CUBO A Mathematical Journal
Vol.10, N
o
¯ 01, (67–75). March 2008
Somewhat Fuzzy Semi α-Irresolute Functions
V. Seenivasan
P.G. Department of Mathematics, Jawahar Science College
Neyveli – 607 803, Tamilnadu, India
email: krishnaseenu@rediffmail.com
G. Balasubramanian
Ramanujan Institute for Advanced Study in Mathematics
University of Madras, Chennai, 600 005, Tamilnadu, India.
and
G. Thangaraj
P.G. Department of Mathematics, Jawahar Science College
Neyveli – 607 803, Tamilnadu, India
ABSTRACT
In this paper the concepts of somewhat fuzzy semi α-irresolute functions and
strongly somewhat fuzzy semi-open functions are introduced and studied. Be-
sides giving characterizations of these functions, some interesting properties of
these functions are also given.
RESUMEN
En este art́ıculo son introducidos y estudiados lon conceptos de funciones semi
fuzzy α-irresoluta y funciones fuertemente fuzzy semi-abiertas. Propiedades de
esta clase de funciones son dadas.
68 V. Seenivasan, G. Balasubramanian and G. Thangaraj CUBO
10, 1 (2008)
Key words and phrases: Somewhat fuzzy semi α-irresolute, fuzzy semi α-irresolute, fuzzy
α-irresolute, somewhat fuzzy semi-open functions.
Math. Subj. Class.: 54A40.
1 Introduction
The fuzzy concept has invaded almost all branches of mathematics ever since the intro-
duction of fuzzy sets by L.A.Zadeh [12]. Fuzzy sets have applications in many fields such as
information [6] and control [8]. The theory of fuzzy topological spaces was introduced and
developed by C.L.Chang [3] and since then various notions in classical topology have been
extended to fuzzy topological spaces. The concept of somewhat continuous functions was
introduced by Karl R.Gentry and Hughes B.Hoyle III in [4] and this concept was studied in
connection with the idea of feeble continuous function and feebly open function introduced
and by Zdenek Frolik in [13]. The concept of semi α-irresolute functions was introduced and
studied in [11]. The concepts of somewhat fuzzy continuous functions and somewhat fuzzy
semi-continuous functions was introduced and studied by G. Thangaraj and G. Balasubra-
manian in [9] and [10] respectively. In this paper we introduce the concepts of somewhat
fuzzy semi α-irresolute functions and strongly somewhat fuzzy semi-open functions and
study their properties.
2 Preliminaries
By a fuzzy topological space we shall mean a non-empty set X together with fuzzy topology
T [3] and denote it by (X, T ). A fuzzy point in X with support x ∈ X and value p (0 < p
≤ 1) is denoted by xp. The complement µ
′
of a fuzzy set µ is 1 − µ, defined by µ
′
(x) = (1X
−µ)(x) = 1 − µ(x), for all x ∈ X [3]. If λ is a fuzzy set in X and µ is a fuzzy set in Y , then
λ × µ is a fuzzy set in X × Y , defined by (λ × µ)(x, y) = min(λ(x), µ(y)), for every (x, y) in
X × Y [1]. A fuzzy topological space X is product related to a fuzzy topological space Y
[1] if for fuzzy sets γ in X and ξ in Y whenever λ
′
(= 1 − λ) � γ and µ
′
( = 1 − µ) � ξ (in
which case (λ
′
× 1) ∨ (1 × µ
′
) ≥ (γ × ξ)) where λ is a fuzzy open set in X and µ is a fuzzy
open set in Y , there exists a fuzzy open set λ1 in X and a fuzzy open set µ1 in Y such that
λ1
′
≥ γ or µ1
′
≥ ξ and (λ1
′
× 1) ∨ (1 × µ
′
1) = (λ
′
× 1) ∨ (1 × µ
′
). Let f : X → Y be a
mapping from X to Y . If λ is a fuzzy set in X, f (λ) is defined by
f (λ) (y) =
{
Sup λ(x),
x∈f −1(y)
if f
−1
(y) 6= φ
0, otherwise,
for each y ∈ Y ; and if µ is a fuzzy set in Y , f −1(µ) is defined by f −1(µ) (x) = µ f (x), for
CUBO
10, 1 (2008)
Somewhat Fuzzy Semi α-Irresolute Functions 69
each x ∈ X [3]. Let f be a mapping from X to Y . Then the graph g of f is mapping from
X to X × Y sending x in X to (x, f (x)). For two mappings f1 : X1 → Y1 and f2 : X2 →
Y2 , we define the product f1 × f2 of f1 and f2 to be a mapping from X1×X2 to Y1 ×Y2
sending (x1, x2) in X1× X2 to (f1(x1), f2 (x2)). For any fuzzy set λ in a fuzzy topological
space, it is shown in [1] that (i) 1 – cl λ = int(1 − λ), (ii) cl(1 − λ) = 1−int λ. A fuzzy set
λ in fuzzy topological space (X, T ) is called fuzzy α-dense if there exists no fuzzy α-closed
set µ such that λ < µ < 1.
Definition 2.1 [5]: A function f : (X, T ) → (Y , S) is said to be fuzzy α-irresolute if
f
−1
(λ) is fuzzy α-open set in (X, T ) for every fuzzy α-open set λ of (Y , S).
Definition 2.2 [7]: A function f from a fuzzy topological space (X, T ) to a fuzzy topological
space (Y , S) is said to be fuzzy semi α-irresolute if f −1(λ) is fuzzy semi-open set in (X,
T ) for each fuzzy α-open set λ in (Y , S).
Definition 2.3 [10]: Let (X, T ) and (Y, S) be fuzzy topological spaces. A function f : (X,
T ) → (Y, S) is called somewhat fuzzy semi-open function if and only if for all λ ∈ T ,
λ 6= 0 there exists a fuzzy semi-open set µ in Y such that µ 6= 0 and µ ≤ f (λ).
Definition 2.4 Let (X, T ) be a fuzzy topological space and λ be any fuzzy set in X.
1. λ is called fuzzy α-open set [2] if λ ≤ int cl intλ
2. λ is called fuzzy semi-open set [1] if λ ≤ cl intλ
The complement of fuzzy α-open (fuzzy semi-open) set is called fuzzy α-closed ( fuzzy
semi-closed) set.
3 Somewhat fuzzy semi α-irresolute functions
The concept of fuzzy semi α-irresolute functions was introduced and studied in [7]. In
this section we shall introduce the concept of somewhat fuzzy semi α-irresolute functions
and study their properties.
Definition 3.1 Let (X,T ) and (Y, S) be any two fuzzy topological spaces. A function f
: (X,T ) → (Y, S) is said to be somewhat fuzzy semi α-irresolute if for any non-zero
fuzzy α-open set λ in Y and f −1(λ ) 6= 0 then there exists a fuzzy semi-open set µ 6= 0 in
X such that µ ≤ f −1(λ ).
70 V. Seenivasan, G. Balasubramanian and G. Thangaraj CUBO
10, 1 (2008)
Clearly every fuzzy semi α-irresolute function is somewhat fuzzy semi α-irresolute, but
the converse is not true as the following example shows:-
Example 3.1 Let µ1, µ2 and µ3 be fuzzy sets on I = [0, 1]
µ1(x) =
{
0, 0 ≤ x ≤ 1
2
2x − 1, 1
2
≤ x ≤ 1,
µ2(x) =
1, 0 ≤ x ≤ 1
4
−4x + 2, 1
4
≤ x ≤ 1
2
0,
1
2
≤ x ≤ 1,
µ3(x) =
{
0, 0 ≤ x ≤ 1
4
1
3
(4x − 1), 1
4
≤ x ≤ 1.
Let S1 = {0, µ2, µ3, (µ2 ∨ µ3), (µ2 ∧ µ3), 1} and S2 = { 0, µ2, 1}. Then S1 and S2 are
fuzzy topologies on I. Let f : (I, S1) → (I,S2) be defined by f (x) = x/2 for each x ∈ I. Let
λ be fuzzy set such that 0 < λ < µ2. Then λ is not fuzzy α-open set in (I, S2). Therefore
the only non-zero fuzzy α-open sets in (I, S2) are 1, µ2 and fuzzy sets ρ such that µ2 < ρ <
1. Now f −1(1) = 1; f −1(µ2) = µ
′́
1 and f
−1
(ρ) = 1. The fuzzy semi-open set µ2 in (I, S1)
is contained in f −1(1), f −1(µ2) and f
−1
(ρ). This proves f is somewhat fuzzy semi
α-irresolute function from (I, S1) to (I, S2). It can be easily seen that int µ1
′
= µ2 and
cl µ2 = µ3
′ in (I, S1). Now µ2 is a fuzzy α-open set in (I, S2). Since µ1
′
� cl int µ1
′ in
(I, S1), µ1
′ is not fuzzy semi-open set in (I, S1). Then f
−1
(µ2) = µ1
′, which is not a fuzzy
semi-open set in (I, S1). Hence f is not fuzzy semi α-irresolute function.
Theorem 3.1 Let f : (X,T ) → (Y, S) and g : (Y, S) → (Z,Q) be any two functions. If f
is somewhat fuzzy semi α-irresolute and g is fuzzy α-irresolute, then g ◦ f is somewhat fuzzy
semi α-irresolute.
Proof: Let λ be non-zero fuzzy α-open set in (Z, Q). Since g is fuzzy α-irresolute, g−1(λ)
6= 0 is fuzzy α-open set in (Y , S). Now (g ◦ f )−1(λ ) = f −1(g−1(λ)) 6= 0. Since g−1(λ) is
fuzzy α-open in (Y , S) and f is somewhat fuzzy semi α-irresolute, we conclude that there
exists a fuzzy semi-open set µ 6= 0 in (X, T ) such that µ ≤ f −1( g−1(λ)) = (g ◦ f )−1(λ ).
Hence g ◦ f is somewhat fuzzy semi α-irresolute.
In above Theorem 3.1 if f is either fuzzy α-irresolute or fuzzy semi α-irresolute and
g is somewhat fuzzy semi α-irresolute, then it is not necessarily true that g ◦ f is somewhat
fuzzy semi α-irresolute as the following example shows:-
Example 3.2 Define f : I → I by f (x) = x/2. Let µ1, µ2 and µ3 be fuzzy sets in I
described in Example 3.1. Let T1 = {0, µ1, µ2, µ1∨µ2, 1}; T2 = { 0, µ
′
2, 1} and T3 = {0,
CUBO
10, 1 (2008)
Somewhat Fuzzy Semi α-Irresolute Functions 71
µ3
′
, 1}. Then T1, T2 and T3 are fuzzy topologies on I. Consider the mapping f : (I, T3) →
(I, T1). It can be easily seen that int µ
′
1 = µ
′
3; cl µ
′
3 = 1; cl µ
′
1 = 1 in (I, T3). Since int
µ
′
1 = µ
′
3 and cl µ
′
3= 1, µ
′
1 is fuzzy semi-open set and also fuzzy α-open set in (I, T3). Let λ,
ρ and δ be fuzzy sets such that 0 < λ < µ1, µ1 < ρ < µ2 and µ2 < δ < (µ1 ∨ µ2). Then λ ,
ρ and δ are not fuzzy α-open sets in (I, T1). Therefore the only fuzzy α-open sets in (I, T1)
are 0, 1 , µ1, µ2, µ1∨µ2 and fuzzy sets µ such that (µ1 ∨ µ2) < µ < 1. Now f
−1
(0) = 0; f
−1
(1) = 1; f −1(µ1) = 0; f
−1
(µ2) = µ
′
1 = f
−1
(µ1 ∨ µ2) and f
−1
(µ) = 1 are fuzzy α-open
sets in (I, T3) and also fuzzy semi-open sets in (I,T3). Therefore f is fuzzy α-irresolute
from (I,T3) to (I,T1) and also f is fuzzy semi α-irresolute from (I, T3) to (I, T1). Let
g : (I, T1) → (I, T2) be defined by g(x) = x, for each x ∈ I. Let λ be fuzzy set such that
0 < λ < µ
′
2. Then λ is not fuzzy α-open set in (I, T2). Therefore the only non-zero fuzzy
α-open set are 1, µ
′
2 and fuzzy sets ρ such that µ
′
2 < ρ < 1. Now g
−1
(1) = 1; g−1( µ
′
2)
= µ
′
2 and g
−1
(ρ) = 1. The fuzzy semi-open set µ1 in (I, T1) is contained in g
−1
(1), g−1(
µ
′
2) and g
−1
(ρ). Therefore g is somewhat fuzzy semi α-irresolute function.
Now consider the functions (g ◦ f ) : (I,T3) → (I, T2). Then (g ◦ f )
−1
(µ
′
2) =
f
−1
(g
−1
(µ
′
2)) = f
−1
(µ
′
2) = µ1 and (g ◦ f )
−1
(1) = f
−1
(g
−1
(1)) = f
−1
(1) = 1. But (g ◦
f )
−1
(µ
′
2) = µ1 and there is no non-zero fuzzy semi-open set in (I, T3) such that it is contained
in (g ◦ f )−1(µ
′
2) = µ1. Therefore (g ◦ f ) is not somewhat fuzzy semi α-irresolute
function.
Definition 3.2 [10]: A fuzzy set λ in fuzzy topological space (X, T ) is called fuzzy semi-
dense if there exists no fuzzy semi-closed set µ such that λ < µ < 1.
Theorem 3.2 : Let (X,T ) and (Y, S) be any two fuzzy topological spaces and f : (X,T )
→ (Y, S) be a function. Then the following assertions are equivalent.
(1) f is somewhat fuzzy semi α-irresolute.
(2) If λ is a fuzzy α-closed set in Y such that f −1(λ ) 6= 1, then there exists a fuzzy
semi-closed set µ 6= 1 in X such that µ ≥ f −1(λ ).
(3) If λ is a fuzzy semi-dense set in X, then f (λ) is fuzzy α-dense set in Y .
Proof: (1) ⇒(2): Suppose f is somewhat fuzzy semi α-irresolute and λ is a fuzzy α-closed
set in Y such that f
−1
(λ) 6= 1. Therefore clearly 1 − λ is fuzzy α-open set in Y , and
f
−1
(1 − λ) = 1 − f −1(λ) 6= 0 (since f −1(λ) 6= 1). By (1), there exists a fuzzy semi-open set
η in X such that η ≤ f −1(1 − λ). That is, η ≤ 1 − f −1(λ) which implies that f −1(λ) ≤
1 − η. Clearly 1 − η is fuzzy semi-closed set and taking µ = 1 − η, we have therefore f −1(λ)
≤ µ. Thus we find that (1) ⇒(2) is proved.
72 V. Seenivasan, G. Balasubramanian and G. Thangaraj CUBO
10, 1 (2008)
(2) ⇒ (3): Let λ be a fuzzy semi-dense set in X and suppose f (λ) is not fuzzy α-dense in
Y . Then there exists a fuzzy α-closed set η(say) in Y such that
f (λ) < η < 1. (A)
Since η < 1, f
−1
(η) 6= 1 and so by (2) there exists a fuzzy semi-closed set δ (δ 6= 1) such
that δ ≥ f −1(η) > f −1(f (λ)) ≥ λ ( from (A)). That is, there exists a fuzzy semi-closed set
δ such that δ > λ which is contradiction to the assumption on λ .Therefore (2) ⇒ (3) is
proved.
(3) ⇒(1): Suppose λ is fuzzy α-open set in Y and f −1(λ) 6= 0 and therefore λ 6= 0. Suppose
there exists no fuzzy semi-open set µ in X such that µ ≤ f −1(λ ). Then 1 − f −1(λ) is fuzzy
set in X such that there is no fuzzy semi-closed set δ in X with 1−f −1(λ) < δ < 1(otherwise
1 − f −1(λ) < δ ⇒ 1 − δ ≤ f −1(λ) and 1 − δ is fuzzy semi-open set, which is a contradiction).
This means 1−f −1(λ) is fuzzy semi-dense in X. Then by (3), f (1−f −1(λ)) is fuzzy α-dense
in Y . But f (1 − f −1(λ)) = f ( f −1(1 − λ)) < 1 −λ < 1(since λ 6= 0). This is contradiction
to the fact that f (1 − f −1(λ)) is fuzzy α-dense. Therefore, there exists a fuzzy semi-open
set µ in X such that µ ≤ f −1(λ). Hence f is somewhat fuzzy semi α-irresolute function.
Theorem 3.3 Let (X1, T1), (X2, T2), (Y1, S1) and (Y2, S2) be fuzzy topological spaces such
that X1 is product related to X2 and Y1 is product related to Y2. Let f1 : X1 → Y1 and f2 :
X2 → Y2 be somewhat fuzzy semi α-irresolute functions. Then f1 × f2 : X1 × X2→ Y1 ×
Y2 is somewhat fuzzy semi α-irresolute function.
Proof: Let λ =
∨
i,j
(λ i × µj ) be fuzzy α-open set in Y1 ×Y2 (where λi and
µj are fuzzy α -open sets in Y1 and Y2, respectively). We can assume that
λi’s and µj’s are not all zeros. If any one is zero, that factor can be omitted.
Now (f1 × f2)
−1(λ) = (f1 × f2)
−1(
∨
i,j
(λi×µj)) =
∨
i,j
(f1 × f2)
−1 (λi ×µj ) =
∨
i,j
(f −11 (λi) × f
−1
2 (µj)). Since f1 : X1 → Y1 is somewhat fuzzy semi α-irresolute
and λi is fuzzy α-open set in Y1 and f
−1
1 (λi) 6= 0, there exists a fuzzy semi-open
set δi in X1 such that δi ≤ f
−1
1 (λi). Also since f2 : X2 → Y2 is somewhat fuzzy
semi α-irresolute and µj is fuzzy α-open set in Y2 and f
−1
2 (µj ) 6= 0, there exists
a fuzzy semi-open set ηj in X2 such that ηj ≤ f
−1
2 (µj ). Therefore δi×ηj ≤
f
−1
1 (λi) × f
−1
2 (µj ) = (f1 ×f2)
−1 (λi ×µj). Then by Theorem 4.3 and Theorem
4.6 in [1]
∨
i,j
(δi×ηj) is a fuzzy semi-open set and
∨
i,j
(δi×ηj) ≤
∨
i,j
(f1 × f2)
−1
(λi ×µj ) = (f1 × f2)
−1(
∨
i,j
(λi×µj )) = (f1 × f2)
−1(λ). This proves f1 × f2 is
somewhat fuzzy semi α-irresolute.
CUBO
10, 1 (2008)
Somewhat Fuzzy Semi α-Irresolute Functions 73
The following lemma which is established in [1] is required to prove Theorem 3.4.
Lemma 3.1 [1]: Let g : X → X ×Y be the graph of a function f : X → Y . If λ is a fuzzy
set in X and µ is a fuzzy set in Y , then g−1(λ ×µ) = λ ∧ f −1(µ).
Theorem 3.4 Let f : (X,T ) → (Y, S) be a function from fuzzy topological space (X,T ) to
another fuzzy topological space (Y, S).If the graph g : X → X ×Y of f is somewhat fuzzy
semi α-irresolute, then f is somewhat fuzzy semi α-irresolute.
Proof: Let λ be a non zero fuzzy α-open set in Y . Then, by Lemma 3.1, we have f −1(λ)
= 1 ∧ f −1(λ) = g−1(1 ×λ). Since g is somewhat fuzzy semi α-irresolute and 1 ×λ 6= 0 is a
fuzzy α-open set in X ×Y , there exists a fuzzy semi-open set µ(6= 0) (say) in X such that µ
≤ g−1(1 ×λ) = f −1(λ). This proves that f is somewhat fuzzy semi α-irresolute function.
4 Strongly somewhat fuzzy semi-open functions
The concept of somewhat fuzzy semi-open function was introduced and studied in[10]. In
this section we shall introduce a strongly notion as follows:-
Definition 4.1 : Let (X, T ) and (Y , S) be any two fuzzy topological spaces. A function f
: (X, T ) → (Y , S) is called strongly somewhat fuzzy semi-open if and only if for each
non-zero fuzzy α-open set λ in (X,T ), there exists a fuzzy semi-open set µ in (Y ,S) such
that µ 6= 0 and µ < f (λ ).
Clearly every strongly somewhat fuzzy semi-open function is somewhat fuzzy semi-open
function. However the converse is not true as the following example shows:-
Example 4.1 Let µ1, µ2 and µ3 be fuzzy sets in I described in Example 3.1. Clearly T1
= {0, µ1, 1} and T2 = {0, µ2, 1} are fuzzy topologies on I. Let f : (I, T1) → (I, T2) be
defined by f (x) = min{2x, 1} for each x ∈ I. It can be easily seen that int µ3 = µ1; cl
µ1 = 1 in ( I, T1). Simple computations gives f (0) = 0; f (1) = 1; f (µ1) = 0. Thus f
is somewhat fuzzy semi-open function. Since µ3≤ int cl int µ3 in (I, T1), µ3 is fuzzy
α-open set in (I, T1). But f (µ3)(y) =
{
0, 0 ≤ y ≤ 1
2
,
1
3
,
1
2
≤ y ≤ 1.
Let λ be any non-zero fuzzy set
such that λ ≤ f (µ3) in (I, T2). Then cl int λ = 0, this shows that λ is not fuzzy semi-open
set. Thus there is no non-zero fuzzy semi-open set such that it is contained in f (µ3). Hence
f is not strongly somewhat fuzzy semi-open functions.
74 V. Seenivasan, G. Balasubramanian and G. Thangaraj CUBO
10, 1 (2008)
Theorem 4.1 Suppose (X, T ) and (Y , S) be fuzzy topological spaces. Let f : (X, T ) →
(Y , S) be an onto function. If f is strongly somewhat fuzzy semi-open function and λ is a
fuzzy semi-dense set in Y , then f −1(λ) is fuzzy α-dense in X.
Proof: Suppose λ is a fuzzy semi-dense set in Y . We want to show that f −1(λ) is a fuzzy
α-dense set in X. Suppose not. Then there exists a fuzzy α-closed set µ in X such that
f
−1
(λ) < µ < 1. Then 1 − f −1(λ) > 1 − µ > 0. f (1 − f −1(λ)) > f (1 − µ) which implies
that f ( f
−1
(1 − λ) > f (1 − µ). That is, f (1 − µ) < f (f −1(1 − λ)) = 1 − λ . Now µ is
fuzzy α-closed set ⇒ 1 − µ is fuzzy α-open set in X. Since f is strongly somewhat fuzzy
semi-open, 1 − µ is fuzzy α-open in X ⇒there exists a fuzzy semi-open set δ 6= 0 in Y such
that δ < f (1 − µ). Therefore δ < f (1 − µ) < 1 − λ ⇒ δ < 1 − λ ⇒ λ < 1 − δ. Now 1 − δ
is fuzzy semi-closed set and λ < 1 − δ ⇒ λ is not a fuzzy semi-dense set in Y , which is a
contradiction to our hypothesis. Therefore f
−1
(λ) must be a fuzzy α-dense in (X, T ).
Theorem 4.2 Suppose (X, T ) and (Y , S) be fuzzy topological spaces. Let f : (X, T ) →
(Y , S) be a 1-1 and onto function. Then the following conditions are equivalent.
(1) f is strongly somewhat fuzzy semi-open.
(2) If λ is a fuzzy α-closed set in X such that f (λ) 6= 1, then there exists a fuzzy semi-
closed set µ in Y such that µ 6= 1 and f (λ) < µ .
Proof: (1)⇒(2). Let λ be a fuzzy α-closed set in X such that f (λ) 6= 1. Then 1 − λ is
fuzzy α-open set and since f is 1-1 and onto f (1 − λ) = 1 − f (λ) 6= 0 [3]. As f is strongly
somewhat fuzzy semi-open, there exists a fuzzy semi-open set η in Y such that η 6= 0 and η
< f (1 − λ) = 1 − f (λ). That is f (λ) < 1 − η = µ (say) and µ is fuzzy semi-closed set. This
proves (1)⇒(2).
(2)⇒(1). Let λ be a fuzzy α-open set in X such that λ 6= 0. Then 1 − λ is fuzzy α-closed
set and 1 − λ 6= 1. Now f (1 − λ) = 1 − f (λ) 6= 1 (for, if 1 − f (λ) = 1, then f (λ) = 0 ⇒ λ =
0). Hence by (2) there exists a fuzzy semi-closed set µ in Y such that µ > f (1 − λ). Then
µ > 1 − f (λ ). That is f (λ) > 1 − µ = δ (say). Clearly δ is fuzzy semi-open set in Y such
that δ < f (λ) and δ 6= 0 (since µ 6= 1). This completes the proof of (2)⇒(1).
Received: November 2006. Revised: August 2007.
CUBO
10, 1 (2008)
Somewhat Fuzzy Semi α-Irresolute Functions 75
References
[1] K.K. Azad, On fuzzy semicontinuity, fuzzy almost continuity and fuzzy weakly conti-
nuity, J. Math. Anal. Appl., 82 (1981), 14–32.
[2] A.S. Bin Shahna, On fuzzy strong semicontinuity and fuzzy pre-continuity, Fuzzy
Sets and Systems, 44 (1991), 303–308.
[3] C.L. Chang, Fuzzy topological spaces, J. Math. Anal.Appl., 24 (1968), 182–190.
[4] Karl R. Gentry and Hughes B. Hoyle III, Somewhat continuous functions,
Czech. Math. Journal, 21 (1971), 5–12.
[5] R. Prasad, S.S. Thakur and R.K. Saraf, Fuzzy α-irresolute mappings, J. Fuzzy
Math., 2 (1994), 335–339.
[6] P. Semets, The degree of belief in a fuzzy event, Information Sciences, 25 (1981),
1–19.
[7] V. Seenivasan and G. Balasubramanian, Fuzzy semi α-irresolute functions, Math-
ematica Bohemica, 132 (2007), 113–123.
[8] M. Sugeno An introductory survey of fuzzy control, Information Sciences, 36 (1985),
59–83.
[9] G. Thangaraj and G. Balasubramanian On somewhat fuzzy continuous functions,
J. Fuzzy Math., 11 (2003), 1–12.
[10] G. Thangaraj and G. Balasubramanian On somewhat fuzzy semicontinuous func-
tions, Kybernetika, 37 (2001), 165–170.
[11] Y. Beceren, On semi α-irresolute functions, J. Indian Acad. Math., 22 (2000), 353–
362.
[12] L.A. Zadeh, Fuzzy sets, Information and control, 8 (1965), 338–353.
[13] Zdenek Frolik, Remarks concerning the invariance of Baire spaces under mappings,
Czech. Math. J., 11 (1961), 381–385.
Fuzzy1.pdf