Download JK Kohli, D. Singh, J. Aggarwal R-SUPERCONTINUOUS

DEMONSTRATIO MATHEMATICA
Vol. XLIII
No 3
2010
J. K. Kohli, D. Singh, J. Aggarwal
R-SUPERCONTINUOUS FUNCTIONS
Abstract. A new strong variant of continuity called ‘R-supercontinuity’ is introduced. Basic properties of R-supercontinuous functions are studied and their place in the
hierarchy of strong variants of continuity that already exist in the literature is elaborated.
It is shown that R-supercontinuity is preserved under the restriction, shrinking and expansion of range, composition of functions, products and the passage to graph function. The
class of R-supercontinuous functions properly contains each of the classes of (i) strongly
θ-continuous functions introduced by Noiri and also studied by Long and Herrington;
(ii) D-supercontinuous functions; and (iii) F -supercontinuous functions; and so include
all z-supercontinuous functions and hence all clopen maps (≡ cl-supercontinuous functions) introduced by Reilly and Vamnamurthy, perfectly continuous functions defined by
Noiri and strongly continuous functions due to Levine. Moreover, the notion of r-quotient
topology is introduced and its interrelations with the usual quotient topology and other
variants of quotient topology in the literature are discussed. Retopologization of the domain of a function satisfying a strong variant of continuity is considered and interrelations
among various coarser topologies so obtained are observed.
1. Introduction
Strong variants of continuity arise in diverse situations in mathematics and applications of mathematics and are interspersed throughout the
literature. In many situations in topology, analysis and in many other
branches of mathematics and its applications, continuity is not enough and
a strong form of continuity is required to meet the demand of a particular
situation. Hence it is of considerable significance both from applications
viewpoint and intrinsic considerations to formulate and study new strong
variants of continuity. In this paper, we introduce one such strong variant
of continuity called ‘R-supercontinuity’ and study the basic properties of
R-supercontinuous functions and elaborate on their place in the hierarchy
Key words and phrases: strongly θ-continuous function, supercontinuous function,
D-supercontinuous function, F -supercontinuous function, R0 -space, r-quotient topology,
δ-quotient topology, θ-quotient topology, r-open set.
2000 Mathematics Subject Classification: 54C08, 54C10, 54D10, 54D20.
This research was partially supported by University Grants Commision, India.
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J. K. Kohli, D. Singh, J. Aggarwal
of strong variants of continuity that already exist in the literature. The
class of R-supercontinuous functions properly includes each of the classes
of D-supercontinuous functions [15], strongly θ-continuous functions introduced by Noiri [28] and F -supercontinuous functions [18]. Thus, in turn the
class of R-supercontinuous functions contain, all z-supercontinuous functions [14], Dδ -supercontinuous functions [16], cl-supercontinuous functions
[33] (≡ clopen maps [30]) initiated by Reilly and Vamanamurthy, perfectly
continuous functions defined by Noiri [29] and strongly continuous functions
due to Levine [22].
Section 2 is devoted to basic definitions and preliminaries. In Section 3,
we introduce the notion of ‘R-supercontinuous function’ and elaborate on its
place in the hierarchy of strong variants of continuity that already exist in
the mathematical literature. We study basic properties of R-supercontinuous
functions in Section 4. It is shown that (i) R-supercontinuity is invariant under restrictions, shrinking and expansion of range and composition
of functions; (ii) a function into a product space is R-supercontinuous if
and only if its composition with each projection map is R-supercontinuous;
and (iii) the graph function g of f is R-supercontinuous if and only if f
is R-supercontinuous and X is an R0 -space. The interplay between Rsupercontinuity and topological properties is considered in Section 5. In
Section 6, properties of graph of an R-supercontinuous function are discussed. The notion of r-quotient topology is introduced in Section 7 and
its interrelations with the standard quotient topology and other variants of
quotient topology in the literature are observed. In Section 8, we consider
the retopologization of the domain of an R-supercontinuous function (or satisfying some other strong form of continuity) in such a way that it is simply
a continuous function. We discuss the interrelations and interplay among
various coarser topologies obtained in this way and substantiate with examples to show that all these coarser topologies in general are distinct. We
conclude the section with alternative proofs of certain results of preceding
sections. Finally an appendix is added in which we include the basics for the
category UPS (= the category whose objects are complete lattices equipped
with Scott topologies and morphisms are Scott continuous maps).
2. Basic definitions and preliminaries
A collection β of subsets of space X is called an open complementary
system [11] if β consists of
S open sets such that for every B ∈ β, there exist
B1 , B2 , . . . ∈ β with B = {X \Bi : i ∈ N}. A subset U of a space X is called
strongly open Fσ -set [11] if there exists a countable open complimentary
system β(U ) with U ∈ β(U ). The complement of a strongly open Fσ -set is
referred to as a strongly closed Gδ -set. A subset H of a space X is called a
R-supercontinuous functions
705
regular Gδ -set [26] if H is an intersection
of a sequence of closed sets whose
T
T
∞
o =
o
interiors contains H, i.e. H = ∞
F
n=1 n
n=1 Fn , where each Fn is a closed
subset of X. The complement of a regular Gδ -set is called a regular Fσ -set.
An open set U of a space X is said to be F -open [18] if for each x ∈ X there
exists a zero set Z in X such that x ∈ Z ⊂ U , equivalently if U is expressible
as a union of zero sets in X. A subset A of a space X is said to be regularly
0
open if it is the interior of its closure, i.e., A = A . A point x ∈ X is called
a θ-adherent point [36] of a set A ⊂ X if every closed neighbourhood of x
intersects A. Let clθ A denote the set of all θ-adherent points of A. The
set A is called θ-closed if A = clθ A. The complement of a θ-closed set is
referred to as θ-open. It is easily verified that a set is θ-open if and only if
it contains a closed neighbourhood of each of its points. A topological space
X is said to be an Alexandroff space [1] if arbitrary intersection of open sets
in X is open. Alexandroff spaces are referred to as saturated spaces in [25].
Several weak and strong variants of continuity occur in the literature.
Here we include definitions of only those strong variants of continuity which
will be dealt with in this paper.
Definitions 2.1. A function f : X → Y from a topological space X into
a topological space Y is said to be
(a) strongly continuous [22] if f (A) ⊂ f (A) for each subset A of X,
(b) perfectly continuous [29] if f −1 (V ) is clopen in X for every open set
V ⊂Y,
(c) cl-supercontinuous [33] (≡ clopen continuous [30]) if for each x ∈ X and
each open set V containing f (x) there is a clopen set U containing x
such that f (U ) ⊂ V ,
(d) z-supercontinuous [14] 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 ,
(e) Dδ -supercontinuous [16] if for each x ∈ X and for each open set V
containing f (x), there exists a regular Fσ -set U containing x such that
f (U ) ⊂ V ,
(f) D-supercontinuous [15] if for each x ∈ X and each open set V containing
f (x) there exists an open Fσ -set V containing x such that f (U ) ⊂ V ,
(g) D ∗ -supercontinuous [32] if for each x ∈ X and each open set V containing f (x) there exists a strongly open Fσ -set V containing x such that
f (U ) ⊂ V ,
(h) strongly θ-continuous [28] 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 ,
(i) F -supercontinuous [18] if for each x ∈ X and each open set V containing
f (x) there exists an F -open set U containing x such that f (U ) ⊂ V ,
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J. K. Kohli, D. Singh, J. Aggarwal
(j) supercontinuous [27] if for each x ∈ X and for each open set V containing
f (x), there exists a regular open set U containing x such that f (U ) ⊂ V .
Definitions 2.2. A topological space X is said to be
(1) an R0 -space ([7], [31]) if for each open set U in X, x ∈ U implies {x} ⊂ U ,
(2) an R1 -space [7] if x, y ∈ X, x 6∈ {y}, then x and y are contained in
disjoint open sets,
(3) a functionally regular space ([2], [4]) if for each closed set A in X and
a point x 6∈ A there exists a continuous real valued function f defined
on X such that f (x) 6∈ f (A); equivalently for each x ∈ X and for each
open set U containing x there exists a zero set Z such that x ∈ Z ⊂ U ,
(4) D-regular space [11] if it has a base of open Fσ - sets,
(5) functionally Hausdorff if for x, y ∈ X, x 6= y there exists a continuous
function f : X → [0, 1] such that f (x) 6= f (y),
(6) semiregular if it has a base of regular open sets.
3. R-supercontinuous functions
An open subset A of a space X is said to be r-open if it is expressible as
a union of closed sets. The complement of an r-open set will be referred to
as r-closed.
It is easily verified that every θ-open set is r-open and an r-open set is
open. However, reverse implications need not be true as is well exhibited by
the following examples.
Example 3.1. Let X denote the space of Smirnov’s deleted sequence
topology [35, p. 86]. Then (−ǫ, ǫ) \ A, where ǫ > 0 and A = { n1 : n ∈ N}, is
an r-open set which is not θ-open.
Example 3.2. Let X = {a, b} denote the two point Sierpinski space with
a as an isolated point. Then {a} is an open set in X which is not r-open.
Similarly if X is the real line R endowed with the right order topology [35,
p. 74], then any ray (a, ∞) (a ∈ R) is an open set in X which is not r-open.
Moreover, if X is an R0 -space; in particular if X is a T1 -space or a
Hausdorff space, then every open set in X is r-open.
Definition 3.3. A function f : X → Y from a topological space X into
topological space Y is said to be R-supercontinuous at a point x ∈ X, if for
each open set V containing f (x) there exists an r-open set U containing x
such that f (U ) ⊂ V . The function f is said to be R-supercontinuous, if it is
R-supercontinuous at each x ∈ X.
The following diagram well illustrates the interrelations that exist between R-supercontinuity and other strong variants of continuity that already exist in the literature and well reflects upon the place enjoyed by
R-supercontinuous functions
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R-supercontinuity in this hierarchy. The following implications are either
well known or follow from definitions.
However, none of the above implications in general is reversible, which is
either well known or follow from the following observations and examples.
Observations and Examples
3.4. If either X or Y is an R0 -space, then every continuous function f : X →
Y is R-supercontinuous. Thus every morphism in a subcategory of TOP
(≡ the category of topological spaces and continuous maps) which is a
subcategory of the subcategory of R0 -spaces is R-supercontinuous. On
the other hand there exist subcategories of TOP in which no morphism
is R-supercontinuous except the constant ones. One such subcategory
of TOP is UPS the category whose objects are complete lattices endowed with Scott topology and whose morphisms are Scott-continuous
maps1 .
3.5. Let X = Y = R be the set of all real numbers with the right ray
topology. Then X is not an R0 -space. The identity function f defined
on X is continuous but not R-supercontinuous. Similarly, if X denotes
a non-discrete Alexandroff space, then the identity mapping defined on
X is continuous but not R-supercontinuous.
3.6. Let X = {(x, y)|y ≥ 0, x, y ∈ Q} and fix some irrational number θ.
The irrational slope topology [35, p. 93] τ on X is generated by ǫneighbourhoods of the form Nǫ ((x, y)) = {(x, y)} ∪ Bǫ (x + y/θ) ∪
Bǫ (x − y/θ), where Bǫ (ξ) = {r ∈ Q : |r − ξ| < ǫ}, Q being the rationals on x-axis. Then (X, τ ) is a Hausdorff space which is not func1
For the definition of Scott topology and Scott continuous maps we refer the interested
reader to [10] or Appendix (Section 9).
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J. K. Kohli, D. Singh, J. Aggarwal
tionally Hausdorff and hence not a functionally regular space. The
identity mapping defined on X is R-supercontinuous but not F -supercontinuous.
3.7. Let (X, τ ) be the topological space formed by adding to the ordinary
closed unit interval [0, 1] another right end point say 1∗ , with the sets
(a, 1) ∪ {1∗ } as a local neighbourhood basis. This topology is known
as Telophase topology [35, p. 92]. Then X is a T1 -space and hence
an R0 -space. The space X is not a semiregular space because the
points 1 and 1∗ do not have a base of regular open sets. The identity
function f defined on X is R-supercontinuous but not supercontinuous.
3.8. Let X = {a, b, c, d} be equipped with topology τ = {φ, X, {a, b}, {d},
{a, b, d}} and let f denote the identity function defined on X. Then f
is a supercontinuous function but not an R-supercontinuous function.
Thus in view of Example 3.7 supercontinuity and R-supercontinuity are
independent notions.
3.9. Let X be the product of the mountain chain space due to Heldermann [11] and N ≡ the set of positive integers) equipped with cofinite topology. The space X is a T1 -space which is neither a regular
space nor a D-regular space. The identity function defined on X is an
R-supercontinuous function but neither a D-supercontinuous function
nor a strongly θ-continuous function.
4. Basic properties of R-supercontinuous functions
Theorem 4.1. For a function f : X → Y from a topological space X into
a topological space Y , the following statements are equivalent
(i)
(ii)
(iii)
(iv)
f is R-supercontinuous.
f − (V ) is r-open for every open set V ⊂ Y .
f −1 (F ) is r-closed for every closed set F ⊂ Y .
f −1 (S) is r-open for every subbasic open set S in Y .
Proof. Easy.
Definition 4.2. Let X be a topological space and let A ⊂ X. A point
x ∈ X is said to be an r-adherent point of A if every r-open set containing
x intersects A. Let AR denote the set of all r-adherent points of the set A.
Then A ⊂ AR and a set A is r-closed if and only if A = AR .
Clearly every adherent point of a set A is an r-adherent point of A and
every r-adherent point of the set A is a θ-adherent point of A. However,
in general the reverse implications are not true as is shown by the following
examples.
R-supercontinuous functions
709
Example 4.3. Let X denotes the two point Sierpinski space considered in
Example 3.2. Then the point a is an r-adherent point of the set {b} but not
an adherent point. Similarly, if X denotes the real line endowed with right
order topology discussed in Example 3.2, then 2 is an r-adherent point of
the singleton {1} but not an adherent point. Infact in this case every point
of X \ {1} is an r-adherent point of the set {1}.
Example 4.4. Let X denote the space of Smirnov’s deleted sequence
topology
in Example 3.1. Then 0 is a θ-adherent point of the set
considered
A = n1 : n ∈ N but not an r-adherent point of A.
Theorem 4.5. For a function f : X → Y , the following statements are
equivalent.
(a) f is R-supercontinuous.
(b) f (AR ) ⊂ f (A) for every set A ⊂ X.
(c) (f −1 (B))R ⊂ f −1 (B) for every set B ⊂ Y .
Proof. (a)⇒(b). Since f (A) is closed in Y , by Theorem 4.1 f −1 (f (A)) is
an r-closed set in X. Again since A ⊂ f −1 (f (A)), AR ⊂ [f −1 (f (A))]R =
f −1 (f (A)) and so f (AR ) ⊂ f (A).
(b)⇒(c). Let B ⊂ Y . Then f ((f −1 (B))R ) ⊂ f (f −1 (B)) ⊂ B and hence
−1
(f (B))R ⊂ f −1 (B).
(c)⇒(a). Let C be any closed subset of Y . Then (f −1 (C))R ⊂ f −1 (C) =
−1
f (C). Again f −1 (C) ⊂ f −1 (C) ⊂ (f −1 (C))R , and hence f −1 (C) =
(f −1 (C))R . Thus f −1 (C) is r-closed and so f is R-supercontinuous.
Definition 4.6. A filter base B is said to R-converge to a point x ∈ X
R
(written as B → x) if every r-open set containing x contains a member of B.
Theorem 4.7. A function f : X → Y is R-supercontinuous if and only if
f (B) → f (x), for each x and for each filter base B in X that R-converges
to x.
R
Proof. Assume that f is R-supercontinuous and let B → x. Let V be an
open set in Y containing f (x). By Theorem 4.1, f −1 (V ) is an r-open set
R
containing x. Since B → x, there exists B ∈ B such that x ∈ B ⊂ f −1 (V )
and so f (B) ⊂ V . This shows that f (B) → f (x).
Conversely, let V be an open subset of Y containing f (x). Let B be the
filter generated by the filter base Nx consisting of all r-open subsets of X
containing x. By hypothesis, f (B) → f (x) and so there exists a member
f (N ) of f (Nx ) such that f (N ) ⊂ V . Choose B ∈ Nx such that B ⊂ N .
Since B is an r-open set containing x and f (B) ⊂ f (N ) ⊂ V, f is an Rsupercontinuous function.
710
J. K. Kohli, D. Singh, J. Aggarwal
Theorem 4.8. If f : X → Y is R-supercontinuous and g : Y → Z
is a continuous function, then the composition g ◦ f is R-supercontinuous.
In particular, composition of two R-supercontinuous functions is R-supercontinuous.
Proof. Let W be an open set in Z. Then g −1 (W ) is an open set in Y . By
R-supercontinuity of f, f −1 (g −1 (W )) = (g ◦ f )−1 (W ) is r-open in X and so
g ◦ f is R-supercontinuous.
Remark 4.9. However, R-supercontinuity of g ◦ f need not imply even
continuity of f . For, let, X = Y = Z = R, where R denotes the set of all real
numbers. Let X, Y and Z be endowed with cocompact topology τ , usual
topology u and cofinite topology τ ∗ , respectively. Let f : (X, τ ) → (Y, τ )
and g : (Y, u) → (Z, τ ∗ ) denote the identity functions. Then g ◦ f and g are
R-supercontinuous but f is not even a continuous function.
Definition 4.10. A function f : X → Y is said to be
(1) R-open (R-closed ) if f (A) is open (closed) in Y for every r-open (rclosed) set A in X.
(2) θ-open if f (A) is open in Y for every θ-open set A in X.
Every open function is R-open and every R-open function is θ-open.
However, reverse implications are not true as is well exhibited by the following examples.
Example 4.11. Let X = {a, b, c} be endowed with topology τ = {φ, X,
{a, b}, {b, c}, {c}, {b}} and let Y = {x, y, z} be equipped with topology τ ∗ =
{φ, Y, {x, y}, {z}}. Let f : (X, τ ) → (Y, τ ∗ ) be defined by f (a) = x, f (b) = y,
f (x) = z. Then f is an R-open function but not an open function.
Example 4.12. Let X denote the space of Smirnov’s deleted sequence
topology considered in Example 3.1. Let Y = {0, 1, 2} be endowed with the
topology τ = {φ, Y, {1}, {0, 1}}. Define a function f : X → Y by


/ (A ∪ {0}),
1, if x ∈
f (x) = 0, if x ∈ A,


2, if x = 0.
It is easily verified that f is a θ-open function but not an R-open function.
Theorem 4.13. Let f : X → Y be an R-open, R-supercontinuous surjection and let g : Y → Z be any function. Then g ◦ f is R-supercontinuous if
and only if g is continuous. Further, if in addition f maps r-open (r-closed)
sets to r-open (r-closed) sets, then g is R-supercontinuous.
Proof. Sufficiency is immediate in view of Theorem 4.8. To prove necessity, suppose that g ◦ f is R-supercontinuous. To show that g is con-
R-supercontinuous functions
711
tinuous, let W be an open (closed) subset of Z. Then (g ◦ f )−1 (W ) =
f −1 (g −1 (W )) is r-open (r-closed) in X. Since f is an R-open (R-closed)
surjection f (f −1 (g −1 (W ))) = g −1 (W ) is an open (closed) set in Y . The last
assertion is immediate in view of Theorem 4.1.
Theorem 4.14. Let f : X → Y be a function. Then the following statements are true.
(a) If f is R-supercontinuous and if A is a subspace of X, then the restriction
function f |A
S : A → X is R-supercontinuous.
Uα , where each Uα is an r-open subset of X. If for each
(b) Let X =
α∈Λ
α ∈ Λ, fα = f |Uα is R-supercontinuous, then f is R-supercontinuous.
n
S
Fi , where each Fi is an r-closed subset of X. If for each
(c) Let X =
i=1
i = 1, . . . , n, fi = f |Ui is R-supercontinuous, then f is R-supercontinuous.
Proof. (a) Let V be any open set in Y . Since f is R-supercontinuous.
f −1 (V ) is an r-open set. Now, since (f |A)−1 = f −1 (V ) ∩ A, it is r-open in
A and so f |A is R-supercontinuous.
(b) Let V be an open subset of Y . Then f −1 (V ) = ∪{fα−1 (V ) : α ∈ Λ}.
Now, R-supercontinuity of fα implies that fα−1 (V ) is r-open in Uα for each
α and hence in X. Since arbitrary union of r-open sets is r-open, f −1 (V ) is
r-open.
n
S
(c) Let F be any closed subset of Y . Then f −1 (F ) =
fi−1 (F ). By
i=1
R-supercontinuity of fi (i = 1, 2, . . . , n), fi−1 (F ) is r-closed in Fi and hence
in X. Now f −1 (F ) being a finite union of r-closed sets is r-closed set. So f
is R-supercontinuous.
Our next result shows that R-supercontinuity is preserved under the
shrinking of its range.
Theorem 4.15. If f : X → Y is R-supercontinuous and f (X) is endowed
with the subspace topology, then f : X → f (X) is R-supercontinuous.
Proof. Since f is R-supercontinuous, for every open subset V of Y, f −1 (V ∩
f (X)) = f −1 (V ) ∩ f −1 (f (X)) = f −1 (V ) ∩ X = f −1 (V ) is r-open. So
f : X → f (X) is R-supercontinuous.
In contrast to Theorem 4.15, it is easily verified that R-supercontinuity
is invariant under the expansion of its range.
Q
Theorem 4.16. Let f : X → α∈Λ Xα be defined byQf (x) = (fα (x))α∈Λ ,
where fα : X → Xα is a function for each α ∈ Λ. Let α∈Λ Xα be endowed
with the product topology. Then f is R-supercontinuous if and only if each
fα is R-supercontinuous.
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J. K. Kohli, D. Singh, J. Aggarwal
Q
Proof. Suppose that the mapping
Q f : X → α∈Λ Xα is R-supercontinuous.
Then fα = πα ◦ f , where πα : α∈Λ Xα → Xα denotes the projection onto
the α-coordinate space Xα . Since projection maps are continuous, in view
of Theorem 4.8, fα is a R-supercontinuous for each α.
Conversely, assume that each fα : X → Xα is R-supercontinuous. To
prove R-supercontinuity of f it suffices to show
Qthat inverse image under f
of
α is r-open.
Q each subbasic open set in the product
Q space α∈Λ X−1
Q Let Uβ ×
X
be
a
subbasic
open
set
in
U
×
X
.
Then
f
α
α
β
α6=β
α6=β Xα =
α∈Λ
f −1 (πβ−1 (Uβ )) = fβ−1 (Uβ ) is r-open in X and so, f is R-supercontinuous.
Q
Q
Theorem 4.17. Let f : α∈Λ Xα → α∈Λ Yα be a mapping defined by
f ((xα )) = (fα (xα )), where fα : Xα → Yα for each α ∈ Λ. Then f is
R-supercontinuous if and only if each fα is R-supercontinuous.
Proof. To prove necessity, let Vβ be any open set in Yβ . Then πβ−1 (Vβ )
Q
Q
= Vβ × α6=β Yα is a subbasic open set in α∈Λ Yα . Now since f is R-superQ
Q
continuous, f −1 (πβ−1 (Vβ )) = fβ−1 (Vβ ) × α6=β Xα is r-open in α∈Λ Xα .
Thus fβ−1 (Vβ ) is an r-open set in Xβ and so fβ is R-supercontinuous.
Q
Conversely, let V = Vβ × α6=β Yα be a subbasic open set in the product
Q
Q
space
Yα . Then f −1 (V ) = f −1 Vβ × α6=β Yα . Now since each fα is
R-supercontinuous, f −1 (V ) is r-open and hence f is R-supercontinuous.
Theorem 4.18. Let f : X → Y be a function and g : X → X × Y
be the graph function defined by g(x) = (x, f (x)) for each x ∈ X. Then
g is R-supercontinuous if and only if f is R-supercontinuous and X is an
R0 -space.
Proof. Suppose that g is R-supercontinuous. By Theorem 4.8, f = πY ◦ g
is R-supercontinuous, where πY : X × Y → Y denotes the projection map.
To show that X is an R0 -space, let U be an open set in X and let x ∈ U .
Then U × Y is an open set containing g(x). By R-supercontinuity of g, there
is an r-open set W containing x such that g(W ) ⊂ U × Y . So W ⊂ U .
Thus U being a union of r-open sets is r-open in X and hence X is an
R0 -space.
Conversely, to prove R-supercontinuity of g, let x ∈ X and let W be an
open set containing g(x). Then there exist open sets U ⊂ X and V ⊂ Y such
that g(x) = (x, f (x)) ⊂ U × V ⊂ W . Since X is an R0 -space, there exists
an r-open set G1 ⊂ X such that x ∈ G1 ⊂ U . Again by R-supercontinuity
of f , there exists an r-open set G2 in X containing x such that f (G2 ) ⊂ V .
Let G = G1 ∩ G2 . Then G is an r-open set containing x such that g(G) ⊂
U × V ⊂ W , which shows that g is R-supercontinuous.
R-supercontinuous functions
713
Remark 4.19. The hypothesis of ‘R0 -space’ in Theorem 4.18 cannot be
omitted. For let X be the real line with the right ray topology [35] and Y
be the real line with indiscrete topology. Let f : X → Y be any function.
Clearly f is R-supercontinuous but the graph function g : X → X × Y is
not R-supercontinuous.
It is well known that the equalizer of two continuous functions into a
Hausdorff space is closed. In case of R-supercontinuous functions we have
the following strong version of the same.
Theorem 4.20. Let f, g : X → Y be R-supercontinuous functions from a
topological space X into a Hausdorff space Y . Then the equalizer E = {x ∈
X : f (x) = g(x)} of the function f and g is an r-closed subset of X.
Proof. To prove that E is r-closed, we shall show that its complement
X \ E is r-open. To this end, let x ∈ X \ E. Then f (x) 6= g(x). In view
of Hausdorffness of Y , there exist disjoint open sets V and W such that
f (x) ∈ V and g(x) ∈ W . Since f and g are R-supercontinuous, f −1 (V ) and
g −1 (W ) are r-open sets containing x. Then U = f −1 (V ) ∩ g −1 (W ) is an
r-open set containing x such that U ∩ E = φ. So X \ E is r-open.
A topological space X is said to be R-compact [19] if every cover of X
by r-open sets has a finite subcover.
Theorem 4.21. [19] If f : X → Y is R-supercontinuous from an Rcompact space X onto a space Y , then Y is compact.
5. Topological properties and R-supercontinuity
Theorem 5.1. Let f : X → Y be an R-supercontinuous open bijection.
Then X and Y are homeomorphic R0 - spaces.
Proof. Let U be an open set in X and let x ∈ X. Since f is an open map,
f (U ) is an open set containing f (x). Now since f is R-supercontinuous,
there exists an r-open set G containing x such that f (G) ⊂ f (U ). Now,
x ∈ f −1 (f (G)) ⊂ f −1 (f (U )). Since f is a bijection, f −1 (f (G)) = G and
f −1 (f (U )) = U . So x ∈ G ⊂ U . Thus U being a union of r-open sets
is r-open and hence X is an R0 -space. Since f is a homeormorphism and
since the property of being an R0 -space is a topological property, Y is an
R0 -space.
Theorem 5.2. Let f : X → Y be an R-supercontinuous injection into a
T0 -space Y . Then X is a T1 -space. Further, if X is an R1 -space, then X is
Hausdorff.
Proof. Let x1 , x2 ∈ X, x1 6= x2 . Then f (x1 ) 6= f (x2 ). Since Y is a T0 -space,
there exists an open set V containing either f (x1 ) or f (x2 ) but not both.
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J. K. Kohli, D. Singh, J. Aggarwal
To be precise, suppose that f (x1 ) ∈ V . Then f −1 (V ) is an r-open set
containing x1 but not x2 . Now x1 ∈ {x1 } ⊂ f −1 (V ) and so X − {x1 } is an
open set containing x2 but not x1 . Thus X is a T1 -space. The last assertion
is immediate in view of the fact that a T0 , R1 -space is Hausdorff [7].
Definition 5.3. [13] A space X is said to be
(i) r-regular if for every r-closed set A and a point x outside A, there exist
open sets U and V in X such that x ∈ U , A ⊂ V and U ∩ V = φ.
(ii) r-completely regular if for every r-closed set A and point x outside A
there exists a continuous function f : X → [0, 1] such that f (x) = 0
and f (A) = 1.
Theorem 5.4. Let X be an r-regular space and let f : X → Y be an
R-supercontinuous open bijection. Then both X and Y are regular spaces.
Proof. Let A be a closed subset of Y and let y 6∈ A. Since f is a bijection, f −1 (y) is a singleton and f −1 (y) 6∈ f −1 (A). Again, since f is Rsupercontinuous f −1 (A) is an r-closed set in X. In view of r-regularity
of X, there exist disjoint open sets U and V such that f −1 (y) ∈ U and
f −1 (A) ⊂ V . Now, since f is an open bijection, f (U ) and f (V ) are disjoint
open sets containing y and A, respectively and so Y is regular. Since f is
a homeomorphism and since regularity is a topological property, X is also
regular.
Theorem 5.5. If f : X → Y is an R-supercontinuous function and if X
is an r-regular space, then f is strongly θ-continuous.
Proof. Let x ∈ X and V be an open set containing f (x). Since f is
R-supercontinuous, there exists an r-open set U containing x such that
f (U ) ⊂ V . Again, since X is an r-regular space, there exists an open set W
such that x ∈ W ⊂ W ⊂ U . Now f (W ) ⊂ f (U ) ⊂ V and so f is strongly
θ-continuous.
Definition 5.6. A function f : X → Y is said to be an R-homeomorphism, if f is a bijection such that f and f −1 are R-supercontinuous.
Theorem 5.7. Let f : X → Y be an R-homeomorphism from an rcompletely regular space X onto Y . Then both the spaces X and Y are
completely regular.
Proof. Let A be a closed set in Y and let y 6∈ A. Then x = f −1 (y) is a
singleton and x does not belongs to the r-closed set f −1 (A). In view of the
r-complete regularity of X, there exists a continuous function g : X → [0, 1]
such that g(x) = 0 and g(f −1 (A)) = 1. Let h = g ◦ f −1 . Since f is an
R-homeomorphism, h is well defined and since f −1 is R-supercontinuous,
R-supercontinuous functions
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h is continuous. Moreover, h(y) = 0 and h(A) = 1. Thus Y is a completely regular space. Now, since f is a homeomorphism and since complete
regularity is a topological invariant, the space X is also completely regular.
6. Properties of graph of an R-supercontinuous function
Let f : X → Y be an R-supercontinuous function. Since every Rsupercontinuous function is continuous, the family {1X , f }, where 1X denotes the identity map on X, separates points and separates points from
closed sets. So the mapping g : X → X × Y defined by g(x) = (x, f (x))
is an embedding of X into X × Y . Thus X is homeomorphic to its graph
G(f ) = g(X). Hence every topological property enjoyed by X is transferred
to its graph G(f ) and conversely.
The next two definitions shed some light on the fact that how the graph
of an R-supercontinuous function f : X → Y is situated in the product space
X ×Y.
Definition 6.1. The graph G(f ) of f : X → Y is called
(i) r-closed with respect to X if for each (x, y) 6∈ G(f ) there exist open sets
U and V containing x and y respectively, such that U is r-open and
(U × V ) ∩ G(f ) = φ; and
(ii) r-closed with respect to X × Y if for each (x, y) 6∈ G(f ) there exist
r-open sets U and V containing x and y respectively, such that (U ×
V ) ∩ G(f ) = φ.
Theorem 6.2. If f : X → Y is R-supercontinuous function and Y is
Hausdorff, then G(f ) is r-closed with respect to X × Y .
Proof. Let x ∈ X and let y 6= f (x). Then there exist disjoint open sets V
and W containing y and f (x) respectively. Since f is R-supercontinuous,
there is an r-open set U containing x such that f (U ) ⊂ W ⊂ Y − V . Then
it is easily verified that U × V is an r-open set containing (x, y) such that
(U × V ) ∩ G(f ) = φ. Consequently, G(f ) is r-closed in X × Y .
Corollary 6.3. If f : X → Y is R-supercontinuous and Y is Hausdorff,
then G(f ), the graph of f is r-closed with respect to X.
7. r-Quotient topology and r-quotient spaces
Let f : X → Y be a surjection defined on a topological space X to
a set Y . The largest topology on Y which makes f continuous is called
the quotient topology on Y . Several variants of quotient topology occur in
the mathematical literature which in general are coarser than the quotient
topology but coincide with it if X is suitably augmented. For example,
see ([14], [15], [16], [27], [32], [33]). The interrelations and interplay among
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J. K. Kohli, D. Singh, J. Aggarwal
certain of these variants of quotient topology are well discussed in [20]. In this
section, we introduce a new variant of quotient topology called ‘r-quotient
topology’ and elaborate on its interrelations with other variants of quotient
topology that already exist in the literature.
Definition 7.1. Let p : X → Y be a function from a topological space X
onto a set Y . The collection of all subsets A ⊂ Y such that p−1 (A) is a
(1) r-open set in X is a topology on Y and is called the r-quotient topology.
The map p is called the r-quotient map and the space Y with r-quotient
topology is called r-quotient space.
(2) cl-open set in X is a topology on Y and is called the cl-quotient topology
[33]. The map p is called the cl-quotient map.
(3) z-open set in X is a topology on Y and is called the z-quotient topology
[14]. The map p is called the z-quotient map.
(4) dδ -open set in X is a topology on Y and is called the Dδ -quotient topology
[16]. The map p is called the Dδ -quotient map.
(5) d∗ -open in X is a topology on Y and is called the D ∗ -quotient topology
[32]. The map p is called the D∗ -quotient map.
(6) d-open set in X is a topology on Y and is called the D-quotient topology
[15]. The map p is called the D-quotient map.
(7) θ-open set in X is a topology on Y and is called the θ-quotient topology
[20]. The map p is called the θ-quotient map.
(8) F -open set in X is a topology on Y and is called the F -quotient topology
[18]. The map p is called the F -quotient map.
(9) δ-open set in X is a topology on Y and is called the δ-quotient topology
[27]. The map p is called the δ-quotient map.
The following diagram well reflects upon the place of r-quotient topology
in the hierarchy of other variants of quotient topology that already exist in
the literature.
However, none of the above inclusions is reversible in general as is shown
by examples in ([14], [15], [16], [20]) and the following examples.
R-supercontinuous functions
717
Example 7.2. Let X = Y be the set of all real numbers and let X be
endowed with the right ray topology τ . Let f denote the identity map
from X onto Y . Then the quotient topology on Y is identical with τ while
r-quotient topology on Y is the indiscrete topology.
Example 7.3. Let (X, τ ) denote the mountain chain space due to Heldermann [11]. Let Y = X and f be the identity map from X onto Y . Since
(X, τ ) is a regular space which is not D-regular, the r-quotient topology and
the θ-quotient topology on Y are identical with τ . However, the D-quotient
topology on Y is strictly coarser than τ .
Example 7.4. Let X = Y be the set of natural numbers. Let X be
endowed with the cofinite topology τc and let f denote the identity map
defined on X. Then the r-quotient topology on Y is identical with τc while
θ-quotient topology on Y is indiscrete.
Theorem 7.5. Let p : X → Y be a surjection from a topological space
(X, τ1 ) onto a topological space (Y, τ2 ), where τ2 is the r-quotient topology.
Then p is R-supercontinuous. Moreover, τ2 is the largest topology on Y which
makes p : X → Y R-supercontinuous.
Proof. The fact that p is R-supercontinuous follows from the definition of
r-quotient topology. To prove the last assertion, let τ ∗ be a topology on Y
such that p : (X, τ1 ) → (Y, τ ∗ ) is R-supercontinuous. Let V be a τ ∗ -open
set in Y . Then since p is R-supercontinuous, p−1 (V ) is r-open in X. Now
by definition of r-quotient topology, V is τ2 -open and so τ ∗ ⊂ τ2 .
The following result shows that a function out of r-quotient space is
continuous if and only if its composition with r-quotient map is R-supercontinuous.
Theorem 7.6. Let p : X → Y be an r-quotient map. Then a function
g : Y → Z is continuous if and only if g ◦ p is R-supercontinuous.
Proof. Suppose that g ◦ p is R-supercontinuous and let W be an open set
in Z. Then (g ◦ p)−1 (W ) = p−1 (g −1 (W )) is r-open in X. Since p is an
r-quotient map, g −1 (W ) is open in Y . Thus g is continuous. Converse is
immediate in view of definition of r-quotient topology.
8. Strong variants of continuity and change of topology
Let f : X → Y be a function from a topological space X into a topological
space Y satisfying one of the strong forms of continuity of Definition 2.1
(a)–(j). If the domain of f is retopologized in an appropriate way, then
f is simply a continuous function. Mathematical literature is replete with
examples of this type. For example, see ([5], [11], [14], [15], [16], [24], [33]).
In this section, we retopologize the domain of an R-supercontinuous function
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J. K. Kohli, D. Singh, J. Aggarwal
in such a way that it is simply a continuous function. Moreover, we elaborate
on the interrelations and interplay among various coarser topologies obtained
in this way for a given topology. As a byproduct we conclude with alternative
proofs of certain results of preceding sections.
8.1. Let (X, τ ) be a topological space. Let
(i) BR denote the set of all r-open subsets of the space (X, τ ). Since
arbitrary union and finite intersections of r-open sets is r-open, the
collection BR is indeed a topology for X. We shall denote this topology
by τR . Clearly τR ⊂ τ .
(ii) Bz denote the collection of all cozero subsets of the space (X, τ ). Since
the intersection of the two cozero sets is a cozero set, the collection
Bz is a base for a topology τz on X [14]. The topology τz has been
referred to by several authors in the literature. See for example, Aull
[3], D’Aristotle [6] and Stephenson [34].
(iii) Bdδ denote the collection of all regular Fσ -sets, the collection Bdδ is a
base for a topology τdδ on X [16].
(iv) Bθ denote the collection of θ-open sets of the space (X, τ ). Since arbitrary union and finite intersections of θ-open sets are θ-open, Bθ is
indeed a topology on X. We shall denote this topology by τθ . The
topology τθ has been extensively referred to in the literature (see [24],
[36]).
(v) Bδ denote the collection of all regular open subsets of the space (X, τ ).
Since the intersection of two regular open sets is regular open, the
collection Bδ constitutes a base for a topology τδ on X and is called
semiregular topology associated with τ . The space (X, τδ ) is often
called the semiregularization of the space and has been extensively
used in the topological literature (see [5, Exercise 20, p. 138], [8]).
(vi) Bd denote the collection of all open Fσ -subsets of the space (X, τ ). Since
the intersection of two open Fσ -sets is an open Fσ -set, the collection
Bd is a base for a topology τd on X ([11], [15]).
(vii) Bd∗ denote the collection of all strongly open Fσ -sets of the space (X, τ ).
Since the intersection of two strongly open Fσ -sets is a strongly open
Fσ -set, the collection Bd∗ constitutes a base for a topology τd∗ on X
([11], [32]).
(viii) BF denote the collection of all F -open subsets of the space (X, τ ).
Since arbitrary union and finite intersection of F -open sets is F -open,
the collection BF is indeed a topology on X denoted by τF [18].
(ix) Bcl denote the collection of all cl-open subsets of the space (X, τ ).
Since the intersection of two cl-open sets is cl-open. The collection Bcl
constitutes a base for a topology τcl on X ([9], [33]).
R-supercontinuous functions
719
The following diagram well illustrates the interrelations that exist among
various coarser topologies on the space (X, τ ) elucidated in the above paragraphs.
τcl
∩
τd∗ ⊃ τz ⊂ τF ⊂ τR
∩
∩
τR ⊃ τd ⊃ τdδ
∩
τ ⊃ τδ ⊃ τθ ⊂ τR ⊂ τ
However, in general none of the above inclusions is reversible as is well
reflected by the existing examples in the mathematical literature and the
following examples.
Examples
8.2. Let (X, τ ) denote the real line endowed with the usual Euclidean topology. Then τz = τ and τcl is the indiscrete topology.
8.3. Let (X, τ ) be the space of irregular lattice topology [35, Example 79, p.
97]. This space is a functionally regular space which is not a semiregular
space. The topology τF is identical with τ but τz is strictly coarser
than τ . Again the topology τδ is strictly weaker than τ . Moreover,
τR = τ and τθ is strictly weaker than τ .
8.4. Let (X, τ ) be the regular space due to Hewitt [12] on which every
continuous real valued function is constant. Then τR is identical with τ
but τF is the indiscrete topology. Moreover, τθ = τ and τ dδ is strictly
coarser than τ .
8.5. Let (X, τ ) denote the space of [17, Example 3.3]. The space (X, τ )
is a functionally Hausdorff Dδ -completely regular space which is not
a completely regular space. Then τdδ is same as τ but τz is strictly
coarser than τ .
8.6. Let (X, τ ) denote the space of [17, Example 3.6]. This space is a regular
space as well as a D-regular space but not a Dδ -completely regular
space. In this case τθ = τd = τ but τdδ is strictly coarser than τ .
8.7. Let (X, τ ) denote the skyline space due to Heldermann [11]. Then X is
a regular space as well as a D-regular space which is not a D-completely
regular space. Then τd = τθ = τ and τd∗ is strictly coarser than τ .
8.8. Let (X, τ ) denote the set of integers endowed with the cofinite topology.
Then (X, τ ) is a D-completely regular space which is not regular. So
τd∗ = τ but τz = τθ is the indiscrete topology.
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J. K. Kohli, D. Singh, J. Aggarwal
Theorem 8.9. A function f : (X, τ ) → (Y, ν) is R-supercontinuous if and
only if f : (X, τR ) → (Y, ν) is continuous.
Many of the results in the preceding sections now follow from Theorem 8.9
and corresponding standard properties of continuous functions.
Theorem 8.10. For a topological space (X, τ ) the following statements are
equivalent.
(a) (X, τ ) is an R0 -space.
(b) Every continuous function from (X, τ ) into a space (Y, ν) is R-supercontinuous.
Proof. (a)⇒(b) is obvious.
(b)⇒(a) Take (Y, ν) = (X, τ ). Then identity function 1x on X is continuous and hence R-supercontinuous. So by above Theorem 1x : (X, τR ) →
(X, τ ) is continuous. Since U ∈ τ implies 1−1
x (U ) = U ∈ τR , therefore
τ ⊂ τR . Thus it follows that τ = τR and hence (X, τ ) is an R0 -space.
Definition 8.11. [21] A function f : X → Y from a topological space
X into a topological space Y is said to be R-continuous at x ∈ X if for
each r-open set V containing f (x) there exists an open set U containing
x such that f (U ) ⊂ V . The function f is said to be R-continuous if it is
R-continuous at each point of X.
Theorem 8.12. Let f : (X, τ ) → (Y, ν) be a function. Then
(a) f is R-continuous if and only if f : (X, τ ) → (Y, νR ) is continuous.
(b) f is R-open if and only if f : (X, τR ) → (Y, ν) is open.
In view of Theorems 8.9 and 8.10, Theorem 4.13 can be restated as
follows:
If f : (X, τR ) → (Y, ν) is a continuous open surjection and g : (Y, ν) →
(Z, σ) is a function, then g is continuous if and only if g ◦ f is continuous.
Further, if f maps open (closed) sets to r-open (r-closed) sets, then g is
R-supercontinuous.
Moreover, r-quotient topology on Y determined by the surjection
f : (X, τ ) → Y in Section 7 is identical with the standard quotient topology
on Y determined by f : (X, τR ) → Y .
9. Appendix (The category UPS)
Details of the concepts and definitions in this section may be found in
[10]. However, for the convenience of the reader we include here the basics
of the category UPS which is a full subcategory of TOP (= the category of
topological spaces and continuous maps).
R-supercontinuous functions
721
Let L be a nonempty set equipped with a transitive relation ≤. We say
that a is a lower bound of a set X ⊆ L, and b is an upper bound if
a ≤ x for each x ∈ X, and
x ≤ b for each x ∈ X, respectively.
If the set of all upper bounds of X has the smallest element, we call this
the supremum of X (sup X). Similarly, if the set of all lower bounds of X
has the largest element, we call this the infimum of X (inf X). The set L is
said to be directed if every finite subset of L has an upper bound in L. Let
↑X = {y ∈ L : x ≤ y for some x ∈ X}. A set X ⊆ L is said to be an upper
set if X = ↑X. A partially ordered set or poset is a nonempty set L equipped
with a reflexive, transitive and antisymmetric relation. A complete lattice
is a poset in which every subset has a supremum and infimum. Let L be a
complete lattice and let U ⊆ L. The set U is said to be Scott open if (i) U is
an upper set and (ii) sup D ∈ U implies U ∩ D 6= φ for every directed set
D ⊆ L. It is easily verified that the collection of all Scott open subsets of L
is a topology for L and is called Scott topology of L and is denoted by σ(L).
It turns out that the space (L, σ(L)) is a T0 -space.
A function f : S → T from a complete lattice S to a complete lattice
T is said to be Scott continuous if f −1 (V ) ∈ σ(S) for every V ∈ σ(T ) or
equivalently f preserves directed supremums, i.e. f (sup D) = sup f (D) for
every directed subset D of S.
The category whose objects are complete lattices endowed with Scott
topologies and whose morphisms are Scott continuous maps is denoted by
UPS (preservation of UP-directed Sups).
Acknowledgement
Authors are thankful to the referee for helpful suggestions which led to
the improvement of an earlier version of the paper and inclusion of Appendix
(Section 9).
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J. K. Kohli
DEPARTMENT OF MATHEMATICS
HINDU COLLEGE UNIVERSITY OF DELHI
DELHI 110007, INDIA
E-mail: jk [email protected]
D. Singh
DEPARTMENT OF MATHEMATICS
SRI AUROBINDO COLLEGE
UNIVERSITY OF DELHI
NEW DELHI 110017, INDIA
E-mail: [email protected]
J. Aggarwal
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF DELHI
DELHI 110007, INDIA
E-mail: [email protected]
Received February 13, 2009; revised version July 7, 2009.