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Faintly b-continuous functions
Arvind Kumar¹, Vinshu², Dr. Bhopal Singh Sharma³
1
Department of Mathematics, University of Delhi, India.
2,3
N.R.E.C. college Khurja (BSR), CCS University, Meerut, India.
Email: [email protected], [email protected], [email protected].
ABSTRACT
In 1961, Levine [8] introduced weakly continuous functions and in 1987, Noiri [14] introduced and studied
weakly 𝛼𝛼-continuous functions. Later on Ekici [5], in 2008, introduced and studied BR-continuous and
hence weakly BR-continuous functions in a similar fashion, by means of b-regular and b-open [3] sets. In
1982 P.E. Long and L.L.Herrington[10] introduced and studied faintly continuous functions. This prompted
us to introduce and study faintly b-continuous and neatly weak b-continuous functions. We studied some
relations between weakly b-continuous functions and faintly b-continuous functions or neatly weak bcontinuous functions.
INTRODUCTION
Levine [8] introduced the concept of a weakly continuous function. In 2008 Ekici [5] has introduced and
studied the class of functions namely BR-continuous functions and weakly BR-continuous functions by
making use of b-regular sets. P.E. Long and L.L.Herrington introduced and studied faintly continuous
functions. In a similar manner here our purpose is to introduce and study generalizations in form of new
classes of functions namely faintly b-continuous and neatly weak b-continuous functions and some relations
between weakly b-continuous functions and faintly b-continuous functions or neatly weak b-continuous
functions. The author [6] has already introduced and studied b-continuous functions. We [20] have already
introduced and studied weakly b-continuous functions.
Let (X, τ) and (Y, 𝜎𝜎) (or X and Y) denote topological spaces. For a subsetA of a space X, the closure A and
the interior of A are denoted by cl(A) and int(A) respectively. A subset A is said to be regular open (resp.
regular closed) if A=int(cl(A)) (resp. A=cl(int(A)). A subset A is said to be preopen [11] (resp. semi open
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[9], b-open [3], 𝛼𝛼-open [12], semi preopen [2] or 𝛽𝛽-open [1]) if A⊂int(cl(A)) (resp. A⊂cl(int(A)),
A⊂int(cl(A))∪cl(int(A)), A⊂int (cl(int(A))), A⊂cl(int(cl(A)))). A subset G of X is called b-neighbourhood
of x∈X if there exists a b-open set B containing x such that B⊂G.
A point x∈X is said to be a 𝛳𝛳-cluster point of A [19] if A∩cl(U)≠𝜙𝜙 for every open set U containing x. The
set of all 𝛳𝛳-cluster points of A is called 𝛳𝛳-closure of A and is denoted by 𝛳𝛳-cl(A). A subset A is called 𝛳𝛳-
closed if 𝛳𝛳-cl(A)=A [19]. The complement of a 𝛳𝛳-closed set is called 𝛳𝛳-open set. The complement of a bopen (resp. preopen, semi open, 𝛼𝛼-open, semi preopen) set is called b-closed (resp. preclosed, semi closed,
𝛼𝛼-closed, semi prelosed). The intersection of all b-closed (resp. preclosed, semi closed, 𝛼𝛼-closed, semi
preclosed) sets of X containing A is called b-closure (resp. preclosure, semi closure, 𝛼𝛼-closure, semi
preclosure) of A and denoted by b-cl(A)(resp. p-cl(A), s-cl(A), 𝛼𝛼-cl(A), sp-cl(A)). The union of all b-open
(resp. preopen, semi open, 𝛼𝛼-open, semi preopen) sets of X contained in A is called b-interior (resp.
preinterior, semi interior, 𝛼𝛼-interior, semi preinterior) of A and denoted by b-int(A) (resp. p-int(A), sint(A), 𝛼𝛼-int(A), sp-int(A)). A subset A is said to be b-regular [3] if it is b-open as well as b-closed.
The family of all b-open (resp. b-regular) sets of X is denoted by BO(X) (resp. BR(X)). A point x∈X is
called b-𝛳𝛳-cluster point [17] of a subset A of X if b-cl(B)∩A≠𝜙𝜙 for every b-open set B containing x. The
set of all b-𝛳𝛳-cluster points of A is called b-𝛳𝛳-closure of A and is denoted by b-𝛳𝛳-cl(A). A subset A of X is
said to be b-𝛳𝛳-closed if A=b-𝛳𝛳-cl(A). The complement of a b-𝛳𝛳-closed set is said to be b-𝛳𝛳-open. A point
x∈X is called b-𝛳𝛳-interior point of A⊂X if there exists a b-regular set U containing x such that U⊂A and is
denoted by x∈b-𝛳𝛳-int(A).
DEFINITIONS AND CHARACTERIZATIONS
Definition 21:- A function f : X → Y is said to be weakly continuous [8] (resp. weakly 𝛼𝛼-continuous
[14]) if for each x∈X and each open set V of Y containing f(x), there exists an open (resp. 𝛼𝛼-open) set U
containing x such that f(U)⊂cl(V).
Definition 2.2:- A function f : X → Y is said to be weakly b-continuous [20] if for each x∈X and each
open set V of Y containing f(x), there exists a b-open set U containing x such that f(U)⊂cl(V).
Definition 2.3:- A function f : X → Y is said to be
(1)(b, s)-open if it maps b-open sets onto semi open sets.
(2) neatly weak b-continuous if for each x in X and each open set V of Y containing f(x), there exists a bopen set U containing x such that int(f(U))⊂cl(V).
Arvind Kumar et al IJSRE Volume 2 Issue 5 May 2014
Theorem 2.4.:- If a function f : X → Y is neatly weak b-continuous and (b, s)-open then f is weakly bcontinuous.
Proof:- Let x be in X and V be any open set in Y containing f(x). Since f is neatly weak
b-
continuous, there exists a b-open set U in X containing x such that int(f(U))⊂cl(V). Since f is (b, s)-open,
therefore f(U) is semi open in Y. Hence f(U)⊂cl(int(f(U))) ⊂cl(V).Thus f is weakly b-continuous.
We can similarly prove
Corollary 2.5:- If f is neatly weak b-continuous carrying b-open sets onto open sets, then also f is weakly bcontinuous.
Definition 2.6:- A function f : X → Y is said to be faintly b-continuous (resp. faintly continuous [10]) if
for each x in X and each 𝛳𝛳-open set V in Y containing f(x) there exists a b-open (resp. open) set U
containing x such that f(U)⊂V.
Theorem 2.7:- Let f : X→Y be a function, then following are equivalent :
(a) f is faintly b-continuous.
(b) f -1(V) is b-open in X for each 𝛳𝛳-open set V in Y.
(c) f -1(V) is b-closed in X for each 𝛳𝛳-closed V in Y.
(d) f(b-cl(A))⊂𝛳𝛳-cl(f(A)) for each subset A of X.
(e) b-cl(f -1(B))⊂f -1(𝛳𝛳-cl(B)) for each subset B of Y.
Proof:- Simple and hence omitted.
Definition 2.8 [6]:- A function f : (X, τ) → (Y, 𝜎𝜎) is said to be b-continuous if and only if inverse image
under f of every open (closed) set is b-open (b-closed).
Theorem 2.9:- For a function f : (X, τ)→(Y, 𝜎𝜎), the following are equivalent :
(a) f is faintly b-continuous.
(b) f : (X, τ)→(Y, 𝜎𝜎 𝛳𝛳 ) is b-continuous where 𝜎𝜎 𝛳𝛳 is the collection of 𝛳𝛳-open sets in space (Y, 𝜎𝜎).
(c) f : (X, τ)→(Y, 𝜎𝜎) is b-continuous if (Y, 𝜎𝜎) is a regular space.
Proof:- It is well known [19] that the collection 𝜎𝜎 𝛳𝛳 of all 𝛳𝛳-open sets in space (Y, 𝜎𝜎) is a topology on Y. So
it is obvious from definitions. Moreover, 𝜎𝜎=𝜎𝜎 𝛳𝛳 if and only if (Y, 𝜎𝜎) is a regular space.
Corollary 2.10:- For a function f : X → (Y, 𝜎𝜎), the following are equivalent provided (Y, 𝜎𝜎) is an almost
regular space.
(a) f is weakly b-continuous.
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(b) f is faintly b-continuous.
If ‘almost regular’ is replaced by ‘regular’ in above corollary then we may add even (c) f is b-continuous
and (d) f is strongly 𝛳𝛳-b-continuous to the list of equivalences.
Proof:- (a) and (b) are obvious for (c) and (d) we have Theorem 3.10 [17] which states that if Y is a regular
space. Then f : X → Y is strongly 𝛳𝛳-b-continuous if and only if f is b-continuous.
The following results about composite of functions are easy to establish.
Theorem 2.11:- For f : X → Y and g : Y → Z, the function gof : X → Z is
(a) b-continuous, whenever f is b-continuous and g is continuous.
(b) b-continuous whenever f is faintly b-continuous and g is strongly 𝛳𝛳-continuous.
(c) strongly 𝛳𝛳-b-continuous whenever f is strongly 𝛳𝛳-b-continuous and g is continuous.
Theorem 2.12:- If f : X → Y is weakly b-continuous and g : Y → Z is continuous, then the composition gof
: X → Z is weakly b-continuous.
Proof:- Let x∈X and A be an open set of Z containing g(f(x)).We have g -1(A) is an open set of Y
containing f(x). Then there exists a b-open set B containing x such that f(B)⊂cl(g-1(A)). Since g is
continuous, so, (gof)(B)⊂g(cl(g-1(A)))⊂cl(A). Thus gof is weakly b-continuous.
We recall that a space X is said to be submaximal [18] if each dense subset of X is open in X. It is further
shown [18] that a space is submaximal if and only if every preopen subset of X is open. A space X is said to
be extremally disconnected [4] if the closure of each open set of X is open. We note [17] that an extermally
disconnected space is exactly the space where every semi open set is 𝛼𝛼-open.
Definition 2.13:- A function f : X → Y is said to be strongly 𝛳𝛳-continuous [13] (resp. strongly 𝛳𝛳-semi
continuous [7], strongly 𝛳𝛳-precontinuous [15], strongly 𝛳𝛳-𝛽𝛽-continuous [16]) if for each x in X and each
open set V of Y containing f(x), there exists an open (resp. semi open, preopen, semi preopen) set U of X
containing x such that f(cl(U))⊂V (resp. f(s-cl(U))⊂V, f(p-cl(U))⊂V, f(sp-cl(U))⊂V).
Theorem 2.14:- The following are equivalent for a function f : X → Y where X is submaximal extremally
disconnected space and Y is regular space.
(a) f is weakly b-continuous.
(b) f is faintly b-continuous.
(c) f is b-continuous.
(d) f is strongly 𝛳𝛳-b-continuous.
(e) f is strongly 𝛳𝛳-continuous.
(f) f is strongly 𝛳𝛳-semi continuous.
Arvind Kumar et al IJSRE Volume 2 Issue 5 May 2014
(g) f is strongly 𝛳𝛳-precontinuous.
(h) f is strongly 𝛳𝛳-semi precontinuous or strongly 𝛳𝛳-𝛽𝛽-continuous.
Proof:- The proof is obvious using the fact that if X is submaximal extremally disconnected space, then
open set, preopen set, semi open set, b-open set and semi preopen (or 𝛽𝛽-open) sets are equivalent concepts
[17].
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