊 where 1 − 𝑉(𝑎) ≥ 0.6, 1 − 𝑉(𝑏) ≥ 0.6and
1𝑋 \𝑊 < 1𝑋 \𝐴.
1𝑋 \𝐴 ≤ 1𝑋 where1𝑋 is the only fuzzy open set in (𝑋, 𝜏). Now,
𝛼𝑐𝑐𝜏 �𝑖𝑖𝑖𝜏 (1𝑋 \𝐴)� = 𝛼𝑐𝑐𝜏 𝐵 = 1𝑋 ≤ 1𝑋 . Therefore, 1𝑋 \𝐴 is
𝑓𝑤𝛼𝑓-closed in (𝑋, 𝜏) and hence 𝑖 is 𝑓𝑤𝛼𝑓-continuous though
it is not 𝑓𝑓𝛼-continuous.
Theorem 3.28.Every 𝑓𝑓𝛼-continuous function is 𝑓𝑓𝑠-continuous.
Proof. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be 𝑓𝑓𝛼-continuous and 𝑉 ∈ 𝜏1 .
As 𝑓 is 𝑓𝑓𝛼-continuous, 𝑓−1 (𝑉) is 𝑓𝑓𝛼-open in (𝑋, 𝜏). By
Proposition 2.12, 𝑓−1 (𝑉) is 𝑓𝑓𝑠-open in (𝑋, 𝜏) and hence 𝑓 is
𝑓𝑓𝑠-continuous.

Remark 3.29. The converse of the above theorem need not be

true as seen from the following example.
Example 3.30.𝑓𝑓𝑠-continuity⇏ 𝑓𝑓𝛼-continuity
Consider Example 3.16. Since 1𝑋 is the only fuzzy open set in
(𝑋, 𝜏) such that 1𝑋 \𝐴 ≤ 1𝑋 , 𝑠𝑐𝑐𝜏 (1𝑋 \𝐴) ≤ 1𝑋 and hence 1𝑋 \𝐴
Now 1𝑋 \𝐵 ∈ 𝜏𝑐 and 𝑖 −1
(1𝑋 \𝐵) = 1𝑋 \𝐵 which is 𝑓𝛼𝑓-open set
is 𝑓𝑓𝑠-closed set in (𝑋, 𝜏). Hence 𝑖 is 𝑓𝑓𝑠-continuous.
in (𝑋, 𝜏). But 𝑐𝑐𝜏 (1𝑋 \𝐵) = 1𝑋 \𝐴 ≰ 1𝑋 \𝐵. Therefore, 𝑖 is not
𝑓𝑓#-continuous. Again, 𝑈(𝑎) ≥ 0.5, 𝑈(𝑏) ≥ 0.6 are 𝑓𝑓𝛼-open
sets in (𝑋, 𝜏) containing 1𝑋 \𝐵and 𝛼𝑐𝑐𝜏 (1𝑋 \𝐵) = 1𝑋 \𝐵 ≤ 𝑈.
Hence 𝑖 is 𝑓𝑓𝛼-continuous.
Theorem 3.21.Every 𝑓𝑓𝛼-continuous function is 𝑓𝑤𝑓𝛼-

continuous.

Proof. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be 𝑓𝑓𝛼-continuous and 𝑉 ∈ 𝜏1. Then 𝑓−1 (𝑉) is 𝑓𝑓𝛼-open in 𝑋. By Proposition 2.11, 𝑓−1 (𝑉) is
𝑓𝑤𝑓𝛼-open in 𝑋 and hence 𝑓 is 𝑓𝑤𝑓𝛼-continuous.

Remark 3.22. The converse of the above theorem need not be

true as seen from the following example.
Example 3.23.𝑓𝑤𝑓𝛼-continuity⇏ 𝑓𝑓𝛼-continuity
Consider Example 3.10. Here 1𝑋 \𝐵 is 𝑓𝑤𝑓𝛼-closed as 1𝑋 is the
only fuzzy α-open set in (𝑋, 𝜏)containing 1𝑋 \𝐵.
Proposition 3.24.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be an 𝑓𝑤𝑓𝛼-continuous
function where (𝑋, 𝜏) is an 𝑓𝑤𝑓𝛼𝑇𝑔𝛼 -space. Then 𝑓 is 𝑓𝑓𝛼-
Proposition 3.31.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be an 𝑓𝑓𝑠-continuous function where (𝑋, 𝜏) is an 𝑓𝑇𝑏 -space. Then 𝑓 is 𝑓𝑓𝛼-continuous.
Proof.Let 𝑉 ∈ 𝜏1 . As 𝑓 is 𝑓𝑓𝑠-continuous, 𝑓−1 (𝑉) is 𝑓𝑓𝑠-open in (𝑋, 𝜏). Then 1𝑋 \𝑓−1(𝑉)is fuzzy closed in (𝑋, 𝜏). Hence
,𝑓−1 (𝑉) is fuzzy open in (𝑋, 𝜏). By Proposition 2.5, 𝑓−1 (𝑉) is
𝑓𝑓𝛼-open in (𝑋, 𝜏) and hence 𝑓 is 𝑓𝑓𝛼-continuous.
Theorem 3.32.Every 𝑓𝑓𝛼-continuous function is 𝑓𝛼𝑓𝛼-

continuous.

Proof.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be 𝑓𝑓𝛼-continuous and 𝑉 ∈ 𝜏1 . Then 𝑓−1 (𝑉) is 𝑓𝑓𝛼-open in (𝑋, 𝜏). By Proposition 2.13,
𝑓−1 (𝑉) is 𝑓𝛼𝑓𝛼-open in (𝑋, 𝜏). Hence 𝑓 is 𝑓𝛼𝑓𝛼-continuous.

Remark 3.33. The converse of the above theorem need not be

true as seen from the following example.
Example 3.34.𝑓𝛼𝑓𝛼-continuity⇏ 𝑓𝑓𝛼-continuity
Consider Example 3.16. The only fuzzy regular open sets in

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ISSN 2229-5518

(𝑋, 𝜏)are 0𝑋 , 1𝑋 . Therefore, 1𝑋 \𝐴 ≤ 1𝑋 ⟹ 𝛼𝑐𝑐𝜏 (1𝑋 \𝐴) = 1𝑋
1𝑋 ⟹ 1𝑋 \𝐴 is 𝑓𝛼𝑓𝛼-closed in (𝑋, 𝜏). Hence 𝑖 is 𝑓𝛼𝑓𝛼-
continuous though it is not 𝑓𝑓𝛼-continuous.
Theorem 3.35.Every 𝑓𝑓𝛼-continuous function is 𝑓𝑓𝑝𝛼-

continuous.

Proof. By Theorem 3.32, every 𝑓𝑓𝛼-continuous function is
𝑓𝛼𝑓𝛼-continuous and again by Proposition 2.5(iii), it is 𝑓𝑓𝑝𝛼-
continuous.

Remark 3.36. The converse of the above theorem need not be true as seen from the following example.

Example 3.37.𝑓𝑓𝑝𝛼-continuity⇏ 𝑓𝑓𝛼-continuity
Consider Example 3.16. The only fuzzy regular open setss in
(𝑋, 𝜏)are 0𝑋 , 1𝑋 . Now1𝑋 \𝐴 ≤ 1𝑋 ⟹ 𝑝𝑐𝑐𝜏 (1𝑋 \𝐴) = 1𝑋
1𝑋 ⟹ 1𝑋 \𝐴 is 𝑓𝑓𝑝𝛼-closed in (𝑋, 𝜏) and hence 𝑖 is 𝑓𝑓𝑝𝛼-
Consider Example 3.40. Here 𝐵 is fuzzy semiopen in (𝑋, 𝜏1 ). But 𝑖 −1(𝐵) = 𝐵 ∉ FSO(𝑋, 𝜏). Therefore, 𝑖 is 𝑓𝑓𝛼-continuous
but not fuzzy semi-continuous.
Remark 3.44. The following two examples show that 𝑓𝑓- continuous function and 𝑓𝑓𝛼-continuous function are
independent notions.
Example 3.45.𝑓𝑓-continuity⇏ 𝑓𝑓𝛼-continuity
Consider Example 3.16. Since 1𝑋 is the only fuzzy open set
such that 1𝑋 \𝐴 ≤ 1𝑋 . Then𝑐𝑐𝜏 (1𝑋 \𝐴) = 1𝑋 and so 1𝑋 \𝐴 is 𝑓𝑓-
closed in (𝑋, 𝜏) and so 𝐴 is 𝑓𝑓-open set in(𝑋, 𝜏). Hence 𝑖 is 𝑓𝑓-
continuous though it is not 𝑓𝑓𝛼-continuous.
Example 3.46.𝑓𝑓𝛼-continuity⇏ 𝑓𝑓-continuity
Let 𝑋 = {𝑎, 𝑏}, 𝜏 = { 0𝑋 , 1𝑋 ,𝐴} , 𝜏1 = {0𝑋 , 1𝑋 , 𝐵} where 𝐴(𝑎) =
0.4, 𝐴(𝑏) = 0.6, 𝐵(𝑎) = 0.7, 𝐵(𝑏) = 0.6. Then (𝑋, 𝜏) and (𝑋, 𝜏1 )
are fts’s. Consider the identity function 𝑖 ∶ (𝑋, 𝜏) → (𝑋, 𝜏1 ).
continuous though it is not 𝑓𝑓𝛼-continuous.
Now 1𝑋 \𝐵 ∈ 𝜏𝑐 . Then 𝑖

−1(1𝑋 \𝐵) = 1𝑋 \𝐵. Now any 𝑓𝑓𝛼-open

Theorem 3.38.If a fuzzy function 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) is 𝑓𝛼- irresolute, then it is 𝑓𝑓𝛼-continuous.
Proof.Let 𝑉 ∈ 𝜏1 . Then 𝑉 ∈ FαO(Y). As 𝑓 is 𝑓𝛼-irresolute,
𝑓−1 (𝑉) ∈ FαO(X). By Proposition 2.8, 𝑓 −1 (𝑉) is 𝑓𝑓𝛼-open in
(𝑋, 𝜏) and hence 𝑓 is 𝑓𝑓𝛼-continuous.

Remark 3.39. The converse of the above theorem need not be

true as seen from the following example.
Example 3.40.𝑓𝑓𝛼-continuity⇏ 𝑓𝛼- continuity
Let 𝑋 = {𝑎, 𝑏}, 𝜏 = { 0𝑋 , 1𝑋 ,𝐴} , 𝜏1 = {0𝑋 , 1𝑋 , 𝐵} where 𝐴(𝑎) =
0.5, 𝐴(𝑏) = 0.4, 𝐵(𝑎) = 0.4, 𝐵(𝑏) = 0.4. Then (𝑋, 𝜏) and (𝑋, 𝜏1 )
are fts’s. Consider the identity function 𝑖 ∶ (𝑋, 𝜏) → (𝑋, 𝜏1 ).
Now 𝑖 −1 (1𝑋 \𝐵) = 1𝑋 \𝐵 and 1𝑋 is the only 𝑓𝑓𝛼-open set in
(𝑋, 𝜏) containing 1𝑋 \𝐵 and so 𝑖 is 𝑓𝑓𝛼-continuous. Now 1𝑋 \𝐵
is fuzzy semiopen set in (𝑋, 𝜏1 ) and 𝑖 −1(1𝑋 \𝐵) = 1𝑋 \𝐵 which
is not fuzzy semiopen in (𝑋, 𝜏). Hence 𝑖 is not 𝑓𝛼-irresolute.

Note 3.41. The following two examples show that fuzzy semi-

continuity and 𝑓𝑓𝛼-continuity are independent notions.
Example 3.42.fuzzy semi-continuity ⇏ 𝑓𝑓𝛼-continuity
Let 𝑋 = {𝑎, 𝑏}, 𝜏 = { 0𝑋 , 1𝑋 ,𝐴} , 𝜏1 = {0𝑋 , 1𝑋 , 𝐵} where 𝐴(𝑎) =
0.5, 𝐴(𝑏) = 0.4, 𝐵(𝑎) = 0.5, 𝐵(𝑏) = 0.5. Then (𝑋, 𝜏) and (𝑋, 𝜏1 )
are fts’s. Consider the identity function 𝑖 ∶ (𝑋, 𝜏) → (𝑋, 𝜏1 ).
Then fuzzy α-open sets in (𝑋, 𝜏) are 0𝑋 , 1𝑋 , 𝐴 and fuzzy α-
closed sets in (𝑋, 𝜏)are 0𝑋 , 1𝑋 , 1𝑋 \𝐴, fuzzy semiopen sets in
(𝑋, 𝜏) are 0𝑋 , 1𝑋 , 𝐴, 𝑉 where 𝐴 ≤ 𝑉 ≤ 1𝑋 \𝐴. 𝑓𝑓𝛼-closed sets in
(𝑋, 𝜏) are 0𝑋 , 1𝑋 , 𝑈, 1𝑋 \𝐴where 𝑈 ≰ 𝐴, 𝑓𝑓𝛼-open sets in (𝑋, 𝜏)
are 0𝑋 , 1𝑋 , 𝐴, 1𝑋 \𝑈 where 1𝑋 \𝑈 ≱ 1𝑋 \𝐴. Now 𝑖 −1 (𝐵) = 𝐵
which is fuzzy semiopen in (𝑋, 𝜏) and so 𝑖 is fuzzy semi-
continuous. Again, 1𝑋 \𝐵 is 𝑓𝑓𝛼-open set such that 𝐵 =
1𝑋 \𝐵 ≤ 1𝑋 \𝐵. But 𝛼𝑐𝑐𝜏 (1𝑋 \𝐵) = 𝛼𝑐𝑐𝜏 𝐵 = 1𝑋 \𝐴 ≰ 1𝑋 \𝐵.
Therefore, 1𝑋 \𝐵 is not 𝑓𝑓𝛼-closed and so 𝐵 is not 𝑓𝑓𝛼-open
in (𝑋, 𝜏) and hence 𝑖 is not 𝑓𝑓𝛼-continuous.
Example 3.43.𝑓𝑓𝛼-continuity fuzzy semi-continuity
set in (𝑋, 𝜏)other than 0𝑋 contains1𝑋 \𝐵 and 𝛼𝑐𝑐𝜏 (1𝑋 \𝐵) = 1𝑋 \𝐵
and hence 𝑖 is 𝑓𝑓𝛼-continuous. But 1𝑋 \𝐵 ≤ 𝐴 and 𝑐𝑐𝜏 (1𝑋 \
𝐵) = 1𝑋 \𝐴 ≰ 𝐴 and so 𝑖 is not 𝑓𝑓-continuous.
Theorem 3.47.A fuzzy function 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) is 𝑓𝑓𝛼-

continuous iff the inverse image of every fuzzy closed set in 𝑌is

𝑓𝑓𝛼-closed in 𝑋.
Proof. Let 𝑓 be 𝑓𝑓𝛼-continuousand 𝐹 ∈ 𝜏𝑐 . Then 1 \𝐹 ∈ 𝜏 .
Since 𝑓 is 𝑓𝑓𝛼-continuous, 𝑓−1 (1𝑋 \𝐹) = 1𝑋 \𝑓−1(𝐹) is 𝑓𝑓𝛼-
open in 𝑋. Hence 𝑓 −1 (𝐹) is 𝑓𝑓𝛼-closed in 𝑋.
Conversely, let us suppose that 𝑓−1 (𝐹)be𝑓𝑓𝛼-closed in 𝑋 for
every fuzzy closed set 𝐹in 𝑌. Let 𝑉 ∈ 𝜏1 . Then 1𝑋 \𝑉 ∈ 𝜏𝑐 . By
assumption, 𝑓−1 (1𝑌 \𝑉) = 1𝑋 \𝑓−1(𝑉) is 𝑓𝑓𝛼-closed in 𝑋 and
so 𝑓−1 (𝑉) is 𝑓𝑓𝛼-open in 𝑋 and hence 𝑓 is 𝑓𝑓𝛼-continuous.
Theorem 3.48.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be an 𝑓𝑓𝛼-continuous, 𝑓-
pre-α-closed function, then 𝑓(𝐴) is 𝑓𝛼𝑓-closed in (𝑌, 𝜏1 ) for every
𝑓𝑓𝛼-closed set 𝐴in (𝑋, 𝜏).
Proof. Let 𝐴 be an 𝑓𝑓𝛼-closed set in 𝑋 and 𝑉 ∈ 𝜏1 be such
that 𝑓(𝐴) ≤ 𝑉. Then 𝐴 ≤ 𝑓 −1 (𝑉). As 𝑓 is 𝑓𝑓𝛼-continuous,
𝑓−1 (𝑉) is 𝑓𝑓𝛼-open in (𝑋, 𝜏). Since 𝐴 is 𝑓𝑓𝛼-closed, and
𝐴 ≤ 𝑓−1 (𝑉), 𝛼𝑐𝑐𝜏 𝐴 ≤ 𝑓−1 (𝑉) ⇒ 𝑓(𝛼𝑐𝑐𝜏 𝐴) ≤ 𝑓𝑓−1 (𝑉) ≤ 𝑉.
Since 𝑓 is 𝑓-pre-α-closed, 𝑓(𝛼𝑐𝑐𝜏 𝐴) is fuzzy α-closed in (𝑌, 𝜏1 ).
Therefore, 𝛼𝑐𝑐𝜏1 �𝑓(𝛼𝑐𝑐𝜏 𝐴)� = 𝑓(𝛼𝑐𝑐𝜏 𝐴) ≤ 𝑉. Now, 𝐴 ≤
𝛼𝑐𝑐𝜏 𝐴 ⇒ 𝑓(𝐴) ≤ 𝑓(𝛼𝑐𝑐𝜏 𝐴) ⇒ 𝛼𝑐𝑐𝜏1 �𝑓(𝐴)� ≤
𝛼𝑐𝑐𝜏1 �𝑓(𝛼𝑐𝑐𝜏 𝐴)� = 𝑓(𝛼𝑐𝑐𝜏 𝐴) ≤ 𝑉. Hence 𝑓(𝐴) is 𝑓𝛼𝑓-closed in
(𝑌, 𝜏1 ).
Theorem 3.49.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be fuzzy continuous, fuzzy
pre-α-closed function, then 𝑓(𝐴) is 𝑓𝛼𝑓-closed in (𝑌, 𝜏1 ) for
every𝑓𝑓𝛼-closed set 𝐴in (𝑋, 𝜏).

Proof. Combining Theorem 3.2 and Theorem 3.8, we say that 𝑓

is 𝑓𝑓𝛼-continuous. Then by Theorem 3.48, 𝑓(𝐴) is 𝑓𝛼𝑓-closed
for every 𝑓𝑓𝛼-closed set 𝐴in 𝑋.
Theorem 3.50.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be an 𝑓𝑓𝛼-continuous, 𝑓-

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pre-α-closed function and (𝑌, 𝜏1 ) is an 𝑓𝛼𝑇𝑏-space, then 𝑓(𝐴) is
Proof. Let 𝐴 ∈ 𝐼𝑋 . Then 𝑐𝑐𝜏 𝑓(𝐴) ∈ 𝜏𝑐
and as 𝑓 is 𝑓𝑓𝛼-
𝑓𝑓𝛼-closed in (𝑌, 𝜏1 ) for every 𝑓𝑓𝛼-closed set 𝐴in (𝑋, 𝜏).
Proof. Let 𝐴 be 𝑓𝑓𝛼-closed in (𝑋, 𝜏) and 𝑉 be any 𝑓𝑓𝛼-open
continuous, 𝑓−1 (𝑐𝑐𝜏 𝑓(𝐴)) is 𝑓𝑓𝛼-closed in (𝑋, 𝜏). Hence by
Result 3.57, 𝑓𝑓𝛼𝑐𝑐𝜏 (𝑓−1 �𝑐𝑐𝜏 𝑓(𝐴))� = 𝑓−1 (𝑐𝑐𝜏 𝑓(𝐴)). Now
set in 𝑌 such that 𝑓(𝐴) ≤ 𝑉. By Proposition 2.14, 𝑉 is 𝑓𝛼𝑓-open
𝑓(𝐴) ≤ 𝑐𝑐𝜏1 𝑓(𝐴) ⇒ 𝐴 ≤ 𝑓

1

𝑓(𝐴) ≤ 𝑓−1

1

(𝑐𝑐𝜏1 𝑓(𝐴)). Therefore,
in 𝑌. Since (𝑌, 𝜏1 )is 𝑓𝛼𝑇𝑏-space, 1𝑋 \𝑉 being 𝑓𝛼𝑓-closed in
(𝑌, 𝜏1 ) is fuzzy closed in (𝑌, 𝜏1 ) and so 𝑉 is fuzzy open in
𝑓−1 (𝑐𝑐𝜏 𝑓(𝐴)) being a 𝑓𝑓𝛼-closed set containing 𝐴. Then
𝑓𝑓𝛼𝑐𝑐𝜏 𝐴 ≤ 𝑓−1 (𝑐𝑐𝜏 𝑓(𝐴)).Therefore, 𝑓(𝑓𝑓𝛼𝑐𝑐𝜏𝐴) ≤ 𝑐𝑐𝜏 𝑓(𝐴).

1 1

(𝑌, 𝜏1 ) . As 𝑓 is𝑓𝑓𝛼-continuous, 𝑓−1 (𝑉) is 𝑓𝑓𝛼-open in (𝑋, 𝜏).
Since 𝐴 is 𝑓𝑓𝛼-closed in (𝑋, 𝜏)and 𝐴 ≤ 𝑓−1 (𝑉), 𝛼𝑐𝑐𝜏 𝐴 ≤
𝑓−1 (𝑉) ⇒ 𝑓(𝛼𝑐𝑐𝜏 𝐴 ) ≤ 𝑓𝑓 −1 (𝑉) ≤ 𝑉. Since 𝑓 is 𝑓-pre-α-
closed, 𝑓(𝛼𝑐𝑐𝜏 𝐴) is fuzzy α-closed in 𝑌. Therefore,
𝛼𝑐𝑐𝜏1 �𝑓(𝛼𝑐𝑐𝜏 𝐴)� = 𝑓(𝛼𝑐𝑐𝜏 𝐴) ≤ 𝑉 and so 𝛼𝑐𝑐𝜏1 �𝑓(𝐴)� ≤
𝛼𝑐𝑐𝜏 �𝑓(𝛼𝑐𝑐𝜏 𝐴)� ≤ 𝑉. Consequently, 𝑓(𝐴) is 𝑓𝑓𝛼-closed in
(𝑌, 𝜏1 ).
Remark 3.51.The composition of two 𝑓𝑓𝛼-continuous
Corollary 3.58.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be a fuzzy continuous function. Then for any 𝐴 ∈ 𝐼𝑋 , 𝑓(𝑓𝑓𝛼𝑐𝑐𝜏 𝐴) ≤ 𝑐𝑐𝜏 𝑓(𝐴).

Proof. The proof follows from the fact that every fuzzy

continuous function is 𝑓𝑓𝛼-continuous and from Theorem
3.57.
functions need not be 𝑓𝑓𝛼-continuous function as seen from
the following example.

4 𝒇𝒇𝜶-OPEN FUNCTIONS AND

FUNCTIONS

𝒇𝒇

𝜶-CLOSED

Example 3.52.Let 𝑋 = {𝑎, 𝑏}, 𝜏 = { 0𝑋 , 1𝑋 ,𝐴} , 𝜏= {0𝑋 , 1𝑋 },
𝜏1 = {0𝑋 , 1𝑋 , 𝐵} where 𝐴(𝑎) = 0.5, 𝐴(𝑏) = 0.4, 𝐵(𝑎) =
0.6, 𝐵(𝑏) = 0.4. Then (𝑋, 𝜏), (𝑋, 𝜏) and (𝑋, 𝜏1 ) are fts’s.
Consider two identity functions𝑖 ∶ (𝑋, 𝜏) → (𝑋, 𝜏)and 𝑖1
(𝑋, 𝜏) → (𝑋, 𝜏1 ). Then clearly 𝑖 and 𝑖1are 𝑓𝑓𝛼-continuous. But
𝑖1 𝜊 𝑖 ∶ (𝑋, 𝜏) → (𝑋, 𝜏1 ) is not 𝑓𝑓𝛼-continuous as seen from
Example 3.10.
Theorem 3.53.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) and 𝑓 ∶ (𝑌, 𝜏1 ) → (𝑍, 𝜏2 ) be two 𝑓𝑓𝛼-continuous functions where (𝑌, 𝜏1 )is 𝑓𝑓𝛼𝑇𝑐-space. Then their composition 𝑓 𝜊 𝑓 ∶ (𝑋, 𝜏) → (𝑍, 𝜏2 ) is an 𝑓𝑓𝛼-continuous

function.

Proof.Let 𝑉 ∈ 𝜏2 . Then 𝑓−1 (𝑉) is 𝑓𝑓𝛼-open in (𝑌, 𝜏1 ). As (𝑌, 𝜏1 ) is 𝑓𝑓𝛼𝑇𝑐 -space, 1𝑌 \𝑓−1(𝑉) is fuzzy closed in (𝑌, 𝜏1 ) and so 𝑓−1 (𝑉) is fuzzy open in (𝑌, 𝜏1 ). Again, as 𝑓 is 𝑓𝑓𝛼- continuous, 𝑓−1 (𝑓−1(𝑉)) is 𝑓𝑓𝛼-open in (𝑋, 𝜏) and so (𝑓𝜊𝑓)−1(𝑉) = 𝑓−1 (𝑓−1 (𝑉)) for every 𝑉 ∈ 𝜏2 . Consequently,
𝑓𝜊𝑓is 𝑓𝑓𝛼-continuous.
Theorem 3.54.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be an 𝑓𝛼-irresolute function
and 𝑓 ∶ (𝑌, 𝜏1 ) → (𝑍, 𝜏2 ) be an 𝑓𝑓𝛼-continuous function in (𝑌, 𝜏1 )
which is 𝑓𝑓𝛼𝑇𝛼-space, then the composition 𝑓𝜊𝑓 ∶ (𝑋, 𝜏) → (𝑍, 𝜏2 )

is 𝑓𝛼-continuous.

Proof. Let 𝑉 ∈ 𝜏2 . As 𝑓 is 𝑓𝑓𝛼-continuous, 𝑓−1 (𝑉) is 𝑓𝑓𝛼-
openin (𝑌, 𝜏1 ). Since (𝑌, 𝜏1 ) is 𝑓𝑓𝛼𝑇𝛼-space, 1𝑋 \𝑓−1 (𝑉) is fuzzy
α-closed in (𝑌, 𝜏1 ) and so ,𝑓−1 (𝑉) is fuzzy α-open in (𝑌, 𝜏1 ).
Since 𝑓 is 𝑓𝛼-irresolute, 𝑓−1 (𝑓−1(𝑉)) = (𝑓𝜊𝑓)−1 (𝑉) ∈ FαO(X).
Hence 𝑓𝜊𝑓 is 𝑓𝛼-continuous.
Definition 3.55.For a fuzzy set𝐴 in an fts (𝑋, 𝜏), 𝑓𝑓𝛼𝑐𝑐𝐴 = ∧
{𝐵 ∶ 𝐴 ≤ 𝐵, 𝐵 is 𝑓𝑓𝛼-closed in (𝑋, 𝜏)}.
Result 3.56. It is clear from Definition 3.56 that 𝑓𝑓𝛼𝑐𝑐𝐴 = 𝐴
for any 𝑓𝑓𝛼-closed set 𝐴 in an fts (𝑋, 𝜏).
In this section two new types of functions viz. 𝑓𝑓𝛼-open function and 𝑓𝑓𝛼-closed function have been introduced and
studied and found the relationship of these two functions with
fuzzy open function and fuzzy closed function.
Definition 4.1. A function𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) is said to be
𝑓𝑓𝛼-open function if the image of every fuzzy open set in
(𝑋, 𝜏) is 𝑓𝑓𝛼-open in (𝑌, 𝜏1 ).
Definition 4.2.A function𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) is said to be 𝑓𝑓𝛼-
closed function if the image of every fuzzy closed set in (𝑋, 𝜏)
is 𝑓𝑓𝛼-closed in (𝑌, 𝜏1 ).
Theorem 4.3.Every fuzzy open function is 𝑓𝑓𝛼-open.
Proof. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be a fuzzy open function and
𝑉 ∈ 𝜏. Then 𝑓(𝑉) is fuzzy open set in (𝑌, 𝜏1 ). By Proposition
2.5, 𝑓(𝑉) is 𝑓𝑓𝛼-open in (𝑌, 𝜏1 ) and hence 𝑓 is 𝑓𝑓𝛼-open
function.

Remark 4.4. The converse of the above theorem need not be true as seen from the following example.

Example 4.5.𝑓𝑓𝛼-open function fuzzy open function
Let 𝑋 = {𝑎, 𝑏}, 𝜏 = { 0𝑋 , 1𝑋 ,𝐴} , 𝜏1 = {0𝑋 , 1𝑋 , 𝐵} where 𝐴(𝑎) =
0.4, 𝐴(𝑏) = 0.6, 𝐵(𝑎) = 0.5, 𝐵(𝑏) = 0.7. Then (𝑋, 𝜏) and (𝑋, 𝜏1 )
are fts’s. Consider the identity function 𝑖 ∶ (𝑋, 𝜏1 ) → (𝑋, 𝜏).
Then 𝑖(𝐵) = 𝐵. We claim that 𝐵 is 𝑓𝑓𝛼-open in (𝑋, 𝜏).
Now 1 − 𝐵(𝑎) = 0.5, 1 − 𝐵(𝑏) = 0.3. As in Example 3.20,
𝑈 ≥ 1𝑋 \𝐵, for all 𝑓𝑓𝛼-open sets 𝑈 in (𝑋, 𝜏) and 𝛼𝑐𝑐𝜏 (1𝑋 \𝐵) =
1𝑋 \𝐵 ≤ 𝑈 and hence 1𝑋 \𝐵 is 𝑓𝑓𝛼-closed in (𝑋, 𝜏) and so 𝐵 is
𝑓𝑓𝛼-open in (𝑋, 𝜏). Consequently, 𝑖is 𝑓𝑓𝛼-open function.
But 𝐵 ∉ 𝜏 and hence 𝑖 is not fuzzy open function.
Theorem 4.6.Every fuzzy closed function is 𝑓𝑓𝛼-closed.
Proof. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be a fuzzy closed function and

𝑐 ∗

Theorem 3.57.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be an 𝑓𝑓𝛼-continuous
function. Then for any𝐴 ∈ 𝐼𝑋 , 𝑓(𝑓𝑓𝛼𝑐𝑐𝜏 𝐴) ≤ 𝑐𝑐𝜏 𝑓(𝐴).
𝑉 ∈ 𝜏𝑐 . Then 𝑓(𝑉) ∈ 𝜏1 . By Proposition 2.5, 𝑓(𝑉) is 𝑓𝑓 𝛼-
closed and hence 𝑓 is 𝑓𝑓𝛼-closed function.

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International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August-2013 979

ISSN 2229-5518

Remark 4.7. The converse of the above theorem need not be true as seen from the following example.

Example 4.8.𝑓𝑓𝛼-closedfunction fuzzy closed function

𝑐

𝑓(𝑓(𝑉)) is 𝑓𝑓𝛼-closed in 𝑍. Since 𝑓 is 𝑓𝑓𝛼-continuous and injective, 𝑓−1 (𝑓𝜊𝑓)(𝑉) = 𝑓−1 𝑓�𝑓(𝑉)� = 𝑓(𝑉) is fuzzy closed in
𝑌. Hence 𝑓 is fuzzy closed function.

Consider Example 4.5. Here 1𝑋 \𝐵 ∈ 𝜏1
and so 𝑖(1𝑋 \𝐵) =
Theorem 4.12. If 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) is 𝑓𝑓 𝛼-closed function,
1𝑋 \𝐵which is 𝑓𝑓𝛼-closed in (𝑋, 𝜏) but is not fuzzy closed in
(𝑋, 𝜏). Hence 𝑖 is 𝑓𝑓𝛼-closed function though it is not fuzzy
closed function.
Theorem 4.9.A function 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) is 𝑓𝑓𝛼-closed iff for each 𝐵 ∈ 𝐼𝑌 and for each 𝐺 ∈ 𝜏with 𝑓 −1 (𝐵) ≤ 𝐺, there exists an
𝑓𝑓𝛼-open set 𝐹 in 𝑌 such that 𝐵 ≤ 𝐹, 𝑓−1 (𝐹) ≤ 𝐺.
Proof. Let 𝐵 ∈ 𝐼𝑌 and 𝐺 ∈ 𝜏be such that 𝑓−1 (𝐵) ≤ 𝐺. Then
1𝑋 \𝐺 ∈ 𝜏𝑐 . As 𝑓 is 𝑓𝑓𝛼-closed function, 𝑓(1𝑋 \𝐺) is 𝑓𝑓𝛼-
closed in 𝑌. Let 𝐹 = 1𝑌 \𝑓(1𝑋 \G). Then 𝐹 is 𝑓𝑓𝛼-open in 𝑌.
Now 1𝑋 \𝐺 ≤ 1𝑋 \𝑓−1 (𝐵) = 𝑓−1 (1𝑌 \𝐵). Therefore, 𝑓(1𝑋 \𝐺) ≤
𝑓𝑓−1 (1𝑌 \𝐵) ≤ 1𝑌 \𝐵 and so 1𝑌 \ 𝑓(1𝑋 \𝐺) ≥ 𝐵 ⇒ 𝐵 ≤ 𝐹and
𝑓−1 (𝐹) = 𝑓−1 �1𝑌 \𝑓(1𝑋 \G)� = 1𝑋 \𝑓−1 𝑓(1𝑋 \𝐺) ⇒ 1𝑋 \𝐺 ≤
𝑓−1 𝑓(1𝑋 \𝐺). Therefore, ≥ 1𝑋 \𝑓−1 𝑓(1𝑋 \G) = 𝑓−1 (𝐹) ⇒
𝑓−1 (𝐹) ≤ 𝐺.
Conversely, let 𝑈 ∈ 𝜏𝑐 . Then 1𝑋 \𝑈 ∈ 𝜏. Now 𝑓−1 �1𝑌 \𝑓(𝑈)� =
1𝑋 \𝑓−1𝑓(𝑈). Since, 𝑈 ≤ 𝑓−1 𝑓(𝑈), 1𝑋 \𝑓−1𝑓(𝑈) ≤ 1𝑋 \𝑈.
Therefore, 𝑓−1 �1𝑌 \𝑓(𝑈)� ≤ 1𝑋 \𝑈, where 1𝑌 \𝑓(𝑈) ∈ 𝐼𝑌 . Then
there exists an 𝑓𝑓𝛼-open set 𝐹 in 𝑌 such that 1𝑌 \𝑓(𝑈) ≤ 𝐹and
𝑓−1 (𝐹) ≤ 1𝑋 \𝑈. Therefore, 𝑈 ≤ 1𝑋 \𝑓−1(𝐹). Hence 1𝑌 \𝐹 ≤
𝑓(𝑈) ≤ 𝑓(1𝑋 \𝑓−1 (𝐹)) ≤ 1𝑌 \𝐹 ⇒ 𝑓(𝑈) = 1𝑌 \𝐹 and so 𝑓(𝑈) is
𝑓𝑓𝛼-closed in 𝑌. Consequently, 𝑓 is 𝑓𝑓𝛼-closed function.
Theorem 4.10.The function 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) is fuzzy closed
function and 𝑓 ∶ (𝑌, 𝜏1 ) → (𝑍, 𝜏2 ) is𝑓𝑓𝛼-closed function, then
their composition 𝑓 𝜊 𝑓 ∶ (𝑋, 𝜏) → (𝑍, 𝜏2 ) is 𝑓𝑓𝛼-closed function.
Proof.Let 𝐺 ∈ 𝜏𝑐 . Then as 𝑓 is fuzzy closed function,𝑓(𝐺) ∈ 𝜏𝑐 .
As 𝑓is 𝑓𝑓𝛼-closed function, 𝑓�𝑓(𝐺)� = (𝑓𝜊𝑓)(𝐺) is 𝑓𝑓𝛼-
closed in (𝑍, 𝜏2 ). Consequently, 𝑓𝜊𝑓 is 𝑓𝑓𝛼-closed function.
Theorem 4.11. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) and 𝑓 ∶ (𝑌, 𝜏1 ) → (𝑍, 𝜏2 )
be such that their composition 𝑓 𝜊 𝑓 ∶ (𝑋, 𝜏) → (𝑍, 𝜏2 ) is an 𝑓𝑓𝛼-

closed function. Then the following statements are true :

(i) If 𝑓 is fuzzy surjective continuous, then 𝑓 is 𝑓𝑓𝛼-

closed function.

(ii) If 𝑓 is fuzzy surjective 𝑓𝑓𝛼-continuous and (𝑋, 𝜏) is an 𝑓𝛼𝑇𝑏 -space, then 𝑓 is 𝑓𝑓𝛼-closed function.
(iii) If 𝑓 is 𝑓𝑓𝛼-continuous and injective, then 𝑓 is fuzzy

closed function.

then 𝑓𝑓𝛼𝑐𝑐𝜏 �𝑓(𝑈)� ≤ 𝑓(𝑐𝑐𝜏 (𝑈)), for every 𝑈 ∈ 𝐼𝑋 .
Proof.Let 𝑈 ∈ 𝐼𝑋 . Then 𝑐𝑐𝜏 𝑈 ∈ 𝜏𝑐 . Since 𝑓 is 𝑓𝑓𝛼-closed,
𝑓(𝑐𝑐𝜏 𝑈) is 𝑓𝑓𝛼-closed set in 𝑌. As 𝑈 ≤ 𝑐𝑐𝜏 𝑈, 𝑓(𝑈) ≤ 𝑓(𝑐𝑐𝜏 𝑈),
by Definition 3.55, 𝑓𝑓𝛼𝑐𝑐𝜏 �𝑓(𝑈)� ≤ 𝑓(𝑐𝑐𝜏 (𝑈)).

REFERENCES

[1] K.K. Azad, “On fuzzy semi-continuity, fuzzy almost continuity and fuzzy weakly continuity,” J. Math. Anal. Appl., 82pp. 14-32, 1981.

[2] A.S. Bin Shahna, “ On fuzzy strong semicontinuity and fuzzy precontinuity,” Fuzzy Sets and Systems, 44 pp. 303-308, 1991.

[3] C.L.Chang, “Fuzzy topological spaces,”J. Math. Anal. Appl., 24 pp.

182-190, 1968.

[4] M.A. FathAlla, “α-continuous mappings in fuzzy topological spaces,"Bull. Cal. Math. Soc., 80 pp. 323-329, 1988.

[5] A.S. Mashhour, M.H. Ghanim and M.A. FathAlla, "On fuzzy noncontinuous mappings," Bull. Cal.Math. Soc., 78 pp. 57-69, 1986.

[6] L.A. Zadeh, “Fuzzy Sets,” Inform. Control, 8 pp. 338-353, 1965.

Proof. (i) Let 𝑉 ∈ 𝜏𝑐 . Since 𝑓 is fuzzy continuous, 𝑓−1
(𝑉) ∈
𝜏𝑐 . Since 𝑓𝜊𝑓 is 𝑓𝑓𝛼-closed function, (𝑓𝜊𝑓)(𝑓−1(𝑉)) is 𝑓𝑓𝛼- closed set in 𝑍. As 𝑓 is surjective, (𝑓𝜊𝑓)(𝑓−1 (𝑉))=
𝑓 �𝑓�𝑓−1 (𝑉)�� = 𝑓(𝑉), proving that 𝑓 is 𝑓𝑓𝛼-closed function.
(ii)Let𝑉 ∈ 𝜏𝑐 . Since 𝑓 is 𝑓𝑓𝛼-continuous, 𝑓

−1 (𝑉) is 𝑓𝑓𝛼-

closed in 𝑋. By Proposition 2.14, 𝑓−1 (𝑉) is 𝑓𝛼𝑓-closed in 𝑋. As
(𝑋, 𝜏) is an 𝑓𝛼𝑇𝑏 -space, ,𝑓−1 (𝑉) is fuzzy closed in 𝑋. As 𝑓𝜊𝑓 is
𝑓𝑓𝛼-closed function, (𝑓𝜊𝑓)�𝑓−1 (𝑉)� = 𝑓(𝑉) (as 𝑓 is
surjective) is 𝑓𝑓𝛼-closed set in 𝑍. Hence 𝑓 is 𝑓𝑓𝛼-closed
function.
(iii)Let𝑉 ∈ 𝜏𝑐 . Since 𝑓𝜊𝑓 is 𝑓𝑓𝛼-closed function, (𝑓𝜊𝑓)(𝑉) =

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