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```Bull. Korean Math. Soc. 39 (2002), No. 3, pp. 441–453
FUZZY r-REGULAR OPEN SETS AND
FUZZY ALMOST r-CONTINUOUS MAPS
Seok Jong Lee and Eun Pyo Lee
Abstract. We introduce the concepts of fuzzy r-regular open sets
and fuzzy almost r-continuous maps in the fuzzy topology of Chattopadhyay. Also we investigate the equivalent conditions of the
fuzzy almost r-continuity.
1. Introduction
Chang  introduced fuzzy topological spaces and other authors continued the investigation of such spaces. Azad  introduced the concepts
of fuzzy regular open set and fuzzy almost continuous maps in Chang’s
fuzzy topology. Chattopadhyay et al.  introduced another definition
of fuzzy topology as a generalization of Chang’s fuzzy topology. By
generalizing the definitions of Azad, we introduce the concepts of fuzzy
r-regular open sets and fuzzy almost r-continuous maps in the fuzzy
topology of Chattopadhyay. Then the concepts introduced by Azad become special cases of our definition. Also we investigate the equivalent
conditions of the fuzzy almost r-continuity.
2. Preliminaries
In this paper, we denote by I the unit interval [0, 1] of the real line and
I0 = (0, 1]. A member µ of I X is called a fuzzy set in X. For any µ ∈ I X ,
µc denotes the complement 1 − µ. By 0̃ and 1̃ we denote constant maps
on X with value 0 and 1, respectively. All other notations are standard
notations of fuzzy set theory.
Received July 24, 2001.
2000 Mathematics Subject Classification: 54A40.
Key words and phrases: fuzzy r-regular open, fuzzy almost r-continuous.
This work was supported partly by the Basic Science Research Institute (BSRI01-4) of Chungbuk National University.
442
Seok Jong Lee and Eun Pyo Lee
A Chang’s fuzzy topology on X is a family T of fuzzy sets in X which
satisfies the following three properties:
(1) 0̃, 1̃ ∈ T .
(2) If µ1 , µ2 ∈ T then µ1 ∧ µ2W∈ T .
(3) If µi ∈ T for each i, then µi ∈ T .
The pair (X, T ) is called a Chang’s fuzzy topological space.
A fuzzy topology on X is a map T : I X → I which satisfies the
following properties:
(1) T (0̃) = T (1̃) = 1,
(2) T (µ
≥ T (µ1 ) ∧ T (µ2 ),
W1 ∧ µ2 ) V
(3) T ( µi ) ≥ T (µi ).
The pair (X, T ) is called a fuzzy topological space.
For each α ∈ (0, 1], a fuzzy point xα in X is a fuzzy set in X defined
by
½
xα (y) =
α
if y = x,
0
if y 6= x.
In this case, x and α are called the support and the value of xα , respectively. A fuzzy point xα is said to belong to a fuzzy set µ in X,
denoted by xα ∈ µ, if α ≤ µ(x). A fuzzy point xα in X is said to be
quasi-coincident with µ, denoted by xα qµ, if α + µ(x) > 1. A fuzzy set
ρ in X is said to be quasi-coincident with a fuzzy set µ in X, denoted
by ρqµ, if there is an x ∈ X such that ρ(x) + µ(x) > 1.
Definition 2.1. () Let µ be a fuzzy set in a fuzzy topological
space (X, T ) and r ∈ I0 . Then µ is called
(1) a fuzzy r-open set in X if T (µ) ≥ r,
(2) a fuzzy r-closed set in X if T (µc ) ≥ r.
Definition 2.2. () Let (X, T ) be a fuzzy topological space. For
each r ∈ I0 and for each µ ∈ I X , the fuzzy r-closure is defined by
V
cl(µ, r) = {ρ ∈ I X : µ ≤ ρ, T (ρc ) ≥ r}.
Definition 2.3. () Let (X, T ) be a fuzzy topological space. For
each r ∈ I0 and for each µ ∈ I X , the fuzzy r-interior is defined by
W
int(µ, r) = {ρ ∈ I X : µ ≥ ρ, T (ρ) ≥ r}.
Fuzzy r-regular open sets and fuzzy almost r-continuous maps
443
Theorem 2.4. () For a fuzzy set µ in a fuzzy topological space
(X, T ) and r ∈ I0 , we have:
(1) int(µ, r)c = cl(µc , r).
(2) cl(µ, r)c = int(µc , r).
Definition 2.5. () Let µ be a fuzzy set in a fuzzy topological
space (X, T ) and r ∈ I0 . Then µ is said to be
(1) fuzzy r-semiopen if there is a fuzzy r-open set ρ in X such that
ρ ≤ µ ≤ cl(ρ, r),
(2) fuzzy r-semiclosed if there is a fuzzy r-closed set ρ in X such
that int(ρ, r) ≤ µ ≤ ρ.
Definition 2.6. () Let xα be a fuzzy point in a fuzzy topological
space (X, T ) and r ∈ I0 . Then a fuzzy set µ in X is called
(1) a fuzzy r-neighborhood of xα if there is a fuzzy r-open set ρ in X
such that xα ∈ ρ ≤ µ,
(2) a fuzzy r-quasi-neighborhood of xα if there is a fuzzy r-open set
ρ in X such that xα qρ ≤ µ.
Definition 2.7. () Let f : (X, T ) → (Y, U) be a map from a fuzzy
topological space X to another fuzzy topological space Y and r ∈ I0 .
Then f is called
(1) a fuzzy r-continuous map if f −1 (µ) is a fuzzy r-open set of X for
each fuzzy r-open set µ in Y ,
(2) a fuzzy r-semicontinuous map if f −1 (µ) is a fuzzy r-semiopen
set of X for each fuzzy r-open set µ in Y ,
(3) a fuzzy r-irresolute map if f −1 (µ) is a fuzzy r-semiopen set of X
for each fuzzy r-semiopen set µ in Y .
All the other nonstandard definitions and notations can be found in
 and .
3. Fuzzy r-regular open sets
We define the notions of fuzzy r-regular open sets and fuzzy r-regular
closed sets, and investigate some of their properties.
Definition 3.1. Let µ be a fuzzy set in a fuzzy topological space
(X, T ) and r ∈ I0 . Then µ is said to be
(1) fuzzy r-regular open if int(cl(µ, r), r) = µ,
(2) fuzzy r-regular closed if cl(int(µ, r), r) = µ.
444
Seok Jong Lee and Eun Pyo Lee
Theorem 3.2. Let µ be a fuzzy set in a fuzzy topological space
(X, T ) and r ∈ I0 . Then µ is fuzzy r-regular open if and only if µc is
fuzzy r-regular closed.
Proof. It follows from Theorem 2.4.
¤
Remark 3.3. Clearly, every fuzzy r-regular open (r-regular closed)
set is fuzzy r-open (r-closed). That the converse need not be true is
shown by the following example. The example also shows that the union
(intersection) of any two fuzzy r-regular open (r-regular closed) sets need
not be fuzzy r-regular open (r-regular closed).
Example 3.4. Let X = I and µ1 , µ2 and µ3 be fuzzy sets in X
defined by
½
0
if 0 ≤ x ≤ 21 ,
µ1 (x) =
2x − 1 if 12 ≤ x ≤ 1;

if 0 ≤ x ≤ 14 ,

1
−4x + 2 if 14 ≤ x ≤ 21 ,
µ2 (x) =


0
if 12 ≤ x ≤ 1;
and
½
µ3 (x) =
if 0 ≤ x ≤ 14 ,
0
1
3 (4x
− 1)
if
1
4
≤ x ≤ 1.
Define T : I X → I by


1
T (µ) =


1
2
if µ = 0̃, 1̃,
if µ = µ1 , µ2 , µ1 ∨ µ2 ,
0
otherwise.
Then clearly T is a fuzzy topology on X.
(1) Clearly, µ1 ∨ µ2 is fuzzy 21 -open. Since int(cl(µ1 ∨ µ2 , 12 ), 12 ) = 1̃ 6=
µ1 ∨ µ2 , µ1 ∨ µ2 is not a fuzzy 21 -regular open set.
(2) Since int(cl(µ1 , 12 ), 12 ) = int(µc2 , 12 ) = µ1 and int(cl(µ2 , 21 ), 12 ) =
int(µc1 , 21 ) = µ2 , µ1 and µ2 are fuzzy 12 -regular open sets. But µ1 ∨ µ2 is
not a fuzzy 21 -regular open set.
(3) In view of Theorem 3.2, µc1 and µc2 are fuzzy 21 -regular closed sets
but µc1 ∧ µc2 = (µ1 ∨ µ2 )c is not a fuzzy 12 -regular closed set.
Fuzzy r-regular open sets and fuzzy almost r-continuous maps
445
Theorem 3.5. (1) The intersection of two fuzzy r-regular open sets
is fuzzy r-regular open.
(2) The union of two fuzzy r-regular closed sets is fuzzy r-regular
closed.
Proof. (1) Let µ and ρ be any two fuzzy r-regular open sets in a
fuzzy topological space X. Then µ and ρ are fuzzy r-open sets and
hence T (µ ∧ ρ) ≥ T (µ) ∧ T (ρ) ≥ r. Thus µ ∧ ρ is a fuzzy r-open set.
Since µ ∧ ρ ≤ cl(µ ∧ ρ, r),
int(cl(µ ∧ ρ, r), r) ≥ int(µ ∧ ρ, r) = µ ∧ ρ.
Now, µ ∧ ρ ≤ µ and µ ∧ ρ ≤ ρ implies
int(cl(µ ∧ ρ, r), r) ≤ int(cl(µ, r), r) = µ
and
int(cl(µ ∧ ρ, r), r) ≤ int(cl(ρ, r), r) = ρ.
Hence int(cl(µ ∧ ρ, r), r) ≤ µ ∧ ρ. Therefore µ ∧ ρ is fuzzy r-regular open.
(2) It follows from (1) and Theorem 3.2.
¤
Theorem 3.6. (1) The fuzzy r-closure of a fuzzy r-open set is fuzzy
r-regular closed.
(2) The fuzzy r-interior of a fuzzy r-closed set is fuzzy r-regular open.
Proof. (1) Let µ be a fuzzy r-open set in a fuzzy topological space
X. Then clearly int(cl(µ, r), r) ≤ cl(µ, r) implies that
cl(int(cl(µ, r), r), r) ≤ cl(cl(µ, r), r) = cl(µ, r).
Since µ is fuzzy r-open, µ = int(µ, r). Also since µ ≤ cl(µ, r), µ =
int(µ, r) ≤ int(cl(µ, r), r). Thus cl(µ, r) ≤ cl(int(cl(µ, r), r), r). Hence
cl(µ, r) is a fuzzy r-regular closed set.
(2) Similar to (1).
¤
Let (X, T ) be a fuzzy topological space. For an r-cut Tr = {µ ∈ I X |
T (µ) ≥ r}, it is obvious that (X, Tr ) is a Chang’s fuzzy topological space
for all r ∈ I0 .
Let (X, T ) be a Chang’s fuzzy topological space and r ∈ I0 . Recall
 that a fuzzy topology T r : I X → I is defined by


 1 if µ = 0̃, 1̃,
r
T (µ) =
r if µ ∈ T − {0̃, 1̃},


0 otherwise.
446
Seok Jong Lee and Eun Pyo Lee
Theorem 3.7. Let µ be a fuzzy set in a fuzzy topological space
(X, T ) and r ∈ I0 . Then µ is fuzzy r-regular open (r-regular closed) in
(X, T ) if and only if µ is fuzzy regular open (regular closed) in (X, Tr ).
Proof. Straightforward.
¤
Theorem 3.8. Let µ be a fuzzy set of a Chang’s fuzzy topological
space (X, T ) and r ∈ I0 . Then µ is a fuzzy regular open (regular closed)
in (X, T ) if and only if µ is fuzzy r-regular open (r-regular closed) in
(X, T r ).
Proof. Straightforward.
¤
4. Fuzzy almost r-continuous maps
We are going to introduce the notions of fuzzy almost r-continuous
maps and investigate some of their properties. Also, we describe the relations among fuzzy almost r-continuous maps, fuzzy r-continuous maps
and fuzzy r-semicontinuous maps.
Definition 4.1. Let f : (X, T ) → (Y, U ) be a map from a fuzzy
topological space X to another fuzzy topological space Y and r ∈ I0 .
Then f is called
(1) a fuzzy almost r-continuous map if f −1 (µ) is a fuzzy r-open set
of X for each fuzzy r-regular open set µ in Y , or equivalently,
f −1 (µ) is a fuzzy r-closed set in X for each fuzzy r-regular closed
set µ in Y ,
(2) a fuzzy almost r-open map if f (ρ) is a fuzzy r-open set in Y for
each fuzzy r-regular open set ρ in X,
(3) a fuzzy almost r-closed map if f (ρ) is a fuzzy r-closed set in Y
for each fuzzy r-regular closed set ρ in X.
Theorem 4.2. Let f : (X, T ) → (Y, U) be a map and r ∈ I0 . Then
the following statements are equivalent:
(1) f is a fuzzy almost r-continuous map.
(2) f −1 (µ) ≤ int(f −1 (int(cl(µ, r), r)), r) for each fuzzy r-open set µ
in Y .
(3) cl(f −1 (cl(int(µ, r), r)), r) ≤ f −1 (µ) for each fuzzy r-closed set µ
in Y .
Fuzzy r-regular open sets and fuzzy almost r-continuous maps
447
Proof. (1) ⇒ (2) Let f be fuzzy almost r-continuous and µ any fuzzy
r-open set in Y . Then
µ = int(µ, r) ≤ int(cl(µ, r), r).
By Theorem 3.6(2), int(cl(µ, r), r) is a fuzzy r-regular open set in Y .
Since f is fuzzy almost r-continuous, f −1 (int(cl(µ, r), r)) is a fuzzy ropen set in X. Hence
f −1 (µ) ≤ f −1 (int(cl(µ, r), r)) = int(f −1 (int(cl(µ, r), r)), r).
(2) ⇒ (3) Let µ be a fuzzy r-closed set of Y . Then µc is a fuzzy
r-open set in Y . By (2),
f −1 (µc ) ≤ int(f −1 (int(cl(µc , r), r)), r).
Hence
f −1 (µ) = f −1 (µc )c ≥ int(f −1 (int(cl(µc , r), r)), r)c
= cl(f −1 (cl(int(µ, r), r)), r).
(3) ⇒ (1) Let µ be a fuzzy r-regular closed set in Y . Then µ is a
fuzzy r-closed set in Y and hence
f −1 (µ) ≤ int(f −1 (int(cl(µ, r), r)), r) = int(f −1 (µ), r).
Thus f −1 (µ) = cl(f −1 (µ), r) and hence f −1 (µ) is a fuzzy r-closed set in
X. Therefore, f is a fuzzy almost r-continuous map.
¤
Theorem 4.3. Let f : (X, T ) → (Y, U) be a map and r ∈ I0 . Then
f is fuzzy almost r-open if and only if f (int(ρ, r)) ≤ int(f (ρ), r) for each
fuzzy r-semiclosed set ρ in X.
Proof. Let f be fuzzy almost r-open and ρ a fuzzy r-semiclosed set
in X. Then int(ρ, r) ≤ int(cl(ρ, r), r) ≤ ρ. Note that cl(ρ, r) is a fuzzy
r-closed set of X. By Theorem 3.6(2), int(cl(ρ, r), r) is a fuzzy r-regular
open set in X. Since f is fuzzy almost r-open, f (int(cl(ρ, r), r)) is a
fuzzy r-open set in X. Thus we have
f (int(ρ, r)) ≤ f (int(cl(ρ, r), r)) = int(f (int(cl(ρ, r), r)), r) ≤ int(f (ρ), r).
448
Seok Jong Lee and Eun Pyo Lee
Conversely, let ρ be a fuzzy r-regular open set of X. Then ρ is fuzzy
r-open and hence int(ρ, r) = ρ. Since int(cl(ρ, r), r) = ρ, ρ is fuzzy
r-semiclosed. So
f (ρ) = f (int(ρ, r)) ≤ int(f (ρ), r) ≤ f (ρ).
Thus f (ρ) = int(f (ρ), r) and hence f (ρ) is a fuzzy r-open set in Y .
¤
The global property of fuzzy almost r-continuity can be rephrased
to the local property in terms of neighborhood and quasi-neighborhood,
respectively, in the following two theorems.
Theorem 4.4. Let f : (X, T ) → (Y, U) be a map and r ∈ I0 . Then
f is fuzzy almost r-continuous if and only if for every fuzzy point xα
in X and every fuzzy r-neighborhood µ of f (xα ), there is a fuzzy rneighborhood ρ of xα such that xα ∈ ρ and f (ρ) ≤ int(cl(µ, r), r).
Proof. Let xα be a fuzzy point in X and µ a fuzzy r-neighborhood of
f (xα ). Then there is a fuzzy r-open set λ in Y such that f (xα ) ∈ λ ≤ µ.
So xα ∈ f −1 (λ) ≤ f −1 (µ). Since f is fuzzy almost r-continuous,
f −1 (λ) ≤ int(f −1 (int(cl(λ, r)), r) ≤ int(f −1 (int(cl(µ, r)), r).
Put ρ = f −1 (int(cl(µ, r), r)). Then xα ∈ f −1 (λ) ≤ int(ρ, r) ≤ ρ. By
Theorem 3.6(2), int(cl(µ, r), r) is fuzzy r-regular open. Since f is fuzzy
almost r-continuous, ρ = f −1 (int(cl(µ, r), r)) is fuzzy r-open. Thus ρ is
a fuzzy r-neighborhood of xα and
f (ρ) = f f −1 (int(cl(µ, r), r)) ≤ int(cl(µ, r), r).
Conversely, let µ be a fuzzy r-regular open set in Y and xα ∈ f −1 (µ).
Then µ is fuzzy r-open and hence µ is a fuzzy r-neighborhood of f (xα ).
By hypothesis, there is a fuzzy r-neighborhood ρxα of xα such that
xα ∈ ρxα and f (ρxα ) ≤ int(cl(µ, r), r) = µ. Since ρxα is a fuzzy rneighborhood of xα , there is a fuzzy r-open set λxα in X such that
xα ∈ λxα ≤ ρxα ≤ f −1 f (ρxα ) ≤ f −1 (µ).
So we have
f −1 (µ) =
≤
W
W
{xα : xα ∈ f −1 (µ)}
{λxα : xα ∈ f −1 (µ)}
≤ f −1 (µ).
W
Thus f −1 (µ) = {λxα : xα ∈ f −1 (µ)} is fuzzy r-open in X and hence
f is almost r-continuous.
¤
Fuzzy r-regular open sets and fuzzy almost r-continuous maps
449
Theorem 4.5. Let f : (X, T ) → (Y, U) be a map and r ∈ I0 . Then f
is a fuzzy almost r-continuous map if and only if for every fuzzy point xα
in X and every fuzzy r-quasi-neighborhood µ of f (xα ), there is a fuzzy
r-quasi-neighborhood ρ of xα such that xα qρ and f (ρ) ≤ int(cl(µ, r), r).
Proof. Let xα be a fuzzy point in X and µ a fuzzy r-quasi-neighborhood of f (xα ). Then there is a fuzzy r-open set λ in Y such that
f (xα )qλ ≤ µ. So xα qf −1 (λ). Since f is fuzzy almost r-continuous,
f −1 (λ) ≤ int(f −1 (int(cl(λ, r)), r) ≤ int(f −1 (int(cl(µ, r)), r).
Put ρ = f −1 (int(cl(µ, r), r)). Then xα qf −1 (λ) ≤ int(ρ, r) ≤ ρ. So xα qρ.
Since int(cl(µ, r), r) is fuzzy r-regular open and f is fuzzy almost rcontinuous, ρ = f −1 (int(cl(µ, r), r)) is fuzzy r-open. Thus ρ is a fuzzy
r-quasi-neighborhood of xα and
f (ρ) = f f −1 (int(cl(µ, r), r)) ≤ int(cl(µ, r), r).
Conversely, let µ be a fuzzy r-regular open set in Y . If f −1 (µ) = 0̃,
then it is obvious. Suppose xα is a fuzzy point in f −1 (µ) such that
α < f −1 (µ)(x). Then α < µ(f (x)) and hence f (x)1−α qµ. So µ is a
fuzzy r-quasi-neighborhood of f (x)1−α = f (x1−α ). By hypothesis, there
is a fuzzy r-quasi-neighborhood ρxα of x1−α such that x1−α qρxα and
f (ρxα ) ≤ int(cl(µ, r), r) = µ. Since ρxα is a fuzzy r-quasi-neighborhood
of x1−α , there is a fuzzy r-open set λxα in X such that
x1−α qλxα ≤ ρxα ≤ f −1 f (ρxα ) ≤ f −1 (µ).
Then α < λxα (x) and hence xα ∈ λxα . So
W
f −1 (µ) = {xα : xα is a fuzzy point in f −1 (µ) such that
≤
W
α < f −1 (µ)(x)}
{λxα : xα is a fuzzy point in f −1 (µ) such that
α < f −1 (µ)(x)}
≤ f −1 (µ)
and hence
f −1 (µ) =
W
{λxα : xα is a fuzzy point in f −1 (µ) such that
α < f −1 (µ)(x)}.
Thus f −1 (µ) is fuzzy r-open in X. Therefore f is fuzzy almost rcontinuous.
¤
450
Seok Jong Lee and Eun Pyo Lee
Theorem 4.6. Let f : (X, T ) → (Y, U ) be fuzzy r-semicontinuous
and fuzzy almost r-open. Then f is fuzzy r-irresolute.
Proof. Let µ be fuzzy r-semiclosed in Y . Then int(cl(µ, r), r) ≤ µ.
Since f is fuzzy r-semicontinuous,
int(cl(f −1 (µ), r), r) ≤ f −1 (cl(µ, r)).
Thus we have
int(cl(f −1 (µ), r), r) = int(int(cl(f −1 (µ), r), r), r) ≤ int(f −1 (cl(µ, r)), r).
Since f is fuzzy r-semicontinuous and cl(µ, r) is fuzzy r-closed, f −1
(cl(µ, r)) is a fuzzy r-semiclosed set in X. Since f is fuzzy almost ropen,
f (int(f −1 (cl(µ, r)), r) ≤ int(f f −1 (cl(µ, r)), r) ≤ int(cl(µ, r), r) ≤ µ.
Hence we have
int(cl(f −1 (µ), r), r) ≤ f −1 f (int(cl(f −1 (µ), r), r)
≤ f −1 f (int(f −1 (cl(µ, r)), r))
≤ f −1 (µ).
Thus f −1 (µ) is fuzzy r-semiclosed in X and hence f is fuzzy r-irresolute.
¤
Remark 4.7. Clearly a fuzzy r-continuous map is a fuzzy almost
r-continuous map. That the converse need not be true is shown by
the following example. Also, the example shows that a fuzzy almost
r-continuous map need not be a fuzzy r-semicontinuous map.
Example 4.8. Let X = I and µ1 , µ2 and µ3 be fuzzy sets in X
defined by
µ1 (x) = x;
µ2 (x) = 1 − x;
and
½
µ3 (x) =
x if 0 ≤ x ≤ 12 ,
0
if
1
2
≤ x ≤ 1.
Fuzzy r-regular open sets and fuzzy almost r-continuous maps
Define T1 : I X → I and T2


1
1
T1 (µ) =
2


0
and
T2 (µ) =
: I X → I by
if µ = 0̃, 1̃
if µ = µ1 , µ2 , µ1 ∨ µ2 , µ1 ∧ µ2
otherwise,


1
if µ = 0̃, 1̃


if µ = µ1 , µ2 , µ3 , µ1 ∨ µ2 , µ1 ∧ µ2
otherwise.
1
2
0
451
Then clearly T1 , T2 are fuzzy topologies on X. Consider the identity map
1X : (X, T1 ) → (X, T2 ). It is clear that µ1 , µ2 , µ1 ∨ µ2 and µ1 ∧ µ2 are
fuzzy 12 -regular open in (X, T2 ) while µ3 is not. Noting that T1 (µ3 ) = 0,
it is obvious that 1X is a fuzzy 21 -almost continuous map which is not
a fuzzy 12 -continuous map. Also, because 0̃ is the only fuzzy 12 -open set
1
contained in µ3 , µ3 = 1−1
X (µ3 ) is not a fuzzy 2 -semiopen set in (X, T1 )
1
and hence 1X is not a fuzzy 2 -semicontinuous map.
Example 4.9. A fuzzy r-semicontinuous map need not be a fuzzy
almost r-continuous map.
Let (X, T ) be a fuzzy topological space as described in Example 3.4
and let f : (X, T ) → (X, T ) be defined by f (x) = x2 . Simple computations give f −1 (0̃) = 0̃, f −1 (1̃) = 1̃, f −1 (µ1 ) = 0̃ and f −1 (µ2 ) = µc1 =
f −1 (µ1 ∨ µ2 ). Since cl(µ2 , 21 ) = µc1 , µc1 is a fuzzy 12 -semiopen set and
hence f is a fuzzy 21 -semicontinuous map. But f −1 (µ2 ) = µc1 and
¡
¡ ¡ ¡
¡
¡ ¡
1 ¢ 1 ¢¢ 1 ¢
1 ¢¢ 1 ¢
int f −1 int cl µ2 , ,
,
= int f −1 int µc1 ,
,
2 2
2
2
2
¡ −1 ¡ ¢ 1 ¢
= int f
µ2 ,
2
¡ c 1¢
= int µ1 ,
= µ2 .
2
Thus f −1 (µ2 ) 6≤ int(f −1 (int(cl(µ2 , 12 ), 12 )), 12 ) and hence f is not a fuzzy
almost 12 -continuous map.
From Example 4.8 and 4.9 we have the following result.
Theorem 4.10. Fuzzy r-semicontinuity and fuzzy almost r-continuity are independent notions.
452
Seok Jong Lee and Eun Pyo Lee
Definition 4.11. Let (X, T ) be a fuzzy topological space and r ∈ I0 .
Then (X, T ) is called a fuzzy r-semiregular space if each fuzzy r-open
set in X is a union of fuzzy r-regular open sets.
Theorem 4.12. Let r ∈ I0 and f : (X, T ) → (Y, U) be a map from
a fuzzy topological space X to a fuzzy r-semiregular space Y . Then f
is fuzzy almost r-continuous if and only if f is fuzzy r-continuous.
Proof. Due to Remark 4.7, it suffices to show that if f is fuzzy almost
r-continuous then it is fuzzy r-continuous. LetWµ be a fuzzy r-open set in
Y . Since (Y, U) is a r-semiregular space, µ = µi , where µi ’s are fuzzy
r-regular open sets in Y . Then since f is a fuzzy almost r-continuous
map, f −1 (µi ) is a fuzzy r-open set for each i. So
W
W
V
T (f −1 (µ)) = T (f −1 ( µi )) = T ( f −1 (µi )) ≥ T (f −1 (µi )) ≥ r.
Thus f −1 (µ) is fuzzy r-open in X and hence f is a fuzzy r-continuous
map.
¤
Theorem 4.13. Let f : (X, T ) → (Y, U) be a map from a fuzzy
topological space X to another fuzzy topological space Y and r ∈ I0 .
Then f is fuzzy almost r-continuous (r-open, r-closed) if and only if
f : (X, Tr ) → (Y, Ur ) is fuzzy almost continuous (open, closed).
Proof. Straightforward.
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Theorem 4.14. Let f : (X, T ) → (Y, U ) be a map from a Chang’s
fuzzy topological space X to another Chang’s fuzzy topological space Y
and r ∈ I0 . Then f is fuzzy almost continuous (open, closed) if and only
if f : (X, T r ) → (Y, U r ) is fuzzy almost r-continuous (r-open, r-closed).
Proof. Straightforward.
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References
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continuity, J. Math. Anal. Appl. 82 (1981), 14–32.
 C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), 182–190.
 K. C. Chattopadhyay and S. K. Samanta, Fuzzy topology: Fuzzy closure operator,
fuzzy compactness and fuzzy connectedness, Fuzzy Sets and Systems 54 (1993),
207–212.
Fuzzy r-regular open sets and fuzzy almost r-continuous maps
453
 K. C. Chattopadhyay, R. N. Hazra and S. K. Samanta, Gradation of openness:
fuzzy topology, Fuzzy Sets and Systems 49 (1992), 237–242.
 Seok Jong Lee and Eun Pyo Lee, Fuzzy r-preopen sets and fuzzy r-precontinuous
maps, Bull. Korean Math. Soc. 36 (1999), 91–108.

, Fuzzy r-continuous and fuzzy r-semicontinuous maps, Int. J. Math. and
Math. Sci. 27 (2001), 53–63.
Seok Jong Lee, Department of Mathematics, Chungbuk National University, Cheongju 361-763, Korea
E-mail: [email protected]
Eun Pyo Lee, Department of Mathematics, Seonam University, Namwon
590-711, Korea
E-mail: [email protected]
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