KoSiKuAGT.dvi
@
Applied General Topology
c© Universidad Politécnica de Valencia
Volume 9, No. 2, 2008
pp. 239-251
Generalizations of Z-supercontinuous functions
and Dδ-supercontinuous functions
J. K. Kohli, D. Singh and Rajesh Kumar
Abstract. Two new classes of functions, called ‘almost z-
supercontinuous functions’ and ‘almost Dδ -supercontinuous functions’
are introduced. The class of almost z-supercontinuous functions prop-
erly includes the class of z-supercontinuous functions (Indian J. Pure
Appl. Math. 33(7), (2002), 1097-1108) as well as the class of al-
most clopen maps due to Ekici (Acta. Math. Hungar. 107(3),
(2005), 193-206) and is properly contained in the class of almost Dδ-
supercontinuous functions which in turn constitutes a proper subclass
of the class of almost strongly θ-continuous functions due to Noiri and
Kang (Indian J. Pure Appl. Math. 15(1), (1984), 1-8) and which in its
turn include all δ-continuous functions of Noiri (J. Korean Math. Soc.
16 (1980), 161-166). Characterizations and basic properties of almost
z-supercontinuous functions and almost Dδ-supercontinuous functions
are discussed and their place in the hierarchy of variants of continu-
ity is elaborated. Moreover, properties of almost strongly θ-continuous
functions are investigated and sufficient conditions for almost strongly
θ-continuous functions to have uθ -closed (θ-closed) graph are formu-
lated.
2000 AMS Classification: Primary: 54C05, 54C08, 54C10; Secondary: 54D10,
54D15, 54D20.
Keywords: (almost) z-supercontinuous function, (almost) Dδ-supercontinuous
function, (almost) strongly θ-continuous function, almost continuous function,
δ-continuous function, faintly continuous function, , uθ-closed graph , θ-closed
graph, uθ-limit point, θ-limit point, z-convergence.
240 J. K. Kohli, D. Singh and R. Kumar
1. Introduction
Among several of the variants of continuity in the literature, some are stronger
than continuity and some are weaker than continuity and yet others are inde-
pendent of continuity. In this paper we introduce two new variants of conti-
nuity which represent generalizations of the notions of z-supercontinuity and
Dδ -supercontinuity and are independent of continuity and coincide with z-
supercontinuity and Dδ-supercontinuity, respectively if the range is a semireg-
ular space. The class of almost z-supercontinuous functions besides containing
the class of z-supercontinuos functions contains the class of almost clopen (≡
almost cl-supercontinuous [34]) functions defined by Ekici [4].
Characterizations and basic properties of almost z-supercontinuous (almost
Dδ-supercontinuous) functions are elaborated in Section 3 and their place in
the hierarchy of variants of continuity is discussed. Section 4 is devoted to the
study of the behaviour of separation axioms under almost z-supercontinuous
(almost Dδ-supercontinuous) functions. In Section 5, characterizations and
properties of almost strongly θ-continuous functions are elaborated. Section 6
is devoted to separation axioms and sufficient conditions for almost strongly
θ-continuous functions to have uθ-closed (θ-closed) graphs are obtained.
2. Preliminaries and Basic Definitions
A subset S of a space X is said to be an H-set [36] or quasi H-closed relative to
X [28] (respectively N -closed relative to X [1]) if for every cover { Uα|α∈Λ} of S
by open sets of X, there exists a finite subset Λo of Λ such that S⊂∪{U α|α∈Λo}
(respectively S⊂∪{(U α)
o|α∈Λo}). A space X is said to be quasi H-closed [28]
(respectively nearly compact [32]) if the set X is quasi H-closed relative to X
(respectively N -closed relative to X). A space X is said to be quasicompact [5]
if every cover of X by cozero sets admits a finite subcover.
A space X is said to be δ-completely regular [13] (almost completely regular
[31]) if for each regular Gδ -set (regularly closed set) F and a point x not in
F there exists a continuous function f : X→[0, 1] such that f (x) = 0 and
f (F ) = 1.
A subset A of a space X is called a regular Gδ-set [21] if A is an intersection of a
sequence of closed sets whose interiors contain A, i.e., if A =
∞⋂
n=1
Fn =
∞⋂
n=1
F on ,
where each Fn is a closed subset of X. The complement of a regular Gδ-set is
called a regular Fσ-set.
A space X is called a Dδ-completely regular ([15], [16]) if it has a base of regular
Fσ-sets.
Definition 2.1. A function f : X→Y from a topological space X into a topo-
logical space Y is said to be almost z-supercontinuous (almost Dδ-supercontinuous)
if for each x∈X and each open set V containing f (x), there exists a cozero set
(regular Fσ-set) U containing x such that f (U )⊂(V )
o.
Generalizations of Z-supercontinuous functions and Dδ -supercontinuous functions 241
Definition 2.2. A set G is said to be δ-open [36] (dδ-open [13], z-open [12])
if for each x∈G, there exists a regular open set (regular Fσ-set, cozero set) H
such that x∈H⊂G, or equivalently, G can be obtained as an arbitrary union of
regular open sets (regular Fσ-sets, cozero sets). The complement of a δ -open
(dδ-open, z-open) set will be referred to as a δ-closed (dδ-closed, z-closed) set.
Definition 2.3. Let X be a topological space and let A⊂X. A point x∈X is
called a δ-adherent [36] ( θ-adherent [36], uθ-adherent ([9], [10]), dδ-adherent
[13], z-adherent [12]) point of A if every regular open set (closed neighborhood,
θ-open set, regular Fσ-set, cozero set) containing x has non-empty intersection
with A. Let Aδ denote the set of all δ-adherent points (clθA the set of all
θ-adherent points, Auθ the set of all uθ-adherent points, [A]dδ the set of all dδ-
adherent points, Az the set of all z-adherent points) of a set A. The set A is
δ-closed (θ-closed, dδ-closed, z-closed) if A = Aδ (A = clθA or A = Auθ , A =
[A]dδ , A = Az ).
Lemma 2.4 ([8], [11]). A subset A of a topological space X is θ-open if and
only if for each x∈A, there is an open set U such that x∈U⊂U⊂A.
Definition 2.5. A space X is called θ-compact [10] ( Dδ-compact [14]) if every
θ-open cover (cover by regular Fσ -sets) of X has a finite subcover.
Definitions 2.6. A function f : X→Y from a topological space X into a
topological space Y is said to be
(a) strongly continuous [18] if f (A) ⊂ f (A) for each subset A of X.
(b) perfectly continuous( [25], [26]) if f −1(V ) is clopen in X for every open
set V ⊂Y .
(c) almost perfectly continuous (≡ regular set connected [3]) if f −1(V ) is
clopen for every regular open set V in Y .
(d) cl-supercontinuous [34] (≡ clopen continuous [29]) if for each open set V
containing f (x) there is a clopen set U containing x such that f (U )⊂V .
(e) almost cl-supercontinuous[17] (≡ almost clopen continuous[4]) if for each
x∈X and each regular open set V containing f (x) there is a clopen set U
containing x such that f (U )⊂V .
(f) z-supercontinuous [12] if for each x∈X and for each open set V containing
f (x), there exists a cozero set U containing x such that f (U )⊂V .
(g) strongly θ-continuous [24] if for each x∈X and for each open set V con-
taining f (x), there exists an open set U containing x such that f (U )⊂V .
(h) supercontinuous [22] if for each x∈X and for each open set V containing
f (x), there exists an open set U containing x such that f (U )o⊂V .
(i) almost strongly θ-continuous [27] if for each x∈X and for each open set
V containing f (x), there exists an open set U containing x such that
f (U )⊂(V )o.
(j) δ-continuous [24] if for each x∈X and for each open set V containing f (x),
there exists an open set U containing x such that f (U )o⊂(V )o.
(k) almost continuous [33] if for each x∈X and for each open set V containing
f (x), there exists an open set U containing x such that f (U )⊂(V )o.
242 J. K. Kohli, D. Singh and R. Kumar
(l) faintly continuous [20] if for each x∈X and for each θ-open set V contain-
ing f (x), there exists an open set U containing x such that f (U )⊂V .
(m) Dδ-supercontinuous [13] if for each x∈X and for each open set V contain-
ing f (x), there exists a regular Fσ set U containing x such that f (U )⊂V .
The following diagram well illustrates the relationships that exist among almost
z-supercontinuous functions, almost Dδ -supercontinuous functions and various
variants of continuity defined above.
However, none of the above implications in general is reversible. Kohli and Ku-
mar [12] showed that a strongly θ-continuous function need not be z-supercontinuous
function. Noiri and Kang [27] gave examples to show that a δ-continuous
function need not be almost strongly θ-continuous and that almost strongly
θ-continuous function need not be strongly θ-continuous. Moreover, Noiri [24]
showed that an almost continuous function need not be δ-continuous.
Example 2.7. Let X = N = Y be the set of positive integers equipped with
cofinite topology. The identity function on X is almost z-supercontinuous but
not Dδ-supercontinuous.
Example 2.8. Let X = Y be the mountain chain space due to Heldermann
[6] which is a regular space. The identity map from X onto Y is strongly
θ-continuous but not almost Dδ-supercontinuous.
Example 2.9. Let X = {x1, x2, x3, x4} and Γ = {X, φ, {x3}, {x1, x2}, {x1, x2, x3}}
Let Y = {y1, y2, y3, y4} and σ = {Y, φ, {y1}, {y3}, {y1, y2}, {y1, y3}, {y1, y2, y3},
{y1, y3, y4}}
Define a function f : (X, Γ) → (Y, σ) as follows: f (x1) = f (x2) = y2 and
f (x3) = f (x4) = y1 Then f is an almost z-supercontinuous functions but not
continuous.
Generalizations of Z-supercontinuous functions and Dδ -supercontinuous functions 243
Example 2.10. Let A = K∪{a+,a−} be the space due to Hewitt [7] which is
Dδ-completely regular. The identity function defined on A is Dδ-supercontinuous
but not almost z-supercontinuous.
Example 2.11. Let X denote the real line endowed with usual topology. The
identity function defined on X is almost z-supercontinuous but not almost cl-
supercontinuous (=almost clopen).
Examples 2.8 and 2.9 show that the notions of almost z-supercontinuous func-
tion (almost Dδ-supercontinuous function) and continuous function are inde-
pendent of each other.
3. Characterizations and Basic Properties of almost
z-Supercontinuous and Dδ-Supercontinuous Functions
Proposition 3.1. For a function f : X→Y from a topological space X into a
topological space Y , the following statements are equivalent:
(a) f is almost z-supercontinuous (almost Dδ-supercontinuous).
(b) The inverse image of every regular open subset of Y is z-open (dδ-open) in
X.
(c) The inverse image of every regular closed subset of Y is z-closed (dδ-closed)
in X.
(d) The inverse image of every δ-open subset of Y is z-open (dδ-open) in X.
(e) The inverse image of every δ-closed subset of Y is z-closed (dδ-closed) in
X.
Proof. It is easy using definitions. �
Theorem 3.2. For a function f : X→Y the following statement are equivalent.
(a) f is almost z-supercontinuous.
(b) f (Az )⊂(f (A))δ for every A⊂X.
(c) (f −1(B))z⊂f
−1(Bδ) for every B⊂Y .
Proof. (a)⇒(b). Let y = f (x) for some x∈Az. To show that f (x)∈(f (A))δ , let
V be any regular open set containing f (x). Then there exists a cozero set U
containing x such that f (U )⊂V . Since x∈Az , U∩A6=φ and so f (U∩A)6=φ which
in turn implies that f (U )∩f (A)6=φ and hence V ∩f (A)6=φ. Thus f (x)∈(f (A))δ.
Hence f (Az )⊂(f (A))δ for every A⊂X.
(b)⇒(c). Let B⊂Y . Then f ((f −1(B))z )⊂(f (f
−1(B)))δ⊂Bδ and so it follows
that (f −1(B))z⊂f
−1(Bδ).
(c)⇒(a). let F be any δ-closed set in Y . Then (f −1(F ))z⊂f
−1(Fδ) = f
−1(F ).
Since f −1(F )⊂(f −1(F ))δ⊂(f
−1(F ))z , so f
−1(F ) = (f −1(F ))z which in turn
implies that f is almost z-supercontinuous. �
Theorem 3.3. For a function f : X→Y the following statement are equivalent.
(a) f is almost Dδ-supercontinuous.
(b) f ([A]dδ )⊂(f (A))δ for every A⊂X.
(c) [f −1(B)]dδ ⊂f
−1(Bδ) for every B⊂Y .
244 J. K. Kohli, D. Singh and R. Kumar
Proof. (a)⇒(b). Let y = f (x) for some x∈[A]dδ . To show that y∈(f (A))δ , let
V be a regular open set containing f (x). Since f is almost Dδ-supercontinuous,
there is a regular Fσ-set U containing x such that f (U )⊂V . Since x∈[A]dδ , U∩A6=φ
and hence f (U∩A)6=φ which in turn implies that f (U )∩f (A)6=φ. Thus V ∩f (A)6=φ
and so y∈(f (A))δ for every A⊂X.
(b)⇒(c). Let B⊂Y . Then f ([f −1(B)]dδ )⊂(f (f
−1(B)))δ⊂Bδ and so it follows
that [f −1(B)]dδ ⊂f
−1(Bδ).
(c)⇒(a). let F be any δ-closed set in Y . Then [f −1(F )]dδ ⊂f
−1(Fδ) = f
−1(F ).
Since f −1(F )⊂[f −1(F )]dδ , f
−1(F ) = [f −1(F )]dδ and so f
−1(F ) is dδ-closed. It
follows that f is almost Dδ-supercontinuous. �
Definition 3.4. A filterbase F is said to z-converge[12] ( dδ-converge[13], δ-
converge[36]) to a point x, written as F
z
→ x(F
dδ
→ x, F
δ
→ x), if every cozero
set (regular Fσ-set, regular open set) containing x contains a member of F.
Theorem 3.5. A function f : X→Y is almost z-supercontinuous (almost Dδ-
supercontinuous) if and only if f (F)
δ
→ f (x) for each x∈X and each filter F
in X that z-converges (dδ-converges) to x.
Proof. We shall prove the result in the case of almost z-supercontinuous func-
tions only. Suppose that f is almost z-supercontinuous and let F be a filter
in X that z-converges to x. Let W be a regular open set containing f (x).
Then x∈f −1(W ) and f −1(W ) is z-open. Let H be a cozero set such that
x∈H⊂f −1(W ) and so f (H)⊂W . Since F z-converges to x, there exists U∈F
such that U⊂H and hence f (U )⊂f (H)⊂W . Thus, f (F)
δ
→ f (x).
Conversely, let W be a regular open set containing f (x). Now, the filter F
generated by the filterbase Bx consisting of cozero sets containing x, z-converges
to x. Since by hypothesis f (F)
δ
→ f (x), there exists a member f (F ) of f (F)
such that f (F )⊂W . Choose B∈Bx such that B⊂F . Since B is a cozero set
containing x and since f (B)⊂f (F )⊂W, f is almost z-supercontinuous. �
Remark 3.6. It is routine to verify that almost z-supercontinuity (almost
Dδ-supercontinuity) is invariant under restrictions and composition of func-
tions and enlargement of range. Moreover, the composition gof is almost
z-supercontinuous whenever f : X→Y is almost z-supercontinuous and g :
Y →Z is δ-continuous. Furthermore, if gof is almost z-supercontinuous and
f is a surjection which maps z-open sets to z-open sets, then g is almost z-
supercontinuous.
The following lemma is due to Singal and Singal [33] and will be used in the
sequel.
Lemma 3.7 (Singal and Singal [33]). Let {Xα : α∈I} be a family of spaces and
let X =
∏
Xα be the product space. If x = (xα)∈X and V is a regular open
subset of X containing x, then there exists a basic regular open set ΠVα such
that x∈ΠVα⊂V , where Vα is regular open in Xα for each α∈I and Vα = Xα
for all α∈I except for a finite number of indices αi, i = 1, 2, . . . , n.
Generalizations of Z-supercontinuous functions and Dδ -supercontinuous functions 245
Theorem 3.8. Let {fα : Xα→Yα} be a family of almost z-supercontinuous (al-
most Dδ -supercontinuous) functions. Let X = ΠXα and Y = ΠYα. Then f :
X→Y defined by f ((xα)) = (fα(xα)) for each (xα)∈X is almost z-supercontinuous
( almost Dδ-supercontinuous).
Proof. Let (xα)∈X and W be a regular open set in Y containing f ((xα)).
By Lemma 3.7 there exists a basic regular open set V = ΠVα such that
f (x)∈V ⊂W , where each Vα is a regular open set in Yα and Vα = Yα for
α∈∆ except for α = α1, α2, . . . , αn. For each i = 1, 2, . . . , n, in view of almost
z-supercontinuity (almost Dδ-supercontinuity) of fαi there exists a cozero set
(regular Fδ-set) Uαi containing xαi such that fαi (Uαi )⊂Vαi . Let U =
∏
Uα,
where Uα = Xα for α 6= αi, (i = 1, 2, . . . , n). Then U is a cozero set (reg-
ular Fδ-set) in X such that (xα)∈U and f (U )⊂V ⊂W . Thus f is almost z-
supercontinuous (almost Dδ -supercontinuous). �
Theorem 3.9. Let f : X→Y be any function. If {Uα : α∈∆} is a cover
of X by cozero sets (regular Fδ-sets) and for each α, fα = f|Uα : Uα→Y
is almost z-supercontinuous (almost Dδ-supercontinuous), then f is almost z-
supercontinuous (almost Dδ-supercontinuous).
Proof. Let V be a regular open set in Y . Then f −1(V ) = ∪{f −1α (V ) : α∈∆}
and since each fα is almost z-supercontinuous (almost Dδ-supercontinuous),
each f −1α (V ) is z-open (dδ-open) in Uα and hence in X. Thus f
−1(V ) being
the union of z-open (dδ -open) sets is z-open (dδ-open). Thus f is almost
z-supercontinuous (almost Dδ-supercontinuous). �
Theorem 3.10. Let f : X→Y be a function and g : X→X×Y , defined by
g(x) = (x, f (x)) for each x∈X, be the graph function. Then g is almost z-
supercontinuous if and only if f is almost z-supercontinuous and X is an almost
completely regular space.
Proof. Suppose that g is almost z-supercontinuous. Let V be a regular open
set in Y . Then p−1y (V ) = X×V is a regular open set in X×Y , where py
is the projection from X×Y onto Y . Therefore f −1(V ) = (pyog)
−1(V ) =
g−1(p−1y (V )) = g
−1(X×V ) is z-open and so f is almost z-supercontinuous.
To prove that X is an almost completely regular space, let F be a regular
closed set and suppose that x /∈F . Then x∈X \ F and g(x)∈(X\F )×Y which
is a regularly open set in X×Y . So there exists a cozero set W in X such
that g(W )⊂(X\F )×Y . Hence x∈W ⊂X\F . Thus X is an almost completely
regular space.
To prove sufficiency, let x∈X and let W be a regular open set containing
g(x). By Lemma 3.7 there exist regular open sets U⊂X and V ⊂Y such
that (x, f (x))∈U×V ⊂W . Since X is almost completely regular, there ex-
ists a cozero set G1 in X containing x such that x∈G1⊂U . Since f is al-
most z-supercontinuous, there exists a cozero set G2 in X containing x such
that f (G2)⊂V . Let G = G1 ∩ G2. Then G is a cozero set containing x and
g(G)⊂U×V ⊂W . This proves that g is almost z-supercontinuous. �
246 J. K. Kohli, D. Singh and R. Kumar
Proposition 3.11. Let f : X→Y be a function defined on a δ-completely
regular space X. Then the graph function G(f ) is almost Dδ-supercontinuous
if and only if f is almost Dδ-upercontinuous.
Proof. It is easy using definitions. �
4. separation axioms related to almost z-supercontinuous
functions and Dδ-supercontinuous functions
Theorem 4.1. An almost z-supercontinuous (almost Dδ -supercontinuous) im-
age of a quasicompact (Dδ-compact) space is nearly compact.
Proof. Let f : X→Y be an almost z-supercontinuous (almost Dδ -supercontinuous)
surjection from a quasicompact (Dδ -compact) space X onto a space Y . Let
V = {Vα : α∈Λ} be a cover of Y by regularly open sets (regular Fδ-sets) in
Y . Since f is almost z-supercontinuous (almost Dδ -supercontinuous), each
f −1(Vα) is z-open (dδ-open) in X and so is a union of cozero sets (regular
Fδ-sets). This in turn yields a cover G of X consisting of cozero sets (reg-
ular Fδ-sets). Since X is quasicompact (Dδ-compact), there is a finite sub-
collection {C1, . . . , Cn} of G which covers X. Suppose Ci⊂f
−1(Vαi ) for some
αi∈Λ(i = 1, . . . , n). Then {Vα1 , . . . , Vαn } is a finite subcollection of V which
covers Y . Thus Y is nearly compact. �
Corollary 4.2. Let f : X→Y be an almost z-supercontinuous (almost Dδ -
supercontinuous) surjection from a quasicompact (Dδ -compact) space onto a
semiregular space Y . Then Y is compact.
Proof. A semiregular nearly compact space is compact. �
Definition 4.3 ([30]). A space X is said to be almost regular if for each regular
closed set A and each point x /∈A, there exist disjoint open sets U and V such
that x∈U, A⊂V .
Theorem 4.4. Let f : X→Y be an almost Dδ-supercontinuous open bijec-
tion onto a space Y . Then Y is an almost regular space. Further, if Y is a
semiregular space, then Y is a regular space.
Proof. Let B be any regularly closed set in Y and let y /∈B. Then f −1(B)∩f −1(y) =
φ. Since f is almost Dδ-supercontinuous, by Proposition 3.1 f
−1(B) is dδ-closed
and so f −1(B) =
⋂
α∈Λ
Zα, where each Zα is a regular Gδ-set. Since f is one-one,
f −1(y) is a singleton and so there exists αo∈Λ, such that f
−1(y) /∈Zαo . Since
Zαo is a regular Gδ-set, Zαo =
∞⋂
i=1
Hi =
∞⋂
i=1
Hoi , where each Hi is a closed set.
So there exists an integer j such that f −1(y) /∈ Hj . Then X \ Hj and H
o
j are
disjoint open sets containing f −1(y) and f −1(B), respectively. Since f is an
open bijection, f (X\Hj ) and f (H
o
j ) are disjoint open sets containing y and B,
respectively. Thus Y is an almost regular space. Since a semiregular almost
regular space is regular, the last assertion is immediate. �
Generalizations of Z-supercontinuous functions and Dδ -supercontinuous functions 247
5. Characterizations and some basic properties of almost
strongly θ-continuous functions
Proposition 5.1. A function f : X→Y is almost strongly θ-continuous if and
only if for each x∈X and each regular open set V containing f (x), there exists
a θ-open set U containing x such that f (U )⊂V .
Proof. It is easy using definitions. �
Theorem 5.2. For a function f : X→Y the following statement are equivalent.
(1) f is almost strongly θ-continuous.
(2) f (Auθ )⊂(f (A))δ for each A⊂X.
(3) (f −1(B))uθ ⊂f
−1(Bδ) for every B⊂Y .
Proof. (a)⇒(b). Since (f (A))δ is δ-closed in Y , by [27, Theorem 3.1, (f )],
f −1((f (A))δ ) is θ-closed in X. Again, since A⊂f
−1((f (A))δ),
Auθ ⊂(f
−1(f (A))δ)uθ = f
−1((f (A))δ ) and so f (Auθ )⊂(f (A))δ.
(b)⇒(c). Let B⊂Y . Then, by hypothesis f ((f −1(B))uθ )⊂(f (f
−1(B))δ⊂Bδ
and so it follows that (f −1(B))uθ ⊂f
−1(Bδ).
(c)⇒(a). Let F be any δ-closed set in Y . Then (f −1(F ))uθ ⊂f
−1(Fδ) = f
−1(F )
which implies that f −1(F ) = (f −1(F ))uθ and so f
−1(F ) is θ-closed. This
proves that f is almost strongly θ-continuous. �
Definition 5.3. ( [9], [10] ): A filter F is said to uθ-converge to a point x,
written as F
uθ
→ x, if every θ-open set containing x contains a member of F.
Theorem 5.4. A function f : X→Y is almost strongly θ-continuous if and
only if f (F)
δ
→ f (x) for each x∈X and each filter in X which uθ-converges to
a point x.
Proof. Suppose that f is almost strongly θ-continuous and let F
uθ
→ x. Let W
be a regular open set in Y containing f (x). Then by Proposition 5.1, f −1(W )
is a θ-open set in X. Since F
uθ
→ x, there exists F ∈F such that F ⊂f −1(W )
and so f (F )⊂W . This shows that f (F)
δ
→ f (x).
Conversely, let V be a regular open subset of Y containing f (x). Now let F be
the filter generated by the filterbase Vx consisting of all θ-open sets containing
x. By hypothesis f (F)
δ
→ f (x) and so there exists a member f (F ) of f (Vx) such
that f (F )⊂V . Choose U∈Vx such that U⊂F which implies that f (U )⊂f (F )
and f (F )⊂V and so f (U )⊂V . Hence f is almost strongly θ-continuous. �
Theorem 5.5. If f : X→Y is faintly continuous and g : Y →Z be almost
strongly θ-continuous. Then gof is almost continuous.
Proof. Let V be a regular open set in Z. By almost strongly θ-continuity of
g, g−1(V ) is θ-open in Y . So (gof )−1(V ) = f −1(g−1(V )) is open in X, since f
is faintly continuous. Hence gof is almost continuous. �
Theorem 5.6. Let f : X→Y be an almost continuous function defined on a
completely regular space X. Then f is almost z-supercontinuous.
248 J. K. Kohli, D. Singh and R. Kumar
Proof. Let V be a regular open set containing f (x). Since f is almost contin-
uous, f −1(V ) is open. Again since X is completely regular space, f −1(V ) is
z-open. Hence f is almost z-supercontinuous. �
Corollary 5.7. If f : X→Y is a δ-continuous function defined on a completely
regular space X, then f is almost z-supercontinuous.
Proof. A δ-continuous function is almost continuous. �
Corollary 5.8. Let f : X→Y be an almost strongly θ-continuous function
defined on a completely regular space X, then f is almost z-supercontinuous.
Proof. An almost strongly θ-continuous function is a δ -continuous function
and hence almost continuous. �
6. separation axioms and almost strongly θ-continuous functions
Definition 6.1 ([2], [10]). A subset S of a space X is said to be θ-set if for
every cover {Uα|α∈Λ} of S by θ-open subsets of X, there exists a finite subset
Λo of Λ such that S⊂∪{Uα|α∈Λo}.
Theorem 6.2. If f : X→Y is almost strongly θ-continuous and A is a θ-set
in X, then f (A) is N -closed relative to Y .
Proof. Let {Uα : α∈Λ} be a cover of f (A) by regular open sets in Y . Since f is
almost strongly θ-continuous, {f −1(Uα) : α∈Λ} is a cover of A, by θ-open sets
in X. Since A is θ-set in X, so A⊂∪{f −1(Uα) : α∈Λo} for some finite subset
Λo of Λ. Thus f (A)⊂∪{Uα : α∈Λo}. Hence f (A) is N -closed relative to Y . �
Corollary 6.3. An almost strongly θ-continuous image of a θ-compact space
is nearly compact.
Corollary 6.4. An almost strongly θ-continuous image of an almost compact
space is nearly compact.
Definition 6.5 ([2], [35]). A topological space X is said to be θ-Hausdorff if
each pair of distinct points are contained in disjoint θ-open sets.
Theorem 6.6. Let f : X→Y be an almost strongly θ-continuous injection into
a Hausdorff space Y . Then X is θ-Hausdorff.
Proof. Let x6=y be two points in X. Since f is one-one, f (x)6=f (y). Since
Y is Hausdorff, there exist disjoint open sets U and V containing f (x) and
f (y), respectively. Now, U∩V = φ which implies that U∩V = φ and so
(U )o∩V = φ which in turn implies that (U )o∩V = φ and thus, (U )o∩(V )o = φ.
Let V1 = (U )
o and V2 = (V )
o, which are regular open sets such that V1∩V2 = φ.
By almost strongly θ-continuity of f, f −1(V1) and f
−1(V2) are disjoint θ-open
sets containing x and y, respectively. Hence X is θ-Hausdorff. �
Definition 6.7. A space X is said to be a δT0-space [17] if for each pair of
distinct points x and y in X there exists a regular open set containing one of
the points x and y but not the other.
Generalizations of Z-supercontinuous functions and Dδ -supercontinuous functions 249
Theorem 6.8. Let f : X→Y be an almost strongly θ-continuous injection into
a δT0-space. Then X is a Hausdorff space.
Proof. Let x1 and x2 be two distinct points in X. Then f (x1) 6= f (x2). Since
Y is a δT0-space, there exists a regular open set V containing one of the points
f (x1) or f (x2) but not the other. To be precise, assume that f (x1)∈V . Since
any union of θ-open sets is θ-open, in view of Proposition 5.1 it follows that
f −1(V ) is a θ-open set containing x1. By Lemma 2.4 there exists an open set
U such that x1∈U ⊂ U ⊂ f
−1(V ). Then U and X\U are disjoint open sets
containing x1 and x2, respectively and so X is Hausdorff. �
Functions with closed graphs are important in functional analysis and several
other areas of mathematics. Several variants of closed graphs occur in literature
(see for example [19], [23]).
Definition 6.9 ([19]). The graph G(f ) of f : X→Y is called θ-closed with
respect to X if for each (x, y) /∈G(f ) there exist open sets U and V containing
x and y, respectively such that (U × V )∩G(f ) = φ.
Definition 6.10 ([19]). The graph G(f ) of f : X→Y is called θ-closed with
respect to X × Y if for each (x, y) /∈G(f ), there exist open sets U and V con-
taining x and y, respectively such that (U × V )∩G(f ) = φ
Definition 6.11. The graph G(f ) of f : X→Y is called uθ-closed with respect
to X × Y if for each (x, y) /∈G(f ), there exist θ-open sets U and V containing
x and y, respectively such that (U × V )∩G(f ) = φ.
Theorem 6.12. Let f : X→Y be a function whose graph is uθ-closed with
respect to X × Y . If K is a θ-set in Y , then f −1(K) is θ-closed in X.
Proof. Let f : X→Y be a function whose graph G(f ) is uθ-closed with respect
to X × Y . Let x∈X\f −1(K). For each y∈K, (x, y) /∈G(f ), there exist θ-open
sets Uy and Vy containing x and y, respectively such that f (Uy)∩Vy = φ. The
family {Vy|y∈K} is a cover of K by θ-open sets of Y . Since K is a θ-set, so
K⊂∪{Vy|y∈Ko} for some finite subset Ko of K. Let U = ∩{Uy|y∈Ko}. Then
U is θ-open set containing x and f (U )∩K = φ which implies that U∩f −1(K) =
φ and hence x /∈(f −1(K))uθ . This shows that f
−1(K) is θ-closed in X. �
Corollary 6.13 ([27]). Let f : X→Y be a function whose graph is θ-closed
with respect to X × Y . If K is quasi H-closed relative to Y , then f −1(K) is
θ-closed in X.
Proof. Since K is quasi H-closed relative to Y , it is a θ-set in Y (see [10]). �
Theorem 6.14. If f : X→Y is an almost strongly θ-continuous function and
Y is a Hausdorff space, then G(f ), the graph of f is θ-closed with respect to
X × Y .
Proof. Let x∈X and let y 6=f (x). Since Y is Hausdorff, there exist disjoint
open sets V and W containing y and f (x), respectively. So V and (W )o are
disjoint sets containing y and f (x), respectively. Since f is almost strongly
250 J. K. Kohli, D. Singh and R. Kumar
θ-continuous, so there is an open set U containing x such that f (U )⊂(W )o.
Then f (U )⊂(W )o⊂Y \V . Consequently, U × V contains no point of G(f ).
Hence G(f ) is θ-closed with respect to X × Y . �
Corollary 6.15. If f : X→Y is an almost strongly θ-continuous function and
Y is Hausdorff, then G(f ), the graph of f is θ-closed with respect to X.
Theorem 6.16. If f : X→Y is an almost strongly θ-continuous function and
Y is an almost regular Hausdorff space, then G(f ), the graph of f is uθ-closed
with respect to X × Y .
Proof. Let x∈X and let y 6=f (x). Since Y is Hausdorff, there exist disjoint
open sets V1 and W1 containing y and f (x), respectively. Thus, there exist
disjoint regular open sets V = (V 1)
o and W = (W 1)
o containing y and f (x),
respectively. Since f is almost strongly θ-continuous, by Proposition 5.1, there
exists a θ-open set U containing x such that f (U )⊂W and so f (U )⊂W ⊂Y \V .
Thus U ×V contains no point of G(f ). Since Y is almost regular, V is a θ-open
set. Thus U × V is a θ-open set and (U × V )∩G(f ) = φ. Hence G(f ) is uθ
-closed with respect to X × Y . �
References
[1] D. Carnahan, Locally nearly compact spaces, Boll. Mat. Un. Ital. 6, no. 4 (1972), 146–
153.
[2] A. K. Das, A note on θ-Hausdorff spaces, Bull. Cal. Math. Soc. 97, no. 1(2005), 15–20.
[3] J. Dontchev, M. Ganster and I. Reilly, More on almost s-continuity, Indian J. Math. 41
(1999), 139–146.
[4] E. Ekici, Generalization of perfectly continuous, regular set-connected and clopen func-
tions, Acta. Math. Hungar. 107, no. 3 (2005), 193–206.
[5] Z-Frolik, Generalizations of compact and Lindelöf spaces, Czechoslovak Math J. 13, no.
84 (1959), 172–217 (Russian) MR 21 # 3821.
[6] N. C. Heldermann, Developability and some new regularity axioms, Can. J. Math. 33,
no. 3(1981), 641–668.
[7] E. Hewitt, On two problems of Urysohn, Ann. of Math. 47, no.3 (1946), 503–509.
[8] J. K. Kohli and A. K. Das, New normality axioms and decompositions of normality,
Glasnik Mat. 37, no. 57 (2002), 105–114.
[9] J. K. Kohli and A. K. Das, On functionally θ-normal spaces, Applied General Topology
6, no. 1 (2005), 1–14.
[10] J. K. Kohli and A. K. Das, A class of spaces containing all generalized absolutely closed
(almost compact) spaces, Applied General Topology 7, no. 2 (2006), 233–244.
[11] J. K. Kohli, A. K. Das and R. Kumar, Weakly functionally θ-normal spaces, θ-shrinking
of covers and partition of unity, Note di Matematica 19 (1999), 293–297.
[12] J. K. Kohli and R. Kumar, z-supercontinuous functions, Indian J. Pure Appl. Math. 33,
no. 7 (2002), 1097–1108.
[13] J. K. Kohli and D. Singh, Dδ-supercontinuous functions, Indian J.Pure Appl. Math. 34,
no. 7 (2003), 1089–1100.
[14] J. K. Kohli and D. Singh, Between compactness and quasicompactness, Acta Math.
Hungar. 106, no. 4 (2005), 317–329.
[15] J. K. Kohli and D. Singh, Between weak continuity and set connectedness, Studii Si
Cercetari Stintifice Seria Mathematica 15 (2005), 55–65.
[16] J. K. Kohli and D. Singh, Between regularity and complete regularity and a factorization
of complete regularity, Studii Si Cercetari Stintifice Seria Mathematica, 17 (2007), 125–
134.
Generalizations of Z-supercontinuous functions and Dδ -supercontinuous functions 251
[17] J. K. Kohli and D. Singh, Almost cl-supercontinuous functions, Applied General Topol-
ogy, to appear.
[18] N. Levine, Strong continuity in topological spaces, Amer. Math. Monthly 67 (1960), 269.
[19] P. E. Long and L. Herrington, Strongly θ-continuous functions, J. Korean Math. Soc. 8
(1981), 21–28.
[20] P. E. Long and L. Herrington, The Tθ-topology and faintly continuous functions, Kyung-
pook Math. J. 22 (1982), 7–14.
[21] J. Mack, Countable paracompactness and weak normality properties, Trans. Amer. Math.
Soc. 148 (1970), 265–272.
[22] B. M. Munshi and D. S. Bassan, Supercontinuous mappings, Indian J. Pure Appl. Math.
13 (1982), 229–236.
[23] T. Noiri, On functions with strongly closed graph, Acta Math. Hungar. 32 (1978), 1–4.
[24] T. Noiri, On δ-continuous functions, J. Korean Math. Soc. 18 (1980), 161–166.
[25] T. Noiri, Supercontinuity and some strong forms of continuity, Indian J. Pure. Appl.
Math. 15, no. 3 (1984), 241–250.
[26] T. Noiri, Strong forms of continuity in topological spaces, Suppl. Rendiconti Circ. Mat.
Palermo, II 12 (1986), 107–113.
[27] T. Noiri and S. M. Kang, On almost strongly θ-continuous functions, Indian J. Pure
Appl. Math. 15, no. 1 (1984), 1–8.
[28] J. R. Porter and J. Thomas, On H-closed spaces and minimal Hausdorff spaces, Trans.
Amer. Math. Soc. 138 (1969), 159–170.
[29] I. L. Reilly and M. K. Vamanamurthy, On super-continuous mappings, Indian J. Pure.
Appl. Math. 14, no. 6 (1983), 767–772.
[30] M. K. Singal and S. P. Arya, On almost regular spaces, Glasnik Mat. 4, no. 24 (1969),
89–99.
[31] M. K. Singal and S. P. Arya, On almost normal and almost completely regular spaces,
Glasnik Mat. 5, no. 25 (1970), 141–152.
[32] M. K. Singal and A. Mathur, On nearly compact spaces, Boll. Un. Mat. Ital. 2, no. 4
(1969), 702–710.
[33] M. K. Singal and A. R. Singal, Almost continuous mappings, Yokohama Math. J. 16
(1968), 63–73.
[34] D. Singh, cl-supercontinuous functions, Applied General Topology 8, no. 2 (2007), 293–
300.
[35] S. Sinharoy and S. Bandyopadhyay, On θ-completely regular spaces and locally θ − H-
closed spaces, Bull. Cal. Math. Soc. 87 (1995), 19–28.
[36] N. K. Veličko, H-closed topological spaces, Amer. Math. Soc. Transl. 78, no. 2 (1968),
103–118.
Received May 2007
Accepted January 2008
J. K. Kohli (jk kohli@yahoo.com)
Dep. of Mathematics, Hindu College, University of Delhi, Delhi 110 007, India
D. Singh (dstopology@rediffmail.com)
Dep. of Mathematics, Sri Aurobindo College, University of Delhi-South Campus,
Delhi 110 017, India
Rajesh Kumar (rkumar2704@yahoo.co.in)
Dep. of Mathematics, Rajdhani College, University of Delhi, Delhi 110 015, India