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 𝑓𝑓∗𝛼 --continuous, 𝑓𝑓∗𝛼- open functions and 𝑓𝑓∗ 𝛼- closed functions are introduced and studied. Again, some important properties of such functions are studied in the newly defined spaces using 𝑓𝑓∗𝛼 –closed sets.
Index Terms—𝑓𝑓∗𝛼 –open sets, 𝑓𝑓∗ 𝛼 –closed sets, 𝑓𝑓∗𝛼 –continuity, 𝑓𝑓∗ 𝛼 –open functions, 𝑓𝑓∗ 𝛼 –closed functions, 𝑓𝑓∗ 𝛼𝑇𝛼 –space,
𝑓𝑓∗ 𝛼𝑇𝑐 space.
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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 𝒇𝒇∗ 𝜶-OPEN SETS AND ITS PROPERTIES
We now recall the following definitions, which are
useful in the sequel.
(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.
(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) 𝑓∗ 𝛼-closed (𝑓𝑓∗ 𝛼-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.
(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) 𝑓𝑓∗ 𝛼𝑇𝑐 -space if every 𝑓𝑓∗ 𝛼-closed set in (𝑋, 𝜏) is
fuzzy closed in (𝑋, 𝜏),
(iv) 𝑓𝑓∗ 𝛼𝑇𝛼 -space if every 𝑓𝑓∗ 𝛼-closed set in (𝑋, 𝜏) is
fuzzy α-closed in (𝑋, 𝜏),
(v) 𝑓𝑤𝑓𝛼𝑇𝑔∗𝛼 -space if every 𝑓𝑤𝑓𝛼-closed set n (𝑋, 𝜏)
is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-
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 𝑓𝑓∗ 𝛼-closed in (𝑋, 𝜏) and hence
𝑉is 𝑓𝑓∗ 𝛼-open in (𝑋, 𝜏).
Proposition 2.6.Every fuzzy regular open set in an fts(𝑋, 𝜏) is
𝑓𝑓∗ 𝛼-open in (𝑋, 𝜏).
proof follows from Proposition 2.5.
Proposition 2.7.Every 𝑓𝑓∗ 𝛼-open set in an fts(𝑋, 𝜏) is 𝑓𝑓𝛼-open in
(𝑋, 𝜏).
Proof. Let 𝑉 be 𝑓𝑓∗ 𝛼-open set in (𝑋, 𝜏). Then 1𝑋 \𝑉 is 𝑓𝑓∗ 𝛼-
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 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-closed set and hence 𝑈 is
𝑓𝑓∗ 𝛼-open set in (𝑋, 𝜏).
Proposition 2.9.Every 𝑓𝑓∗ 𝛼-open set is 𝑓𝛼𝑓-open n (𝑋, 𝜏).
Proof. Let 𝑈 be an 𝑓𝑓∗ 𝛼-open set in (𝑋, 𝜏). Then 1𝑋 \𝑈 is 𝑓𝑓∗ 𝛼-
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 𝑓𝑓∗ 𝛼-open and hence by Proposition 2.7, 𝑉 is 𝑓𝑓𝛼-open in (𝑋, 𝜏). Since 1𝑋 \𝑈 is 𝑓𝑓∗ 𝛼- 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 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼—closed and
hence 𝐴 is 𝑓𝑓∗ 𝛼-open in (𝑋, 𝜏).
Proposition 2.11.Every 𝑓𝑓∗ 𝛼-open set is 𝑓𝑤𝑓𝛼-open in (𝑋, 𝜏).
Proof. Let 𝑈 be 𝑓𝑓∗ 𝛼-open in (𝑋, 𝜏). Then 1𝑋 \𝑈 is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-open set is 𝑓𝑓𝑠-open set in (𝑋, 𝜏).
Proof. Let 𝑈 be 𝑓𝑓∗ 𝛼-open in (𝑋, 𝜏). Then 1𝑋 \𝑈 is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-open set is 𝑓𝛼𝑓𝛼-open in (𝑋, 𝜏).
Proof.Let 𝐴 be 𝑓𝑓∗ 𝛼-open in 𝑋. Then 1𝑋 \𝐴 is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-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.
3𝒇𝒇∗ 𝜶-CONTINUOUS FUNCTIONS
In this section the concept of 𝑓𝑓∗ 𝛼-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 (𝑓𝑓∗ 𝛼 –continuous, for short) if𝑓−1 (𝑉)is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-continuous function is𝑓𝑓𝛼-continuous.
Proof. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be 𝑓𝑓∗ 𝛼-continuous and 𝑉 ∈ 𝜏1 .
Then 𝑓−1 (𝑉) is 𝑓𝑓∗ 𝛼-open in 𝑋. By Proposition 2.7, 𝑓−1 (𝑉) is
𝑓𝑓𝛼-open in 𝑋 and hence 𝑓 is 𝑓𝑓𝛼-continuous.
true as seen from the following example.
Example 3.10.𝑓𝑓𝛼-continuity⇏ 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-closed as 1𝑋 \𝐵 is 𝑓𝑓𝛼-open
in (𝑋, 𝜏)and 𝛼𝑐𝑐 (1𝑋 \𝐵) = 1𝑋 \𝐴 ≰ 1𝑋 \𝐵. Hence 𝑖 is not 𝑓𝑓∗ 𝛼-
continuous.
Theorem 3.11. Every 𝑓𝛼-continuous function is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-open
is 𝑓𝑓∗ 𝛼-continuous.
Proof.Let 𝑉 ∈ 𝜏1 . Then 𝑓−1 (𝑉) ∈ 𝜏. By Proposition 2.5, 𝑓−1 (𝑉)
is 𝑓𝑓∗ 𝛼-open in 𝑋 and hence 𝑓 is 𝑓𝑓∗ 𝛼-continuous.
true as seen from the following example.
Example 3.4. 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-continuous.
true as seen from the following example.
Example 3.13.𝑓𝑓∗ 𝛼-continuity⇏ 𝑓𝛼-continuity
Consider Example 3.4. Here 𝑖 is 𝑓𝑓∗ 𝛼-continuous. Now 𝐵 ∈
𝜏1, 𝑖 −1 (𝐵) = 𝐵 ∉ FαO(𝑋, 𝜏). Hence 𝑖 is not 𝑓𝛼-continuous.
Theorem 3.14.Every 𝑓𝑓∗ 𝛼-continuous function is 𝑓𝛼𝑓-continuous.
Proof. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 )be 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-open in 𝑋. By Proposition 2.9, 𝑓−1
(𝑉) is
(𝑋, 𝜏) containing 1𝑋 \𝐵 and so 𝑖 is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-
continuous.
Proof. Let𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be fuzzy completely continuous
𝑓𝛼𝑓-open in 𝑋 and hence 𝑓 is 𝑓𝛼𝑓-continuous.
true as seen from the following example.
Example 3.16.𝑓𝛼𝑓-continuity⇏ 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-
open in 𝑋. Consequently, 𝑓 is 𝑓𝑓∗ 𝛼-continuous.
true in general as seen from the following example. Example 3.7.𝑓𝑓∗ 𝛼-continuity ⇏ fuzzy completely continuity Consider Example 3.4. Here 𝑖 is 𝑓𝑓∗ 𝛼-continuous. Now
identity function 𝑖 ∶ (𝑋, 𝜏) → (𝑋, 𝜏1 ). We claim that 𝑖 is 𝑓𝛼𝑓-
continuous but not 𝑓𝑓∗ 𝛼-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 .
IJSER © 2013
International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August-2013 976
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 𝑓𝑓∗ 𝛼-closed in (𝑋, 𝜏) and so 𝑖 is not
𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-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
𝑓𝑓∗ 𝛼-open in (𝑋, 𝜏) and hence 𝑓 is 𝑓𝑓∗ 𝛼-continuous.
Theorem 3.18.Every 𝑓𝑓#-continuous function is 𝑓𝑓∗ 𝛼-continuous.
continuous.
Proof.Let𝑉 ∈ 𝜏1 . As 𝑓 is 𝑓𝑤𝑓𝛼-continuous, 𝑓−1 (𝑉) is 𝑓𝑤𝑓𝛼- open in (𝑋, 𝜏). As (𝑋, 𝜏) is 𝑓𝑤𝑓𝛼𝑇𝑔∗ 𝛼 -space, 1𝑋 \𝑓−1 (𝑉) is 𝑓𝑓∗ 𝛼- closed in (𝑋, 𝜏)and hence 𝑓−1 (𝑉) is 𝑓𝑓∗ 𝛼-open in (𝑋, 𝜏). Consequently, 𝑓 is 𝑓𝑓∗ 𝛼-continuous.
Theorem 3.25.Every 𝑓𝑓∗ 𝛼-continuous function is 𝑓𝑤𝛼𝑓-
continuous.
Proof. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be 𝑓𝑓∗ 𝛼-continuous function. By
Theorem 3.21, 𝑓 is 𝑓𝑤𝑓𝛼-continuous. Then by Proposition
2.15(ii), 𝑓 is 𝑓𝑤𝛼𝑓-continuous.
true as seen from the following example.
Example 3.27.𝑓𝑤𝛼𝑓-continuity⇏ 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-open in (𝑋, 𝜏) and hence 𝑓 is
𝑓𝑓∗ 𝛼-continuous.
true as seen from the following example.
Example 3.20.𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-continuous.
Theorem 3.28.Every 𝑓𝑓∗ 𝛼-continuous function is 𝑓𝑓𝑠-continuous.
Proof. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be 𝑓𝑓∗ 𝛼-continuous and 𝑉 ∈ 𝜏1 .
As 𝑓 is 𝑓𝑓∗ 𝛼-continuous, 𝑓−1 (𝑉) is 𝑓𝑓∗ 𝛼-open in (𝑋, 𝜏). By
Proposition 2.12, 𝑓−1 (𝑉) is 𝑓𝑓𝑠-open in (𝑋, 𝜏) and hence 𝑓 is
𝑓𝑓𝑠-continuous.
true as seen from the following example.
Example 3.30.𝑓𝑓𝑠-continuity⇏ 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-continuous.
Theorem 3.21.Every 𝑓𝑓∗ 𝛼-continuous function is 𝑓𝑤𝑓𝛼-
continuous.
Proof. Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be 𝑓𝑓∗ 𝛼-continuous and 𝑉 ∈ 𝜏1. Then 𝑓−1 (𝑉) is 𝑓𝑓∗ 𝛼-open in 𝑋. By Proposition 2.11, 𝑓−1 (𝑉) is
𝑓𝑤𝑓𝛼-open in 𝑋 and hence 𝑓 is 𝑓𝑤𝑓𝛼-continuous.
true as seen from the following example.
Example 3.23.𝑓𝑤𝑓𝛼-continuity⇏ 𝑓𝑓∗ 𝛼-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 𝑓𝑤𝑓𝛼𝑇𝑔∗𝛼 -space. Then 𝑓 is 𝑓𝑓∗ 𝛼-
Proposition 3.31.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be an 𝑓𝑓𝑠-continuous function where (𝑋, 𝜏) is an 𝑓𝑇𝑏 -space. Then 𝑓 is 𝑓𝑓∗ 𝛼-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
𝑓𝑓∗ 𝛼-open in (𝑋, 𝜏) and hence 𝑓 is 𝑓𝑓∗ 𝛼-continuous.
Theorem 3.32.Every 𝑓𝑓∗ 𝛼-continuous function is 𝑓𝛼𝑓𝛼-
continuous.
Proof.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be 𝑓𝑓∗ 𝛼-continuous and 𝑉 ∈ 𝜏1 . Then 𝑓−1 (𝑉) is 𝑓𝑓∗ 𝛼-open in (𝑋, 𝜏). By Proposition 2.13,
𝑓−1 (𝑉) is 𝑓𝛼𝑓𝛼-open in (𝑋, 𝜏). Hence 𝑓 is 𝑓𝛼𝑓𝛼-continuous.
true as seen from the following example.
Example 3.34.𝑓𝛼𝑓𝛼-continuity⇏ 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-continuous.
Theorem 3.35.Every 𝑓𝑓∗ 𝛼-continuous function is 𝑓𝑓𝑝𝛼-
continuous.
Proof. By Theorem 3.32, every 𝑓𝑓∗ 𝛼-continuous function is
𝑓𝛼𝑓𝛼-continuous and again by Proposition 2.5(iii), it is 𝑓𝑓𝑝𝛼-
continuous.
Example 3.37.𝑓𝑓𝑝𝛼-continuity⇏ 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-continuous
but not fuzzy semi-continuous.
Remark 3.44. The following two examples show that 𝑓𝑓- continuous function and 𝑓𝑓∗ 𝛼-continuous function are
independent notions.
Example 3.45.𝑓𝑓-continuity⇏ 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-continuous.
Example 3.46.𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-open in
(𝑋, 𝜏) and hence 𝑓 is 𝑓𝑓∗ 𝛼-continuous.
true as seen from the following example.
Example 3.40.𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-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.
continuity and 𝑓𝑓∗ 𝛼-continuity are independent notions.
Example 3.42.fuzzy semi-continuity ⇏ 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-closed and so 𝐵 is not 𝑓𝑓∗ 𝛼-open
in (𝑋, 𝜏) and hence 𝑖 is not 𝑓𝑓∗ 𝛼-continuous.
Example 3.43.𝑓𝑓∗ 𝛼-continuity ⇏ fuzzy semi-continuity
set in (𝑋, 𝜏)other than 0𝑋 contains1𝑋 \𝐵 and 𝛼𝑐𝑐𝜏 (1𝑋 \𝐵) = 1𝑋 \𝐵
and hence 𝑖 is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-
continuous iff the inverse image of every fuzzy closed set in 𝑌is
𝑓𝑓∗ 𝛼-closed in 𝑋.
Proof. Let 𝑓 be 𝑓𝑓∗ 𝛼-continuousand 𝐹 ∈ 𝜏𝑐 . Then 1 \𝐹 ∈ 𝜏 .
Since 𝑓 is 𝑓𝑓∗ 𝛼-continuous, 𝑓−1 (1𝑋 \𝐹) = 1𝑋 \𝑓−1(𝐹) is 𝑓𝑓∗ 𝛼-
open in 𝑋. Hence 𝑓 −1 (𝐹) is 𝑓𝑓∗ 𝛼-closed in 𝑋.
Conversely, let us suppose that 𝑓−1 (𝐹)be𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-closed in 𝑋 and
so 𝑓−1 (𝑉) is 𝑓𝑓∗ 𝛼-open in 𝑋 and hence 𝑓 is 𝑓𝑓∗ 𝛼-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
𝑓𝑓∗ 𝛼-closed set 𝐴in (𝑋, 𝜏).
Proof. Let 𝐴 be an 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-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
every𝑓𝑓∗ 𝛼-closed set 𝐴in (𝑋, 𝜏).
is 𝑓𝑓𝛼-continuous. Then by Theorem 3.48, 𝑓(𝐴) is 𝑓𝛼𝑓-closed
for every 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-
𝑓𝑓∗ 𝛼-closed in (𝑌, 𝜏1 ) for every 𝑓𝑓∗ 𝛼-closed set 𝐴in (𝑋, 𝜏).
Proof. Let 𝐴 be 𝑓𝑓∗ 𝛼-closed in (𝑋, 𝜏) and 𝑉 be any 𝑓𝑓𝛼-open
continuous, 𝑓−1 (𝑐𝑐𝜏 𝑓(𝐴)) is 𝑓𝑓∗ 𝛼-closed in (𝑋, 𝜏). Hence by
Result 3.57, 𝑓𝑓∗ 𝛼𝑐𝑐𝜏 (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 𝑓𝑓∗ 𝛼-closed set containing 𝐴. Then
𝑓𝑓∗ 𝛼𝑐𝑐𝜏 𝐴 ≤ 𝑓−1 (𝑐𝑐𝜏 𝑓(𝐴)).Therefore, 𝑓(𝑓𝑓∗ 𝛼𝑐𝑐𝜏𝐴) ≤ 𝑐𝑐𝜏 𝑓(𝐴).
1 1
(𝑌, 𝜏1 ) . As 𝑓 is𝑓𝑓𝛼-continuous, 𝑓−1 (𝑉) is 𝑓𝑓𝛼-open in (𝑋, 𝜏).
Since 𝐴 is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-closed in
(𝑌, 𝜏1 ).
Remark 3.51.The composition of two 𝑓𝑓∗ 𝛼-continuous
Corollary 3.58.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be a fuzzy continuous function. Then for any 𝐴 ∈ 𝐼𝑋 , 𝑓(𝑓𝑓∗𝛼𝑐𝑐𝜏 𝐴) ≤ 𝑐𝑐𝜏 𝑓(𝐴).
continuous function is 𝑓𝑓∗ 𝛼-continuous and from Theorem
3.57.
functions need not be 𝑓𝑓∗ 𝛼-continuous function as seen from
the following example.
4 𝒇𝒇∗ 𝜶-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 𝑓𝑓∗ 𝛼-continuous. But
𝑖1 𝜊 𝑖 ∶ (𝑋, 𝜏) → (𝑋, 𝜏1 ) is not 𝑓𝑓∗ 𝛼-continuous as seen from
Example 3.10.
Theorem 3.53.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) and 𝑓 ∶ (𝑌, 𝜏1 ) → (𝑍, 𝜏2 ) be two 𝑓𝑓∗ 𝛼-continuous functions where (𝑌, 𝜏1 )is 𝑓𝑓∗ 𝛼𝑇𝑐-space. Then their composition 𝑓 𝜊 𝑓 ∶ (𝑋, 𝜏) → (𝑍, 𝜏2 ) is an 𝑓𝑓∗ 𝛼-continuous
function.
Proof.Let 𝑉 ∈ 𝜏2 . Then 𝑓−1 (𝑉) is 𝑓𝑓∗ 𝛼-open in (𝑌, 𝜏1 ). As (𝑌, 𝜏1 ) is 𝑓𝑓∗ 𝛼𝑇𝑐 -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 𝑓𝑓∗ 𝛼- continuous, 𝑓−1 (𝑓−1(𝑉)) is 𝑓𝑓∗ 𝛼-open in (𝑋, 𝜏) and so (𝑓𝜊𝑓)−1(𝑉) = 𝑓−1 (𝑓−1 (𝑉)) for every 𝑉 ∈ 𝜏2 . Consequently,
𝑓𝜊𝑓is 𝑓𝑓∗ 𝛼-continuous.
Theorem 3.54.Let 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) be an 𝑓𝛼-irresolute function
and 𝑓 ∶ (𝑌, 𝜏1 ) → (𝑍, 𝜏2 ) be an 𝑓𝑓∗ 𝛼-continuous function in (𝑌, 𝜏1 )
which is 𝑓𝑓∗ 𝛼𝑇𝛼-space, then the composition 𝑓𝜊𝑓 ∶ (𝑋, 𝜏) → (𝑍, 𝜏2 )
is 𝑓𝛼-continuous.
Proof. Let 𝑉 ∈ 𝜏2 . As 𝑓 is 𝑓𝑓∗ 𝛼-continuous, 𝑓−1 (𝑉) is 𝑓𝑓∗ 𝛼-
openin (𝑌, 𝜏1 ). Since (𝑌, 𝜏1 ) is 𝑓𝑓∗ 𝛼𝑇𝛼-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), 𝑓𝑓∗ 𝛼𝑐𝑐𝐴 = ∧
{𝐵 ∶ 𝐴 ≤ 𝐵, 𝐵 is 𝑓𝑓∗ 𝛼-closed in (𝑋, 𝜏)}.
Result 3.56. It is clear from Definition 3.56 that 𝑓𝑓∗ 𝛼𝑐𝑐𝐴 = 𝐴
for any 𝑓𝑓∗ 𝛼-closed set 𝐴 in an fts (𝑋, 𝜏).
In this section two new types of functions viz. 𝑓𝑓∗ 𝛼-open function and 𝑓𝑓∗ 𝛼-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
𝑓𝑓∗ 𝛼-open function if the image of every fuzzy open set in
(𝑋, 𝜏) is 𝑓𝑓∗ 𝛼-open in (𝑌, 𝜏1 ).
Definition 4.2.A function𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) is said to be 𝑓𝑓∗ 𝛼-
closed function if the image of every fuzzy closed set in (𝑋, 𝜏)
is 𝑓𝑓∗ 𝛼-closed in (𝑌, 𝜏1 ).
Theorem 4.3.Every fuzzy open function is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-open in (𝑌, 𝜏1 ) and hence 𝑓 is 𝑓𝑓∗ 𝛼-open
function.
Example 4.5.𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-closed in (𝑋, 𝜏) and so 𝐵 is
𝑓𝑓∗ 𝛼-open in (𝑋, 𝜏). Consequently, 𝑖is 𝑓𝑓∗ 𝛼-open function.
But 𝐵 ∉ 𝜏 and hence 𝑖 is not fuzzy open function.
Theorem 4.6.Every fuzzy closed function is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-continuous
function. Then for any𝐴 ∈ 𝐼𝑋 , 𝑓(𝑓𝑓∗𝛼𝑐𝑐𝜏 𝐴) ≤ 𝑐𝑐𝜏 𝑓(𝐴).
𝑉 ∈ 𝜏𝑐 . Then 𝑓(𝑉) ∈ 𝜏1 . By Proposition 2.5, 𝑓(𝑉) is 𝑓𝑓 𝛼-
closed and hence 𝑓 is 𝑓𝑓∗ 𝛼-closed function.
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Example 4.8.𝑓𝑓∗ 𝛼-closedfunction ⇏ fuzzy closed function
𝑐
𝑓(𝑓(𝑉)) is 𝑓𝑓∗ 𝛼-closed in 𝑍. Since 𝑓 is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-closed in (𝑋, 𝜏) but is not fuzzy closed in
(𝑋, 𝜏). Hence 𝑖 is 𝑓𝑓∗ 𝛼-closed function though it is not fuzzy
closed function.
Theorem 4.9.A function 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) is 𝑓𝑓∗ 𝛼-closed iff for each 𝐵 ∈ 𝐼𝑌 and for each 𝐺 ∈ 𝜏with 𝑓 −1 (𝐵) ≤ 𝐺, there exists an
𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-closed function, 𝑓(1𝑋 \𝐺) is 𝑓𝑓∗ 𝛼-
closed in 𝑌. Let 𝐹 = 1𝑌 \𝑓(1𝑋 \G). Then 𝐹 is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-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
𝑓𝑓∗ 𝛼-closed in 𝑌. Consequently, 𝑓 is 𝑓𝑓∗ 𝛼-closed function.
Theorem 4.10.The function 𝑓 ∶ (𝑋, 𝜏) → (𝑌, 𝜏1 ) is fuzzy closed
function and 𝑓 ∶ (𝑌, 𝜏1 ) → (𝑍, 𝜏2 ) is𝑓𝑓∗ 𝛼-closed function, then
their composition 𝑓 𝜊 𝑓 ∶ (𝑋, 𝜏) → (𝑍, 𝜏2 ) is 𝑓𝑓∗ 𝛼-closed function.
Proof.Let 𝐺 ∈ 𝜏𝑐 . Then as 𝑓 is fuzzy closed function,𝑓(𝐺) ∈ 𝜏𝑐 .
As 𝑓is 𝑓𝑓∗ 𝛼-closed function, 𝑓�𝑓(𝐺)� = (𝑓𝜊𝑓)(𝐺) is 𝑓𝑓∗ 𝛼-
closed in (𝑍, 𝜏2 ). Consequently, 𝑓𝜊𝑓 is 𝑓𝑓∗ 𝛼-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 𝑓𝑓∗ 𝛼-
closed function. Then the following statements are true :
(i) If 𝑓 is fuzzy surjective continuous, then 𝑓 is 𝑓𝑓∗ 𝛼-
closed function.
(ii) If 𝑓 is fuzzy surjective 𝑓𝑓𝛼-continuous and (𝑋, 𝜏) is an 𝑓𝛼𝑇𝑏 -space, then 𝑓 is 𝑓𝑓∗ 𝛼-closed function.
(iii) If 𝑓 is 𝑓𝑓∗ 𝛼-continuous and injective, then 𝑓 is fuzzy
closed function.
then 𝑓𝑓∗ 𝛼𝑐𝑐𝜏 �𝑓(𝑈)� ≤ 𝑓(𝑐𝑐𝜏 (𝑈)), for every 𝑈 ∈ 𝐼𝑋 .
Proof.Let 𝑈 ∈ 𝐼𝑋 . Then 𝑐𝑐𝜏 𝑈 ∈ 𝜏𝑐 . Since 𝑓 is 𝑓𝑓∗ 𝛼-closed,
𝑓(𝑐𝑐𝜏 𝑈) is 𝑓𝑓∗ 𝛼-closed set in 𝑌. As 𝑈 ≤ 𝑐𝑐𝜏 𝑈, 𝑓(𝑈) ≤ 𝑓(𝑐𝑐𝜏 𝑈),
by Definition 3.55, 𝑓𝑓∗ 𝛼𝑐𝑐𝜏 �𝑓(𝑈)� ≤ 𝑓(𝑐𝑐𝜏 (𝑈)).
[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 𝑓𝑓∗ 𝛼-closed function, (𝑓𝜊𝑓)(𝑓−1(𝑉)) is 𝑓𝑓∗ 𝛼- closed set in 𝑍. As 𝑓 is surjective, (𝑓𝜊𝑓)(𝑓−1 (𝑉))=
𝑓 �𝑓�𝑓−1 (𝑉)�� = 𝑓(𝑉), proving that 𝑓 is 𝑓𝑓∗ 𝛼-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
𝑓𝑓∗ 𝛼-closed function, (𝑓𝜊𝑓)�𝑓−1 (𝑉)� = 𝑓(𝑉) (as 𝑓 is
surjective) is 𝑓𝑓∗ 𝛼-closed set in 𝑍. Hence 𝑓 is 𝑓𝑓∗ 𝛼-closed
function.
(iii)Let𝑉 ∈ 𝜏𝑐 . Since 𝑓𝜊𝑓 is 𝑓𝑓∗ 𝛼-closed function, (𝑓𝜊𝑓)(𝑉) =
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