On weakly (μ, λ)-open functions
We study some characterizations and properties of almost (μ, λ)-open functions. Some conditions are presented under which an almost (μ, λ)-open function is equivalent to a (μ, λ)-open function.
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nasplib_isofts_kiev_ua-123456789-1663082025-02-09T13:56:33Z On weakly (μ, λ)-open functions Про слабко (μ, λ)-відкриті функції Roy, B. Короткі повідомлення We study some characterizations and properties of almost (μ, λ)-open functions. Some conditions are presented under which an almost (μ, λ)-open function is equivalent to a (μ, λ)-open function. Вивчаються дєякі характеристики та властивості майже (μ, λ)-відкритих Функцій. Наведено дєякі умови, за яких майже (μ, λ)-відкрита функція еквівалентна (μ, λ)-відкритій функції. This work is supported by UGC, New Delhi 2014 Article On weakly (μ, λ)-open functions / B. Roy // Український математичний журнал. — 2014. — Т. 66, № 10. — С. 1425–1430. — Бібліогр.: 19 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/166308 517.93 en Український математичний журнал application/pdf Інститут математики НАН України |
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Короткі повідомлення Короткі повідомлення Roy, B. On weakly (μ, λ)-open functions Український математичний журнал |
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We study some characterizations and properties of almost (μ, λ)-open functions. Some conditions are presented under which an almost (μ, λ)-open function is equivalent to a (μ, λ)-open function. |
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On weakly (μ, λ)-open functions |
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On weakly (μ, λ)-open functions |
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On weakly (μ, λ)-open functions |
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on weakly (μ, λ)-open functions |
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On weakly (μ, λ)-open functions / B. Roy // Український математичний журнал. — 2014. — Т. 66, № 10. — С. 1425–1430. — Бібліогр.: 19 назв. — англ. |
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AT royb onweaklymlopenfunctions AT royb proslabkomlvídkritífunkcíí |
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2025-11-26T13:27:18Z |
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UDC 517.93
B. Roy (Women’s Christian College, Kolkata, India)
ON WEAKLY (µ, λ)-OPEN FUNCTIONS*
ПРО СЛАБКО (µ, λ)-ВIДКРИТI ФУНКЦIЇ
In the paper, some characterizations and properties of almost (µ, λ)-open functions are investigated. Some conditions are
given under which an almost (µ, λ)-open function is equivalent to a (µ, λ)-open function.
Вивчаються деякi характеристики та властивостi майже (µ, λ)-вiдкритих функцiй. Наведено деякi умови, за яких
майже (µ, λ)-вiдкрита функцiя еквiвалентна (µ, λ)-вiдкритiй функцiї.
1. Introduction. During the last few years, different forms of open sets have been studied. A
significant contribution to the theory of generalized open sets was extended by A. Császár [4, 5, 7, 8].
In [3], he introduced the concept of generalized neighbourhood systems and generalized topological
spaces. He also introduced the concept of continuous functions and associated interior and closure
operators on generalized topological spaces and had shown that the fundamental definitions and major
part of many statements and constructions in set topology can be formulated by replacing topology
with the help of generalized topology. The notion of (µ, λ)-open function was studied by Ekici [11].
Al-Omari and Noiri [1] introduced the definition of almost (µ, λ)-open functions. We investigate
several characterizations of these functions. An endeavour has been made to obtain several conditions
for an almost (µ, λ)-open function to be a (µ, λ)-open function.
We recall some notions defined in [3]. Let X be a nonempty set and expX be the power set
of X. We call a class µ j expX a generalized topology [3] (briefly, GT) if ∅ ∈ µ and union of
elements of µ belongs to µ. A set X, with a GT µ on it is said to be a generalized topological space
(briefly, GTS) and is denoted by (X,µ). A GT µ on X is said to be strong [8] if X ∈ µ.
For a GTS (X,µ), the elements of µ are called µ-open sets and the complement of µ-open sets are
called µ-closed sets. For A j X, we denote by cµ(A) the intersection of all µ-closed sets containing
A, i.e., the smallest µ-closed set containing A; and by iµ(A) the union of all µ-open sets contained
in A, i.e., the largest µ-open set contained in A (see [3, 6]).
According to [3, 9], a GT is said to be a quasitopology (briefly, QT) iff M,M
′ ∈ µ implies
M ∩M ′ ∈ µ. A QT on X coincides with a topology on a subset X0 j X.
It is easy to observe that iµ and cµ are idempotent and monotonic, where γ : expX → expX
is said to be idempotent iff for each A j X, γ(γ(A)) = γ(A), and monotonic iff γ(A) j γ(B)
whenever A j B j X. It is also well known from [6, 7] that if µ is a GT on X and A j X, x ∈ X,
then x ∈ cµ(A) iff (x ∈M ∈ µ ⇒M ∩A 6= ∅) and that cµ(X \A) = X \ iµ(A).
2. Almost (µ, λ)-open and (µ, λ)-open functions.
Definition 2.1. Let (X,µ) be a GTS. The µ(θ)-closure [7] (resp. µ(θ)-interior [7]) of a subset
A in a GTS (X,µ) is denoted by c
µ(θ)
(A) (resp. i
µ(θ)
(A)) and is defined to be the set {x ∈ X :
cµ(U) ∩ A 6= ∅ for each U ∈ µ(x)} (resp. {x ∈ X: there exists U ∈ µ(x) such that cµ(U) j A}),
where µ(x) = {U ∈ µ : x ∈ U}.
Theorem 2.1 [15]. In a GTS (X,µ), X\c
µ(θ)
(A) = i
µ(θ)
(X\A) andX\i
µ(θ)
(A) = c
µ(θ)
(X\A).
* This work is supported by UGC, New Delhi.
c© B. ROY, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10 1425
1426 B. ROY
Definition 2.2. A function f : (X,µ) → (Y, λ) is said to be almost (µ, λ)-open [1] if for each
U ∈ µ, f(U) j i
λ
(f(cµ(U))).
Theorem 2.2. For a function f : (X,µ) → (Y, λ), the following properties are equivalent:
(a) f is almost (µ, λ)-open,
(b) f(i
µ(θ)
(A)) j i
λ
(f(A)) for any subset A of X,
(c) i
µ(θ)
(f−1(B)) j f−1(i
λ
(B)) for every subset B of Y,
(d) f−1(c
λ
(B)) j c
µ(θ)
(f−1(B)) for every subset B of Y,
(e) for each x ∈ X and each µ-open set U containing x, there exists a λ-open set V containing
f(x) such that V j f(cµ(U)).
Proof. (a) ⇒ (b). Let A be any subset of X and x ∈ i
µ(θ)
(A). Then there exists a µ-open set U
containing x such that cµ(U) j A. Thus f(x) ∈ f(U) j f(cµ(U)) j f(A). Since f is almost (µ, λ)-
open, f(U) j i
λ
(f(cµ(U))) j i
λ
(f(A)) and so x ∈ f−1(i
λ
(f(A))). Thus i
µ(θ)
(A) j f−1(i
λ
(f(A)))
and hence f(i
µ(θ)
(A)) j i
λ
(f(A)).
(b) ⇒ (c). Let B be any subset of Y. Then by (b), f(i
µ(θ)
(f−1(B))) j i
λ
(B). Thus
i
µ(θ)
(f−1(B)) j f−1(i
λ
(B)).
(c) ⇒ (d). Let B be any subset of Y. Then X \ c
µ(θ)
(f−1(B)) = i
µ(θ)
(X \f−1(B)) (by Theorem
2.1) = i
µ(θ)
(f−1(Y \ B)) j f−1(i
λ
(Y \ B)) (by (c)) = f−1(Y \ c
λ
(B)) = X \ f−1(c
λ
(B)). Thus
f−1(c
λ
(B)) j c
µ(θ)
(f−1(B)).
(d) ⇒ (e). Let x ∈ X and U be any µ-open set containing x. Let B = Y \ f(cµ(U)).
Then by (d), f−1(c
λ
(Y \ f(cµ(U)))) j c
µ(θ)
(f−1(Y \ f(cµ(U)))). Now, f−1(c
λ
(Y \ f(cµ(U)))) =
= X \f−1(i
λ
(f(cµ(U)))). Also, c
µ(θ)
(f−1(Y \f(cµ(U)))) = c
µ(θ)
(X \f−1(f(cµ(U)))) j c
µ(θ)
(X \
cµ(U)) = X \ i
µ(θ)
(cµ(U)) (by Theorem 2.1) j X \ U. Therefore, U j f−1(i
λ
(f(cµ(U)))) and
thus f(U) j i
λ
(f(cµ(U))). Since f(x) ∈ f(U), there exists a V ∈ λ containing f(x) such that
V j f(cµ(U)).
(e) ⇒ (a). Let U be a µ-open set containing x. Then by (e), there exists a λ-open set V containing
f(x) such that V j f(cµ(U)). Thus f(x) ∈ V j i
λ
(f(cµ(U))) for each x ∈ U. Therefore we obtain,
f(U) j i
λ
(f(cµ(U))). Hence f is almost (µ, λ)-open.
Example 2.1. The concept of weak BR-openness as a natural dual to the weak BR-continuity
due to Ekici [12] was introduced and studied in [2].
Theorem 2.3. For a bijective function f : (X,µ) → (Y, λ) the following properties are equiva-
lent:
(a) f is almost (µ, λ)-open,
(b) c
λ
(f(iµ(F ))) j f(F ) for each µ-closed set F in X,
(c) c
λ
(f(U)) j f(cµ(U)) for each U ∈ µ.
Proof. (a) ⇒ (b). Let F be a µ-closed subset of X. Then X \ F is µ-open and Y \ f(F ) =
= f(X\F ) j i
λ
(f(cµ(X\F ))) (by (a)) = i
λ
(f(X\iµ(F ))) = i
λ
(Y \f(iµ(F ))) = Y \c
λ
(f(iµ(F ))).
Thus c
λ
(f(iµ(F ))) j f(F ).
(b) ⇒ (c). Let U be any µ-open set in X. Then we have c
λ
(f(U)) = c
λ
(f(iµ(U))) j
j c
λ
(f(iµ(cµ(U)))) j f(cµ(U)) (by (b)).
(c) ⇒ (a). Let U be a µ-open set in X. Then we have, Y \ i
λ
(f(cµ(U))) = c
λ
(Y \ f(cµ(U))) =
= c
λ
(f(X \ cµ(U))) j f(cµ(X \ cµ(U)))) (by (c)) = f(X \ iµ(cµ(U))) j f(X \ U) = Y \ f(U).
Therefore f(U) j i
λ
(f(cµ(U))). Hence f is almost (µ, λ)-open.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
ON WEAKLY (µ, λ)-OPEN FUNCTIONS 1427
Example 2.2. Let X = {a, b, c}, µ = {∅, {a}, {a, b}} and λ = {∅, {c}, {a, b}, {b, c}, X}.
Then (X,µ) and (X,λ) are two GTS’s. Consider the map f : (X,µ) → (X,λ) defined by f(a) =
= f(c) = c and f(b) = b. Then f is clearly not a bijection but f is almost (µ, λ)-open. We note that
c
λ
(f({a, b})) " f(cµ({a, b})) (see Theorem 2.3(c)).
Definition 2.3. A function f : (X,µ) → (Y, λ) is said to be (µ, λ)-open [11, 14] if f(U) is
λ-open for each µ-open set U in X.
Remark 2.1. It follows from Definitions 2.2 and 2.3 that every (µ, λ)-open function is almost
(µ, λ)-open but the converse is not true as follows from the next example.
Example 2.3. ConsiderX = {a, b, c}, µ = {∅, {a}, {c}, {a, c}} and λ = {∅, {a, b}, {b, c}, X}.
Then µ and λ are two GT’s on X. Now the identity mapping f : (X,µ) → (X,λ) is almost (µ, λ)-
open but not (µ, λ)-open.
Definition 2.4. A function f : (X,µ) → (Y, λ) is said to be strongly (µ, λ)-continuous iff
f(cµ(A)) j f(A) for each subset A of X.
Theorem 2.4. If a function f : (X,µ) → (Y, λ) is strongly (µ, λ)-continuous and almost (µ, λ)-
open then f is (µ, λ)-open.
Proof. Let U be any µ-open set in X. We have to show that f(U) is λ-open in Y. Since f is
almost (µ, λ)-open and strongly (µ, λ)-continuous, f(U) j i
λ
(f(cµ(U))) j i
λ
(f(U)). Thus f(U)
is λ-open.
Definition 2.5. A GTS (X,µ) is said to be µ-regular [13, 16] iff for each x ∈ X and each U ∈ µ
containing x there exists a V ∈ µ containing x such that x ∈ V j cµ(V ) j U.
Theorem 2.5. Let (X,µ) be µ-regular. Then the function f : (X,µ) → (Y, λ) is (µ, λ)-open iff
f is almost (µ, λ)-open.
Proof. One part of the theorem is trivial due to Remark 2.1. For the converse, let f be almost
(µ, λ)-open and U be any µ-open set in X. Then for each x ∈ U, there exists a Vx ∈ µ containing
x such that x ∈ Vx j cµ(Vx) j U. Hence U = ∪{Vx : x ∈ U} = ∪{cµ(Vx) : x ∈ U} and
hence f(U) = ∪{f(Vx) : x ∈ U} j ∪{i
λ
(f(cµ(Vx))) : x ∈ U} j i
λ
(∪{f(cµ(Vx))) : x ∈ U} j
j i
λ
(f(∪{cµ(Vx) : x ∈ U})) = i
λ
(f(U)). Thus f is (µ, λ)-open.
Definition 2.6. A function f : (X,µ) → (Y, λ) is said to satisfy the almost (µ, λ)-open interiority
condition if for each U ∈ µ, i
λ
(f(cµ(U))) j f(U).
Theorem 2.6. If a function f : (X,µ) → (Y, λ) is almost (µ, λ)-open and satisfy the almost
(µ, λ)-open interiority condition, then f is (µ, λ)-open.
Proof. Let U be a µ-open set in X. Since f is almost (µ, λ)-open, f(U) j i
λ
(f(cµ(U))) =
= i
λ
(i
λ
(f(cµ(U)))) j i
λ
(f(U)). Hence f(U) = i
λ
(f(U)), i.e., f(U) is λ-open.
Example 2.4. Let X = {a, b, c}, µ = {∅, {a}} and λ = {∅, {a}, {a, b}}. Then the identity
map f : (X,µ) → (X,λ) is (µ, λ)-open but it does not satisfy almost (µ, λ)-interiority condition.
For any subset A of a GTS (X,µ), the µ-frontier [17] of A is denoted by Frµ(A) and defined
by Frµ(A) = cµ(A) ∩ cµ(X \A).
Definition 2.7. A function f : (X,µ) → (Y, λ) is said to be complementary weakly (µ, λ)-open
if f(Frµ(U)) is λ-closed for each U ∈ µ.
That the notions of complementary weakly (µ, λ)-open and almost (µ, λ)-open functions are
independent as shown in the next two examples.
Example 2.5. (a) Let X = {a, b, c}, µ = {∅, {c}} and λ = {∅, {a, b}, {a, c}, X}. Then it is
easy to see that the identity map f : (X,µ) → (X,λ) is almost (µ, λ)-open but not complementary
weakly (µ, λ)-open.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
1428 B. ROY
(b) Let X = {a, b, c}, µ = {∅, {a}, {a, b}, {b, c}, X} and λ = {∅, {a}, {a, b}, X}. Consider the
function f : (X,µ) → (X,λ) defined by f(a) = b and f(b) = f(c) = c. Then it can be checked that
f is complementary weakly (µ, λ)-open but not almost (µ, λ)-open.
Theorem 2.7. If f : (X,µ) → (Y, λ) is an almost (µ, λ)-open and complementary weakly (µ, λ)-
open bijection and (Y, λ) is a QT then f is (µ, λ)-open.
Proof. Let x ∈ X and U be a µ-open set containing x. Since f is almost (µ, λ)-open, by
Theorem 2.2 there exists V ∈ λ containing f(x) such that V j f(cµ(U)). Now, Frµ(U) =
= cµ(U)∩ cµ(X \U) = cµ(U)∩ (X \U). Since x ∈ U, x 6∈ Frµ(U) and hence f(x) 6∈ f(Frµ(U)).
Put W = V ∩ (Y \ f(Frµ(U))). Since f is complementary weakly (µ, λ)-open and (Y, λ) is a QT,
W is a λ-open set containing f(x). It is now sufficient to show that W j f(U). Let y ∈ W. Then
y ∈ V j f(cµ(U)) and y 6∈ f(Frµ(U)) = f(cµ(U) ∩ (X \U)) = f(cµ(U)) ∩ (Y \ f(U)). Thus we
have y ∈ (Y \ f(cµ(U))) ∪ f(U). Thus y ∈ f(U). Hence f is (µ, λ)-open.
Example 2.6. Consider the Example 2.5(b). Then (Y, λ) is a QT and f is complementary weakly
(µ, λ)-open which is not bijective. Also f is not (µ, λ)-open.
Definition 2.8. A function f : (X,µ) → (Y, λ) is said to be contra (µ, λ)-closed if f(F ) is
λ-open for every µ-closed set F in (X,µ).
Theorem 2.8. If a function f : (X,µ) → (Y, λ) is contra (µ, λ)-closed, then f is almost (µ, λ)-
open.
Proof. Let U be a µ-open set in (X,µ). Then f(U) j f(cµ(U)) = i
λ
(f(cµ(U))). Thus f is
almost (µ, λ)-open.
The converse of the above theorem need not be true as shown in the next example.
Example 2.7. Let X = {a, b, c}, µ = {∅, {a}, {a, b}} and λ = {∅, {a, b}, {a, c}, X}. Then µ
and λ are two GT’s on X. The identity map f : (X,µ) → (X,λ) is almost (µ, λ)-open but not contra
(µ, λ)-closed.
3. Some properties of almost (µ, λ)-open functions.
Definition 3.1. A GTS (X,µ) is said to be µ-hyperconnected [11] if cµ(U) = X for each
nonempty µ-open set U.
Theorem 3.1. Let (X,µ) be µ-hyperconnected. If for a function f : (X,µ) → (Y, λ), f(X) is
λ-open in (Y, λ), then f is almost (µ, λ)-open. If µ is a strong GT then the converse is also true.
Proof. Let f(X) be λ-open in (Y, λ). Let U ∈ µ. Then f(U) j f(X) = i
λ
(f(X)) =
= i
λ
(f(cµ(U))). Thus f(U) j i
λ
(f(cµ(U))). Hence f is almost (µ, λ)-open.
Conversely, let f be an almost (µ, λ)-open function. Since µ is strong, X ∈ µ and thus
f(X) j i
λ
(f(cµ(X))) = i
λ
(f(X)). Thus f(X) is λ-open.
Example 3.1. Let X = {a, b, c}, µ = {∅, {a}, {a, b}} and λ = {∅, {a}, {c}, {a, c}}. Then µ
and λ are two GT’s on X such that X 6∈ µ. Consider the function f : (X,µ) → (X,λ) defined by
f(a) = a; f(b) = c and f(c) = b. It is easy to check that f is almost (µ, λ)-open and (X,µ) is
µ-hyperconnected but f(X) is not λ-open.
Definition 3.2. A GTS (X,µ) is said to be µ-connected [19] if X can not be written as the union
of two nonempty disjoint µ-open sets of X.
Theorem 3.2. If f : (X,µ) → (Y, λ) is an almost (µ, λ)-open bijection and (Y, λ) is λ-
connected, then (X,µ) is µ-connected.
Proof. Suppose that (X,µ) is not µ-connected. Then there exist disjoint µ-open sets U1 and U2
such that X = U1 ∪ U2. Hence we have f(U1) ∩ f(U2) = ∅ and Y = f(U1) ∪ f(U2). Since f is
almost (µ, λ)-open, f(Ui) j i
λ
(f(cµ(Ui))) for i = 1, 2. Since each Ui is µ-closed, Ui = cµ(Ui),
and hence f(Ui) = i
λ
(f(Ui)) for i = 1, 2. So, f(Ui) are λ-open for i = 1, 2. Thus Y has been
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
ON WEAKLY (µ, λ)-OPEN FUNCTIONS 1429
decomposed into two nonempty disjoint λ-open sets which contradicts that (Y, λ) is λ-connected.
Thus (X,µ) is µ-connected.
Remark 3.1 [11]. If a GTS (X,µ) is µ-hyperconnected, then (X,µ) is µ-connected.
Corollary 3.1. If f : (X,µ) → (Y, λ) is an almost (µ, λ)-open bijection and (Y, λ) is λ-
hyperconnected, then (X,µ) is connected.
Example 3.2. LetX = {a, b, c}, µ = {∅, {c}, {a, b}, {b, c}, X} and λ = {∅, {a}, {a, b}, {a, c},
X}. Then (X,µ) and (X,λ) are two GTS’s. Then the map f : (X,µ) → (X,λ) defined by
f(a) = f(c) = a and f(b) = b is clearly not a bijection but f is almost (µ, λ)-open and (X,λ) is
λ-connected but (X,µ) is not µ-connected.
Definition 3.3. A subset A of a GTS (X,µ) is said to be weakly µ-compact (briefly ωµ-compact)
[18] if every cover of A by µ-open subsets of X has a finite subfamily the union of whose µ-closures
covers A.
Definition 3.4. A GTS (X,µ) is said to be extremally µ-disconnected [10] if the µ-closure of
any µ-open set is µ-open.
Lemma 3.1. If a function f : (X,µ) → (Y, σ) is (µ, λ)-open then for eachB j Y, f−1(c
λ
(B)) j
j cµ(f−1(B)).
Theorem 3.3. Let (X,µ) be an extremally µ-disconnected space where µ is a QT. Let f :
(X,µ) → (Y, λ) be an one-to-one (µ, λ)-open, almost (µ, λ)-open mapping such that f−1(y) is
ωµ-compact for each y ∈ Y. Then for every ω
λ
-compact subset G of Y, f−1(G) is ωµ-compact.
Proof. Let {Vα : α ∈ Λ} be a µ-open cover of f−1(G). Then for each y ∈ G, f−1(y) j
j ∪{cµ(Vα) : α ∈ Λy} = Hy for some finite subset Λy of Λ. Then Hy is µ-open as X is extremally
µ-disconnected. So by Theorem 2.2 ((a) ⇔ (e)), there exists a λ-open set Uy containing y ∈
∈ Y such that f−1(Uy) j cµ(Hy). Then {Uy : y ∈ G} is a cover of G by λ-open subsets of
Y. Thus by ω
λ
-compactness of G, there exists a finite subset K of G such that G j ∪{c
λ
(Uy) :
y ∈ K}. Hence by Lemma 3.1, f−1(G) j ∪{cµ(f−1(Uy)) : y ∈ K} j ∪{cµ(Hy) : y ∈ K}. Thus
f−1(G) j ∪{cµ(Vα) : α ∈ Λy , y ∈ K}. Hence f−1(G) is ωµ-compact.
Conclusion. The definitions of various types of weakly open functions may be introduced from
the definition of weakly (µ, λ)-open functions by replacing the generalized topologies µ and λ (resp.
on X and Y ) suitably.
1. Al-Omari A., Noiri T. A unified theory for contra-(µ, λ)-continuous functions in generalized topological spaces //
Acta Math. hung. – 2012. – 135. – P. 31 – 41.
2. Caldas M., Ekici E., Jafari S., Latif R. M. On weak BR-open functions and their characterizations in topological
spaces // Demonstr. math. – 2011. – 44, № 1. – P. 159 – 168.
3. Császár Á. Generalized topology, generalized continuity // Acta Math. hung. – 2002. – 96. – P. 351 – 357.
4. Császár Á. γ-connected sets // Acta Math. hung. – 2003. – 101. – P. 273 – 279.
5. Császár Á. Separation axioms for generalized topologies // Acta Math. hung. – 2004. – 104. – P. 63 – 69.
6. Császár Á. Generalized open sets in generalized topologies // Acta Math. hung. – 2005. – 106. – P. 53 – 66.
7. Császár Á. δ- and θ-modifications of generalized topologies // Acta Math. hung. – 2008. – 120. – P. 275 – 279.
8. Császár Á. On generalized neighbourhood systems // Acta Math. hung. – 2008. – 121. – P. 359 – 400.
9. Császár Á. Remarks on quasi-topologies // Acta Math. hung. – 2008. – 119. – P. 197 – 200.
10. Császár Á. Extremally disconnected generalized topologies // Ann. Univ. Sci. Budapest. – 2004. – 47. – P. 91 – 96.
11. Ekici E. Generalized hyperconnectedness // Acta Math. hung. – 2011. – 133. – P. 140 – 147.
12. Ekici E. Generalization of weakly clopen and strongly θ-b-continuous functions // Chaos, Solitons and Fractals. –
2008. – 38. – P. 79 – 88.
13. Kim Y. K., Min W. K. Remarks on enlargements of generalized topologies // Acta Math. hung. – 2011. – 130. –
P. 390 – 395.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
1430 B. ROY
14. Min W. K. Some results on generalized topological spaces and generalized systems // Acta Math. hung. – 2005. –
108. – P. 171 – 181.
15. Min W. K. A note on θ(g, g
′
)-continuity in generalized topological spaces // Acta Math. hung. – 2009. – 125. –
P. 387 – 393.
16. Roy B. On a type of generalized open sets // Appl. Gen. Top. – 2011. – 12. – P. 163 – 173.
17. Roy B. On faintly continuous functions via generalized topology // Chinese J. Math. – 2013. – Article ID 412391. –
6 p.
18. Sarsak M. S. Weakly µ-compact spaces // Demonstr. math. – 2012. – 45, № 4. – P. 929 – 938.
19. Shen R.-X. A note on generalized connectedness // Acta Math. hung. – 2009. – 122. – P. 231 – 235.
Received 05.10.12,
after revision — 07.01.14
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
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