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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 14, no. 1, 2013

pp. 1-15

Upper and lower cl-supercontinuous

multifunctions

J. K. Kohli and Chaman Prakash Arya

Abstract

The notion of cl-supercontinuity (≡ clopen continuity) of functions is
extended to the realm of multifunctions. Basic properties of upper
(lower) cl-supercontinuous multifunctions are studied and their place
in the hierarchy of strong variants of continuity of multifunctions is
discussed. Examples are included to reflect upon the distinctiveness of
upper (lower) cl-supercontinuity of multifunctions from that of other
strong variants of continuity of multifunctions which already exist in
the literature.

2010 MSC: 54C05, 54C10, 54C60, 54D20.

Keywords: upper(lower)cl-supercontinuous multifunction, strongly contin-
uous multifunction, upper(lower) perfectly continuous multi-
function, upper(lower) z-supercontinuous multifunction, up-
per(lower) D-supercontinuous multifunction, upper(lower) su-
percontinuous multifunction, cl-open set, zero dimensional
space, cl-para-Lindelöf space, cl-paracompact space, nonmingled
multifunction.

1. Introduction

Several weak and strong variants of continuity occur in the lore of mathemat-
ical literature which have been studied by host of authors. The strong variants
of continuity with which we shall be dealing in this paper include strong con-
tinuity due to Levine [17], perfect continuity introduced by Noiri [19], clopen
continuity (cl-supercontinuity) defined by Reilly and Vamanamurthy [20], and
studied by Singh[21], and Kohli and Singh [15], complete continuity initiated
by Arya and Gupta [5] and z-supercontinuity introduced by Kohli and Kumar



2 J. K. Kohli and C. P. Arya

[14]. Multifunctions arise naturally in many areas of mathematics and ap-
plications of mathematics and have wide ranging applications in optimization
theory, control theory, game theory, mathematical economics, dynamical sys-
tems and differential inclusions. Recently, there has been considerable interest
in trying to extend the notions and results of weak and strong variants of conti-
nuity of functions to the realm of multifunctions (see [1], [2], [3], [4], [5], [8], [9],
[10], [16], [22], [27]). The present paper is written in continuation of the same
theme. In this paper we extend the notion of cl-supercontinuity of functions to
the framework of multifunctions and introduce the notions of upper and lower
cl-supercontinuous multifunctions and elaborate on their place in the hierarchy
of strong variants of continuity of multifunctions. In the process we extend cer-
tain result of Singh [21] pertaining to cl-supercontinuous functions to the setting
of multifunctions. It turns out that class of upper (lower) cl-supercontinuous
multifunctions properly includes the class of upper (lower) perfectly continuous
multifunctions and so includes all strongly continuous multifunctions [12] and
is strictly contained in the class of upper (lower) z-supercontinuous multifunc-
tions [3]. Section 2 is devoted to preliminaries and basic definitions, wherein we
introduce the notions of upper and lower cl-supercontinuous multifunctions and
discuss the interrelations that exist among them and other strong variants of
continuity of multifunctions that already exist in the literature. Examples are
included to reflect upon the distintiveness of the notions so introduced and other
strong variants of continuity of multifunctions in the literature. In Section 3 we
obtain characterizations and study basic properties of upper cl-supercontinuous
multifunctions. It turns out that upper cl-supercontinuity of multifunctions is
preserved under the shrinking and expansion of range, composition of multi-
functions, union of multifunctions, restriction to a subspace, and the passage
to the graph multifunction. Further, we formulate a sufficient condition for
the intersection of two multifunctions to be cl-supercontinuous. Moreover, we
prove that the graph of an upper cl-supercontinuous multifunction with closed
values into a regular space is cl-closed with respect to X. Furthermore, an
upper cl-supercontinuous multifunction maps mildly compact sets to compact
sets. Finally it is shown that a closed, open, upper cl-supercontinuous multi-
function with paracompact values maps cl-paracompact sets to paracompact
sets. Section 4 is devoted to the study of lower cl-supercontinuous multifunc-
tions, wherein characterizations of lower cl-supercontinuity are obtained. It
is shown that lower cl-supercontinuity is preserved under the shrinking and
expansion of range, union of multifunctions, restriction to a subspace and pas-
sage to the graph multifunction. Further, it is shown that a product of multi-
functions is lower cl-supercontinuous if and only if each multifunction is lower
cl-supercontinuous.

2. Preliminaries and Basic Definitions

Throughout the paper we essentially follow the notations and terminology
of L. Górniewicz. Let X and Y be nonempty sets. Then ϕ : X ⊸ Y is called a
multifunction from X into Y if for each x ∈ X, ϕ(x) is a nonempty subset of



Upper and lower cl-supercontinuous multifunctions 3

Y. Let B be a subset of Y. Then the set ϕ−1+ (B) = {x ∈ X : ϕ(x) ∩ B 6= ∅} is
called large inverse image[6]1 of B and the set ϕ−1− (B) = {x ∈ X : ϕ(x) ⊂
B} is called small inverse image of B. The set Γϕ = {(x,y) ∈ X × Y |y ∈
ϕ(x)} is called the graph of the multifunction. Let A be subset of X. Then
ϕ(A) = ∪{ϕ(x) : x ∈ A} is called image of A. A multifunction ϕ : X ⊸ Y
is upper semicontinuous (respectively lower semicontinuous) if ϕ−1− (U)

(respectively ϕ−1+ (U)) is an open set in X for every open set U in Y. A subset
U of a topological space X is called a cl-open set if it can be expressed as the
union of clopen sets. The complement of a cl-open set will be referred to as
a cl-closed set. A subset A of a space X is called regular open if it is the

interior of its closure, i.e., A = A
◦
. A collection β of subsets of a space X is

called an open complementary system [7] if β consists of open sets such
that for each B ∈ β, there exist B1,B2, ...,∈ β with B =

⋃
{X \ Bi : i ∈ Z

+}.
A subset U of a space X is called strongly open Fσ-set [7] if there exists a
countable open complementary system β(U) with U ∈ β(U). A subset H of a
space X is called a regular Gδ-set [18] if H is the intersection of a sequence
of closed sets whose interiors contain H, i.e., if H =

⋂∞
i=1 Fi =

⋂∞
i=1 F

◦
i , where

each Fi is a closed subset of X. The complement of a regular Gδ-set is called a
regular Fσ-set. Let X be a topological space and let A ⊂ X. A point x ∈ X
is called a θ-adherent point [25] of A if every closed neighbourhood of x
intersects A. Let clθA denote the set of all θ-adherent point of A. The set A is
called θ-closed if A = clθA. The complement of a θ-closed set is referred to as
a θ-open set. A point x ∈ X is said to be a cl-adherent point of A if every
clopen set containing x intersects A. Let [A]cl denote the set of all cl-adherent
points of A. Then a set A is cl-closed if and only if A=[A]cl. A subset A of
a space X is said to be cl-closed if it is the intersection of clopen sets. The
complement of a cl-closed set is referred to as a cl-open set.

Definition 2.1 ([13]). A multifunction ϕ : X ⊸ Y from a topological space
X into a topological space Y is said to be
(1) strongly continuous if ϕ−1− (B) is clopen in X for every subset B ⊂ Y.

(2)upper perfectly continuous if ϕ−1− (V ) is clopen in X for every open set
V ⊂ Y.
(3) lower perfectly continuous if ϕ−1+ (V ) is clopen in X for every open set
V ⊂ Y.
(4) upper completely continuous if ϕ−1− (V ) is regular open in X for every
open set V ⊂ Y.
(5) lower completely continuous if ϕ−1+ (V ) is regular open in X for every
open set V ⊂ Y.

1However,what we call “large inverse image ϕ
−1
+

(B)” some authors call it ‘lower inverse

image’ and denote it by ϕ−(B); and similarly they call “small inverse image ϕ
−1
−

(B)” as

‘upper inverse image’ and employ the notation ϕ+(B) for the same.



4 J. K. Kohli and C. P. Arya

Definition 2.2. A multifunction ϕ : X ⊸ Y from a topological space X into
a topological space Y is said to be
(1) upper z-supercontinuous [3] if for each x ∈ X and each open set V
containing ϕ(x), there exists a cozero set U containing x such that ϕ(U) ⊂ V.
(2) lower z-supercontinuous[3] if for each x ∈ X and each open set V with
ϕ(x) ∩V 6= ∅, there exists a cozero set U containing x such that ϕ(z) ∩V 6= ∅
for each z ∈ U.
(3) upper Dδ-supercontinuous [4] if for each x ∈ X and each open set
V containing ϕ(x), there exists a regular Fσ-set U containing x such that
ϕ(U) ⊂ V.
(4) lower Dδ-supercontinuous [4] if for each x ∈ X and each open set V
with ϕ(x) ∩ V 6= ∅ , there exists a regular Fσ-set U containing x such that
ϕ(z) ∩ V 6= ∅ for each z ∈ U.
(5) upper D-supercontinuous [1] if for each x ∈ X and each open set V
containing ϕ(x), there exists an open Fσ-set U containing x such that ϕ(U) ⊂ V.
(6) lower D-supercontinuous [1] if for each x ∈ X and each open set V with
ϕ(x)∩V 6= ∅, there exists an open Fσ-set U containing x such that ϕ(z)∩V 6= ∅
for each z ∈ U.
(7) upper D∗-supercontinuous[12] if for each x ∈ X and each open set V
containing ϕ(x), there exists a strongly open Fσ-set U containing x such that
ϕ(U) ⊂ V.
(8) lower D∗-supercontinuous [12] if for each x ∈ X and each open set V
with ϕ(x) ∩ V 6= ∅, there exists a strongly open Fσ-set U containing x such
that ϕ(z) ∩ V 6= ∅ for each z ∈ U.
(9) upper strongly θ-continuous [16] if for each x ∈ X and each open set V
containing ϕ(x), there exists a θ-open set U containing x such that ϕ(U) ⊂ V.
(10) lower strongly θ-continuous[16] if for each x ∈ X and each open set
V with ϕ(x) ∩ V 6= φ, there exists a θ-open set U containing x such that
ϕ(z) ∩ V 6= ∅ for each z ∈ U.

Definition 2.3 ([21]). The graph Γϕ of a multifunction ϕ : X ⊸ Y is said to
be cl-closed with respect to X if for each (x,y) 6∈ Γϕ there exist a clopen set
U containing x and an open set V containing y such that (U × V ) ∩ Γϕ = ∅.

Definition 2.4 ([26]). A multifunction ϕ : X ⊸ Y is said to have nonmingled
point images provided that for x,y ∈ X with x 6= y, the image sets ϕ(x) and
ϕ(y) are either disjoint or identical.

Definition 2.5. A space X is said to be
(a) mildly compact[23] if for every clopen cover of X has a finite subcover.
In [24] Sostak calls mildly compact spaces as clustered spaces.
(b) cl-paracompact (cl-para-Lindelöf) if every clopen cover of X has locally
finite (locally countable) open refinement which covers X.



Upper and lower cl-supercontinuous multifunctions 5

Definition 2.6. We say that a multifunction ϕ : X ⊸ Y is
(a) upper cl-supercontinuous at x ∈ X if for each open set V with ϕ(x) ⊂ V,
there exists a clopen set U containing x such that ϕ(U) ⊂ V. The multifunction
is said to be upper cl-supercontinuous if it is upper cl-supercontinuous at each
x ∈ X.
(b) lower cl-supercontinuous at x ∈ X if for each open set V with ϕ(x)∩V 6=
φ, there exists a clopen set U containing x such that ϕ(z) ∩ V 6= ∅ for each
z ∈ U. The multifunction is said to be lower cl-supercontinuous if it is lower
cl-supercontinuous at each x ∈ X.

The following diagram well illustrates the interrelations that exist among
various strong variants of continuity of multifunctions defined in Definition
2.1, 2.2 and 2.6.

Figure 1.

However, none of the above implications is reversible as is well illustrated
by the examples in the sequel and the examples in ([1], [2], [3], [4], [12], [16]).

Example 2.7. Let X = {a,b,c} with the topology ℑX = {∅,X,{a},{b,c}}
and let Y = {x,y} with the topology ℑY = {∅,Y,{y}}. Define a multifunction
ϕ : (X,ℑX) ⊸ (Y,ℑY ) by ϕ(a) = {y}, ϕ(b) = {x,y}, ϕ(c) = {x}. Then the
multifunction is upper perfectly continuous but not lower perfectly continuous.
Again, for {x} ⊂ Y , ϕ−1− ({x}) = {c} is not clopen which implies that the
multifunction ϕ is not strongly continuous.



6 J. K. Kohli and C. P. Arya

Example 2.8. Let X = {a,b,c} with the topology ℑX = {∅,X,{a},{c},{a,c},
{a,b}} and let Y = {x,y} with the topology ℑY = {∅,Y,{y}}. Define a mul-
tifunction ϕ : (X,ℑX) ⊸ (Y,ℑY ) by ϕ(a) = {y}, ϕ(b) = {x,y}, ϕ(c) = {y}.
Then clearly ϕ is lower perfectly continuous. But for {y} ⊂ Y ϕ−1− ({y}) = {a}
is not clopen, which implies the multifunction ϕ is not strongly continuous.

Example 2.9. Let X = ℜ, set of real numbers with upper limit topology
ℑ and let Y be same as X with usual topology U. Define a multifunction
ϕ : (X,ℑ) ⊸ (Y,U) by ϕ(x) = {x} for each x ∈ X. Then clearly ϕ is upper
(lower) cl-supercontinuous. But for ϕ−1− (a,b) = (a,b) = ϕ

−1
+ (a,b) is not clopen

in X, which implies that ϕ is not upper (lower) perfectly continuous.

Example 2.10. Let X be a completely regular space which is not zero dimen-
sional and let Y be same as X. Then the identity mapping ϕ : X ⊸ Y defined
by ϕ(x) = {x} for each x ∈ X, is upper (lower) z-supercontinuous but not
upper (lower) cl-supercontinuous.

3. Properties of Upper cl-Supercontinuous Multifunctions

Theorem 3.1. For a multifunction ϕ : X ⊸ Y from a topological space X
into a topological Y the following statements are equivalent:
(a) ϕ is upper cl-supercontinuous.
(b) ϕ−1− (B) is a cl-open set in X for every open set B in Y.

(c) ϕ−1+ (B) is a cl-closed in X for every closed set B in Y.

(d) [ϕ−1+ (B)]cl ⊂ ϕ
−1
+ (B) for every subset B of Y.

Proof. a)⇒ (b). Let B be an open subset of Y . To show that ϕ−1− (B) is cl-open

in X, let x ∈ ϕ−1− (B). Then ϕ(x) ⊂ B. Since ϕ is upper cl-supercontinuous,
therefore, there exists a clopen set H containing x such that ϕ(H) ⊂ B. Hence
x ∈ H ⊂ ϕ−1− (B) and so is a cl-open set in X.
(b)⇒ (c). Let B be a closed subset of Y . Then Y \ B is an open subset of
Y. By (b), ϕ−1− (Y \ B) is cl-open set in X. Since ϕ

−1
− (Y \ B) = X \ ϕ

−1
+ (B),

ϕ
−1
+ (B) is a cl-closed set in X.

(c)⇒ (d). Since B is closed, ϕ−1+ (B) is a cl-closed set containing ϕ
−1
+ (B)

Therefore [ϕ−1+ (B)]cl ⊂ ϕ
−1
+ (B) .

(d)⇒ (a). Let x ∈ X and let V be an open set in Y such that ϕ(x) ⊂ V. Then

ϕ(x)∩(Y \V ) = ∅ and (Y \ V ) = Y \V. Hence [ϕ−1+ (Y \V )]cl ⊂ ϕ
−1
+ (Y \V ) =

X\ϕ−1− (V ). Since ϕ
−1
+ (Y \V ) is cl-closed, its complement ϕ

−1
− (V ) is cl-open set

containing x. So there is a clopen set U containing x and contained in ϕ−1− (V ),
whence ϕ(U) ⊂ V. Thus ϕ is upper cl-supercontinuous. �

Theorem 3.2. If a multifunction ϕ : X ⊸ Y is upper cl-supercontinuous and
ϕ(X) is endowed with the subspace topology, then, the multifunction ϕ : X ⊸
ϕ(X) is upper cl-supercontinuous.

Proof. Since ϕ is upper cl-supercontinuous for every open set V of Y, ϕ−1− (V ∩

ϕ(X)) = ϕ−1− (V ) ∩ ϕ
−1
− (ϕ(X)) = ϕ

−1
− (V ) ∩ X = ϕ

−1
− (V ) is cl-open and hence

ϕ : X ⊸ ϕ(X) is cl-supercontinuous. �



Upper and lower cl-supercontinuous multifunctions 7

Theorem 3.3. If ϕ : X ⊸ Y is upper cl-supercontinuous and ψ : Y ⊸ Z is up-
per semicontinuous, then ψoϕ is upper cl-supercontinuous. In particular, com-
position of upper cl-supercontinuous multifunctions is upper cl-supercontinuous.

Proof. Let W be an open set in Z. Since ψ is upper semicontinuous, ψ−1− (W) is

an open set in Y . Again, since ϕ is upper cl- supercontinuous, ϕ−1− (ψ
−1
− (W)) =

(ψoϕ)−1− (W) is a cl-open set in X. Thus ψoϕ : X ⊸ Z is upper cl-supercontinuous.
�

In contrast to Theorem 3.2, the following corollary shows that upper cl-
supercontinuity of a multifunction remains invariant under extension of its
range.

Corollary 3.4. Let ϕ : X ⊸ Y be upper cl-supercontinuous. If Z is a space
containing Y as a subspace, then ψ : X ⊸ Z defined by ψ(x) = ϕ(x) for each
x ∈ X is upper cl-supercontinuous.

Proof. Let W be an open set in Z. Then W ∩ Y is an open set in Y. Since
ϕ : X ⊸ Y is upper cl-supercontinuous, ϕ−1− (W ∩ Y ) is cl-open set in X. Now

ψ
−1
− (W) = {x ∈ X : ψ(x) ⊂ W} = {x ∈ X : ϕ(x) ⊂ W ∩ Y }. Thus ψ : X ⊸ Z

is upper cl-supercontinuous. �

Theorem 3.5. If ϕ : X ⊸ Y and ψ : X ⊸ Y are upper cl-supercontinuous
multifunctions, then ϕ ∪ ψ : X ⊸ Y defined by (ϕ ∪ ψ)(x) = ϕ(x) ∪ ψ(x) for
each x ∈ X, is upper cl-supercontinuous.

Proof. Let U be an open set in Y. Since ϕ and ψ are upper cl-supercontinuous,
ϕ
−1
− (U) and ψ

−1
− (U) are cl-open sets in X. Since (ϕ ∪ ψ)

−1
− (U) = ϕ

−1
− (U) ∩

ψ
−1
− (U) and since finite intersection of cl-open sets is cl-open, (ϕ ∪ ψ)

−1
− (U) is

cl-open in X. Thus ϕ ∪ ψ is upper cl-supercontinuous. �

In general intersection of two upper cl-supercontinuous multifunctions need
not be upper cl-supercontinuous. However, in the following theorem we formu-
late a sufficient condition for the intersection of two multifunctions to be upper
cl-supercontinuous.

Theorem 3.6. Let ϕ : X ⊸ Y and ψ : X ⊸ Y be multifunctions from a
space X into a Hausdorff space Y such that ϕ(x) is compact for each x ∈ X
satisfying
(1) ϕ is upper cl-supercontinuous, and
(2) the graph Γψ of ψ is cl-closed with respect to X. Then the multifunction ϕ∩ψ
defined by (ϕ∩ψ)(x) = ϕ(x)∩ψ(x) for each x ∈ X, is upper cl-supercontinuous.

Proof. Let x0 ∈ X and V be an open set containing ϕ(x0) ∩ ψ(x0). It suffices
to find a clopen set U containing x0 such that (ϕ ∩ ψ)(U) ⊂ V. If V ⊃ ϕ(x0),
it follows from upper cl-supercontinuity of ϕ. If not, then consider the set
K = ϕ(x0) \ V which is compact. Now for each y ∈ K, y ∈ Y \ ψ(x0). This
implies that (x0,y) ∈ X ×Y \Γψ. Since the graph of ψ is cl-closed with respect
to X, there exist clopen set Uycontaining x0 and an open set Vy containing y



8 J. K. Kohli and C. P. Arya

such that Γψ ∩ (Uy × Vy) = ∅. Therefore, for each x ∈ Uy, ψ(x) ∩ Vy = ∅
Since K is compact, there exist finitely many in y1,y1, ...,yn in K such that
K ⊂ ∪ni=1Vyi. Let W = ∪

n
i=1Vyi. Then V ∪ W is an open set containing ϕ(x0).

Since ϕ is upper cl-supercontinuous, there exists a clopen set U0 containing
x0 such that ϕ(U0) ⊂ V ∪ W. Let U = U0 ∩ (∩

n
i=1Uyi). Then U is a clopen

set containing x0. Hence for each z ∈ U, ϕ(z) ⊂ V ∪ W and ψ(z) ∩ W = ∅.
Therefore, (ϕ(z) ∩ ψ(z)) ∩ W = ∅ for each z ∈ U. This proves that ϕ ∩ ψ is
upper cl-supercontinuous at x0 �

Corollary 3.7. Let ψ : X ⊸ Y be a multifunction from a space X into a
compact Hausdorff space Y such that the graph Γψ of ψ is cl-closed with respect
to X. Then ψ is upper cl-supercontinuous.

Proof. Let the multifunction ϕ : X ⊸ Y be defined by ϕ(x) = Y for each
x ∈ X. Now an application of Theorem 3.6 yields the desired result. �

Theorem 3.8. Let ϕ : X ⊸ Y be any multifunction. Then the following
statements are true:
(a) If ϕ : X ⊸ Y is upper cl-supercontinuous and A ⊂ X, then the restriction
ϕ|A : A ⊸ Y is upper cl-supercontinuous.
(b) If {Uα : α ∈△} is a cl-open cover of X and if for each α, the restriction
ϕα = ϕ|Uα : Uα ⊸ Y is upper cl-supercontinuous, then ϕ : X ⊸ Y is upper
cl-supercontinuous.

Proof. (a) Let W be an open set in Y. Since ϕ : X ⊸ Y is upper cl-supercontinuous,
ϕ
−1
− (W) is a cl-open set in X. Now ϕ|A(W) = {x ∈ A | ϕ(x) ⊂ W} = {x ∈

A | x ∈ ϕ−1− (W)} = A ∩ ϕ
−1
− (W), which is cl-open in X and so ϕ|A is upper

cl-supercontinuous.
(b) Let W be an open set in Y. Since ϕα = ϕ|Uα : Uα ⊸ Y is upper cl-

supercontinuous, (ϕα)
−1
− (W) is a cl-open set in Uα and consequently cl-open

in X. Since ϕ−1− (W) = ∪α∈
∧(ϕα)

−1
− (W) and since the union of cl-open set is

cl-open, ϕ−1− (W) is cl-open set in X. In view of Theorem 3.1, ϕ : X ⊸ Y is
upper cl-supercontinuous. �

Theorem 3.9. Let ϕ : X ⊸ Y be a multifunction and let g : X ⊸ X × Y
defined by g(x) = {(x,y) ∈ X × Y |y ∈ ϕ(x)} for each x ∈ X, be the graph mul-
tifunction. If g is upper cl-supercontinuous, then ϕ is upper cl-supercontinuous
and the space X is zero dimensional. Furthermore, if in addition ϕ(x) is
compact for each x ∈ X and X is zero dimensional, then g is upper cl-
supercontinuous whenever ϕ is.

Proof. Suppose that g is upper cl-supercontinuous. By Theorem3.3, the multi-
function ϕ = pyog is upper cl-supercontinuous, where py : X ×Y ⊸ Y denotes
the projection mapping. To show that X is zero dimensional, let U be an open
set in X and let x ∈ U. Then U ×Y is an open set in X ×Y and g(x) ⊂ U ×Y.
Since g is upper cl-supercontinuous, there exists a clopen set W containing x
such that g(W) ⊂ U × Y and so W ⊂ g−1− (U × Y ) = U. Hence x ∈ W ⊂ U and



Upper and lower cl-supercontinuous multifunctions 9

thus X is zero dimensional.
Conversely, suppose that X is zero dimensional, the multifunction ϕ is upper
cl-supercontinuous and ϕ(x) is compact for each x ∈ X. Let W be an open set
containing g(x) = {x} × ϕ(x). Then by Wallace theorem [10, p.142] there exist
open sets U in X, V in Y and g(x) ⊂ U × V ⊂ W. So x ∈ U and ϕ(x) ⊂ V.
Since X is zero dimensional there exists a clopen set containing x such that
x ∈ G1 ⊂ U. Again since ϕ is upper cl-supercontinuous, there exists a clopen
set G2 containing x such that ϕ(G2) ⊂ V . Let G = G1 ∩ G2. Then G is a
clopen set containing x and it is easily verified that g(G) ⊂ U × V ⊂ W. This
proves that g is upper cl-supercontinuous. �

The following theorem gives sufficient conditions for the graph of a multi-
function to be cl-closed with respect to X.

Theorem 3.10. If ϕ : X ⊸ Y is upper cl-supercontinuous, where Y is a
regular space and ϕ(x) is closed for each x ∈ X, then the graph Γϕ of ϕ is a
cl-closed with respect to X.

Proof. Let (x,y) 6∈ Γϕ. Then y 6∈ ϕ(x). Since Y is regular, there exist disjoint
open sets Vy and Vϕ(x) containing y and ϕ(x), respectively. Since ϕ is upper
cl-supercontinuous, there exists a clopen set Ux containing x such that ϕ(Ux) ⊂
Vϕ(x). We assert that (Ux × Vy) ∩ Γϕ = ∅. For, if (h,k) ∈ (Ux × Vy) ∩ Γϕ, then
h ∈ ϕ−1− (Vϕ(x)), k ∈ Vy and k ∈ ϕ(h). Hence ϕ(h) ⊂ Vϕ(x) and k ∈ ϕ(h) ∩ Vy
which contradicts the fact that Vy and Vϕ(x) are disjoint. Thus the graph Γϕ
of ϕ is a cl-closed with respect to X. �

The following theorem is a sort of partial converse to Theorem3.10 and shows
that the multifunctions which have cl-closed graph with respect to X have nice
properties.

Theorem 3.11. If ϕ : X ⊸ Y is a multifunction with cl-closed graph with
respect to X and K ⊂ Y is compact, then ϕ−1+ (K) is cl-closed in X. Further,
if in addition Y is compact, then ϕ is upper cl-supercontinuous.

Proof. To prove that ϕ−1+ (K) is cl-closed, we shall show that X \ ϕ
−1
+ (K)

is cl-open. To this end, let x ∈ X \ ϕ−1+ (K). Then ϕ(x) ∩ K = ∅. Since
Γϕ is cl-closed with respect to X, for each y ∈ K there exist clopen set Uy
containing x and an open set Vy containing y such that (Uy × Vy) ∩ Γϕ = ∅.
The collection Ω = {Vy|y ∈ K} is an open cover of the compact set K. So there
exists a finite subset {y1, ...,yn} of K such that K ⊂ ∪

n
i=1Vyi = V (say). Let

U = ∩ni=1Uyi. Then U is a clopen set containing x and since ϕ(U) ∩ K = ∅.
Thus U ⊂ X \ ϕ−1+ (K) and so X \ ϕ

−1
+ (K) is cl-open as desired. The last

assertion is immediate in view of Theorem3.1 and the fact that a closed subset
of a compact space is compact. �

Corollary 3.12. If ϕ : X ⊸ Y is a multifunction with ϕ(X) ⊂ K, where K is
compact and the graph Γϕ of ϕ is cl-closed with respect to X, then ϕ is upper
cl-supercontinuous.



10 J. K. Kohli and C. P. Arya

Theorem 3.13. Let ϕ : X ⊸ Y be an upper cl-supercontinuous multifunction
such that ϕ(x) is compact for each x ∈ X. If A is a mildly compact set in X,
then ϕ(A) is compact.

Proof. Let Ω be an open cover of ϕ(A). Then Ω is also an open cover of
ϕ(a) for each a ∈ A. Since each ϕ(a) is compact, there exists a finite sub-
set βa ⊂ Ω such that ϕ(a) ⊂ ∪B∈βaB = Va (say). Since ϕ is upper cl-
supercontinuous, there exists a clopen set Ua containing a such that ϕ(Ua) ⊂ Va
and so Ua ⊂ ϕ

−1
− (Va). Let Q = {Ua|a ∈ A}. Then Q is a clopen covering of

A. Since A is mildly compact, there exists a finite subset {a1, ...,an} of A such
that A ⊂ ∪ni=1Uai ⊂ ∪

n
i=1ϕ

−1
− (Vai). Therefore ϕ(A) ⊂ ϕ(∪

n
i=1ϕ

−1
− (Vai)) =

∪ni=1ϕ(ϕ
−1
− (Vai)) ⊂ ∪

n
i=1Vai, where Vai = ∪B∈βai B, i = 1, ...,n and each βai is

finite. Thus ϕ(A) is compact. �

We may recall that a space X is called a P-space if every Gδ-set in X is
open in X.

Theorem 3.14. Let ϕ : X ⊸ Y be a closed, open, and upper cl-supercontinuous,
nonmingled multifunction from a space X into a P-space Y such that ϕ(x) is
para-Lindelöf for each x ∈ X. If A is a cl-para-Lindelöf set in X, then ϕ(A)
is para-Lindelöf set in Y. In particular, if X is cl-para-Lindelöf and ϕ is onto,
then Y is para-Lindelöf.

Proof. Let Ψ be an open cover of ϕ(A). Then Ψ is also an open covering of
ϕ(x) for each x ∈ A. Since ϕ(x) is para-Lindelöf, Ψ has a locally countable
open refinement ψx such that ϕ(x) ⊂ ∪ψx = Vx (say). Since ϕ is upper cl-
supercontinuous, there exists a clopen set Ux containing x such that ϕ(Ux) ⊂
Vx. Now u = {Ux | x ∈ A}is a clopen cover of A. Since A is cl-para-Lindelöf,
u has a locally countable open refinement Ω = {Wα | α ∈ Λ} such that
A ⊂ ∪α∈ΛWα. So for each α ∈ Λ there exists a xα ∈ A such that Wα ⊂ Uxα
and hence ϕ(Wα) ⊂ ϕ(Uα) ⊂ ∪ψxα. Let ℜα = {ϕ(Wα) ∩ V | V ∈ ψxα} and let
ℜ = {R | R ∈ ℜα,α ∈ Λ}. We shall show that ℜ is a locally countable open
refinement of Ψ. Since ϕ is open, ϕ(Wα) is open and so each R ∈ ℜ is open.
Let R ∈ ℜ. Then R ∈ ℜα for some α ∈ Λ, i.e. R = ϕ(Wα) ∩ V ⊂ V ⊂ U
for some U ∈ Ψ. This shows that ℜ is an open refinement of Ψ. To show that
ℜ is locally countable, let y ∈ ϕ(A). Then y ∈ ϕ(x) for some x ∈ A. Since
Ω is locally countable, for each x ∈ A we can choose an open neighborhood
Gx of x which intersects only countably many members Wα1,Wα2, ...Wαn... of
Ψ. Since ϕ is a nonmingled open multifunction, it follows that H0 = ϕ(Gx)
is an open neighborhood of y which intersects only countably many members
ϕ(Wα1),ϕ(Wα2 ))...ϕ(Wαn )... of the family {ϕ(Wα) | α ∈ Λ}. Furthermore
each ℜαk (k = 1, ...,n, ...) is locally countable, hence there exists an open
neighborhood Hk (k = 1, ...,n, ...) of y which intersects only countably many
members of ℜαk (k = 1, ...,n, ...). Finally let H = ∩

∞
k=1Hk. Since Y is P-space,

H is an open neighborhood of y which intersects at most countably many
member of ℜ, and so ℜ is locally countable. Moreover, ϕ(A) ⊂ ϕ(∪α∈ΛWα) =



Upper and lower cl-supercontinuous multifunctions 11

∪α∈Λϕ(Wα) ⊂ ∪α∈Λ(∪ℜα) = ∪{R : R ∈ ℜ}. Hence ℜ is a locally countable
open refinement of Ψ that covers ϕ(A). Thus ϕ(A) is para-Lindelöf. �

Theorem 3.15. Let ϕ : X ⊸ Y be a closed, open, upper cl-supercontinuous
nonmingled multifunction from a space X into a space Y such that ϕ(x) is
paracompact for each x ∈ X. If A is a cl-paracompact, then ϕ(A) is para-
compact. In particular, if X is cl-paracompact space and ϕ is onto, then Y is
paracompact.

Proof. Proof of Theorem 3.15 is similar (even simpler) to that of Theorem 3.14
and hence omitted. �

4. Properties of Lower cl-Supercontinuous Multifunctions

Theorem 4.1. For a multifunction ϕ : X ⊸ Y from a topological space X
into a topological space Y the following statements are equivalent.
(a) ϕ is lower cl-supercontinuous.
(b) ϕ−1+ (B) is a cl-open set in X for every open set B in Y.

(c) ϕ−1− (B) is a cl-closed in X for every closed set B in Y.
(d) For each x ∈ X and for each open set V with ϕ(x) ∩ V 6= ∅ there exists a
cl-open set U containing x such that ϕ(z) ∩ V 6= φ for each z ∈ U.

Proof. (a)⇒(b). Let B be an open subset of Y. To show that ϕ−1+ (B) is cl-open

in X, let x ∈ ϕ−1+ (B). Then ϕ(x) ∩B 6= ∅. Since ϕ is lower cl-supercontinuous,
there exists a clopen set H containing x such that ϕ(h)∩B 6= ∅ for each h ∈ H.
Hence x ∈ H ⊂ ϕ−1+ (B) and so ϕ

−1
+ (B) is a cl-open set in X being a union of

clopen sets.
(b)⇒(c). Let B be a closed subset of Y. Then Y \B is an open subset of Y. By
(b), ϕ−1+ (Y \B) is a cl-open set in X. Since ϕ

−1
+ (Y \B) = X \ϕ

−1
− (B), ϕ

−1
− (B)

is a cl-closed set in X.
(c)⇒(d). Let x ∈ X and let V be an open set in Y with ϕ(x) ∩ V 6= ∅. Then
Y \V is a closed set in Y with ϕ(x) * (Y \V ).Therefore, By (c), ϕ−1− (Y \V ) =
X \ ϕ−1+ (V ) is a cl-closed set in X not containing x and so ϕ

−1
+ (V ) is a cl-open

set in X containing x. Let U = ϕ−1+ (V ). Then U is a cl-open set containing x
such that ϕ(z) ∩ V 6= ∅ for each z ∈ U.
The assertion (d)⇒(a) is trivial, since every cl-open set is the union of clopen
sets. �

Theorem 4.2. A multifunction ϕ : X ⊸ Y is lower cl-supercontinuous if and

only if ϕ([A]cl) ⊂ ϕ(A) for every subset A of X.

Proof. Suppose that ϕ : X ⊸ Y is lower cl-supercontinuous. Let A be subset of
X. Then ϕ(A) is a closed subset of Y. By Theorem 4.1 ϕ−1− (ϕ(A)) is a cl-closed

set in X. Since A ⊂ ϕ−1− (ϕ(A)) and since [A]cl ⊂ [ϕ
−1
− (ϕ(A))]cl = ϕ

−1
− (ϕ(A)),

ϕ([A]cl) ⊂ ϕ(ϕ
−1
− (ϕ(A))) ⊂ ϕ(A).

Conversely, suppose that ϕ([A]cl) ⊂ ϕ(A) for every subset A of X and let F be
a closed set in Y . Then ϕ−1− (F) is subset of X. By hypothesis, ϕ([ϕ

−1
− (F)]cl) ⊂



12 J. K. Kohli and C. P. Arya

ϕ(ϕ−1− (F)) ⊂ F = F and ϕ
−1
− (ϕ([ϕ

−1
− (F)]cl)) ⊂ ϕ

−1
− (F) so which in its turn

implies that [ϕ−1− (F)]cl ⊂ ϕ
−1
− (F). Hence ϕ

−1
− (F) = [ϕ

−1
− (F)]cl and so in view

of Theorem 4.1 ϕ : X ⊸ Y is lower cl-supercontinuous. �

Theorem 4.3. A multifunction ϕ : X ⊸ Y is lower cl-supercontinuous if and
only if [ϕ−1− (B)]cl ⊂ ϕ

−1
− (B) for every subset B of Y.

Proof. Suppose that ϕ : X ⊸ Y is lower cl-supercontinuous. Let B ⊂ Y.
Then B is a closed subset of Y. By Theorem 4.1, ϕ−1− (B) is a cl-closed subset

of X. Since, ϕ−1− (B) ⊂ ϕ
−1
− (B), [ϕ

−1
− (B)]cl ⊂ [ϕ

−1
− (B)]cl = ϕ

−1
− (B). That is

[ϕ−1− (B)]cl ⊂ ϕ
−1
− (B).

Conversely Suppose that [ϕ−1− (B)]cl ⊂ ϕ
−1
− (B) for every B ⊂ Y. Let F be

any closed subset of Y. By hypothesis [ϕ−1− (F)]cl ⊂ ϕ
−1
− (F) = ϕ

−1
− (F). Hence

[ϕ−1− (F)]cl = ϕ
−1
− (F) and so in view of Theorem 4.1 ϕ is lower cl-supercontinuous.

�

The following theorem shows that lower cl-supercontinuity of a multifunction
remains invariant under the shrinking of its range.

Theorem 4.4. If ϕ : X ⊸ Y is lower cl-supercontinuous and ϕ(X) is endowed
with subspace topology, then ϕ : X ⊸ ϕ(X) is lower cl-supercontinuous.

Theorem 4.5. If ϕ : X ⊸ Y is lower cl-supercontinuous and ψ : Y ⊸ Z
is lower semicontinuous, then ψoϕ is lower cl-supercontinuous. In particu-
lar, composition of two lower cl-supercontinuous multifunctions is upper cl-
supercontinuous.

Proof. Let W be an open set in Z. Since ψ is upper semi continuous, ψ−1+ (W)

is an open set in Y. Again since ϕ is lower cl-supercontinuous, ϕ−1+ (ψ
−1
+ (W))is

cl-open in X, and so (ψoϕ)−1+ (W) = ϕ
−1
+ (ψ

−1
+ (W)) is a cl-open set in X. Thus

ψoψ : X ⊸ Z is lower cl-supercontinuous. �

In contrast to Theorem 4.4 the following corollary shows that lower cl-
supercontinuity of a multifunction is preserved under the expansion of its range.

Corollary 4.6. Let ϕ : X ⊸ Y be lower cl-supercontinuous. If Z is a space
containing Y as a subspace, then ψ : X ⊸ Z defined by ψ(x) = ϕ(x) for x ∈ X
is lower cl-supercontinuous.

Proof. Let W be an open set in Z. Then W ∩ Y is an open set in Y. Since
ϕ : X ⊸ Y is lower cl-supercontinuous, ϕ−1+ (W ∩ Y ) is cl-open set in X. Now,

ψ
−1
+ (W) = {x ∈ X : ψ(x) ∩ W 6= ∅} = {x ∈ X : ϕ(x) ∩ (W ∩ Y ) 6= ∅} =
ϕ
−1
+ (W ∩ Y ). Thus ψ : X ⊸ Z is lower cl-supercontinuous. �

Theorem 4.7. If ϕ : X ⊸ Y and ψ : X ⊸ Y are lower cl-supercontinuous
multifunctions, then the multifunction ϕ ∪ ψ : X ⊸ Y defined by (ϕ ∪ ψ)(x) =
ϕ(x) ∪ ψ(x) for each x ∈ X, is lower cl-supercontinuous.



Upper and lower cl-supercontinuous multifunctions 13

Proof. Let U be an open set in Y. Then ϕ−1+ (U) and ψ
−1
+ (U) are cl-open sets

in X. Since (ϕ ∪ ψ)−1+ (U) = ϕ
−1
+ (U) ∪ ψ

−1
+ (U) and since any union of cl-

open sets is cl-open, (ϕ ∪ ψ)−1+ (U) is cl-open in X. Thus ϕ ∪ ψ is lower cl-
supercontinuous. �

Theorem 4.8. Let ϕ : X ⊸ Y be any multifunction. Then the following
statements are true:
(a) If ϕ : X ⊸ Y is lower cl-supercontinuous and A ⊂ X, then the restriction
ϕ|A : A ⊸ Y is lower cl-supercontinuous.
(b) If {Uα : α ∈ ∆} is a cl-open cover of X and for each α, the restriction
ϕα = ϕ|Uα : Uα ⊸ Y is lower cl-supercontinuous, then ϕ : X ⊸ Y is lower
cl-supercontinuous.

Proof. (a) Let W be an open set in Y. Since ϕ : X ⊸ Y is lower cl-supercontinuous,
ϕ
−1
+ (W) is a cl-open set in X. Now, (ϕ|A)

−1
+ (W) = {x ∈ A | ϕ(x) ∩ W 6= ∅} =

{x ∈ A | x ∈ ϕ−1+ (W)} = A ∩ ϕ
−1
+ (W), which is cl-open in X and so ϕ|A is

lower cl-supercontinuous.
(b) Let W be an open set in Y. Since ϕα = ϕ|Uα : Uα ⊸ Y is lower cl-

supercontinuous, (ϕα)
−1
+ (W) is a cl-open set in Uα and consequently cl-open in

X. Since ϕ−1+ (W) = ∪α∈∆(ϕα)
−1
+ (W) and since any union of cl-open sets is cl-

open, ϕ−1+ (W) is cl-open set in X. Thus ϕ : X ⊸ Y is lower cl-supercontinuous.
�

Theorem 4.9. Let {ϕα : X ⊸ Xα|α ∈ Λ} be a family of multifunctions and
let ϕ : X ⊸

∏
α∈Λ Xα be defined by ϕ(x) =

∏
α∈Λ ϕα(x). Then ϕ is lower cl-

supercontinuous if and only if each ϕα : X ⊸ Xα is lower cl-supercontinuous.

Proof. Let ϕ : X ⊸
∏
α∈Λ Xα be lower cl-supercontinuous. Let pβ :

∏
α∈Λ Xα

−→ Xβ be the projection map onto Xβ. Then pβ being a single valued con-
tinuous function is lower semicontinuous. By Theorem4.5 ϕβ = pβoϕ is lower
cl-supercontinuous for each β ∈ Λ.
Conversely, suppose that ϕβ : X ⊸ Xβ is a lower cl-supercontinuous for each
β ∈ Λ. Since the finite intersections and arbitrary union of cl-open sets is cl-
open, therefore, in view of Theorem 4.1 it suffices to prove that ϕ−1+ (B) is
a cl-open set for every basic open set B in the product space

∏
α∈Λ Xα. Let

B = Uα1 ×Uα2 ×...×UαN ×(
∏
α6=α1,α2,...,αN

Xα) be a basic open set in
∏
α∈Λ Xα

Now it is easily verified that ϕ−1+ (Uα1 × ... × UαN × (
∏
α6=α1,α2,...,αN

Xα)) =

(ϕα1)
−1
+ (Uα1)∩...∩(ϕαN )

−1
+ (UαN ). Since each ϕαi is cl-supercontinuous, ϕ

−1
+ (B)

is cl-open in X being the finite intersection of cl-open sets. Thus ϕ is lower
cl-supercontinuous. �

Theorem 4.10. For each α ∈ ∆ let ϕα : Xα ⊸ Yα be a multifunction and let
ϕ :

∏
α∈Λ Xα ⊸

∏
α∈Λ Yα be a multifunction defined by ϕ(x) =

∏
ϕα(xα) for

each x = (xα) ∈
∏
α∈Λ Xα. Then ϕ is lower cl-supercontinuous if and only if

each ϕα is lower cl-supercontinuous.



14 J. K. Kohli and C. P. Arya

Proof. Suppose that ϕ :
∏
α∈Λ Xα ⊸

∏
α∈Λ Yα is lower cl-supercontinuous. Let

Uβ be an open set in Yβ.Then Uβ ×
∏
α6=β Yα is a subbasic open set in

∏
α∈Λ Yα.

So in view of Theorem 4.1, ϕ−1+ (Uβ ×
∏
α6=β Yα) is a cl-open set in

∏
α∈Λ Xα.

Now it is easily verified that ϕ−1+ (Uβ ×
∏
α6=β Yα) = (ϕβ)

−1
+ (Uβ) ×

∏
α6=β Xα,

and so (ϕβ)
−1
+ (Uβ) is cl-open in Xβ. This proves that each ϕβ is lower cl-

supercontinuous.
Conversely suppose that ϕα : Xα ⊸ Yα is lower cl-supercontinuous for each
α ∈ Λ and let B = Vα1 × Vα2 × ... × VαN × (

∏
α6=α1,α2,...,αN

Yα) be a ba-

sic open set in
∏
α∈Λ Yα. Then ϕ

−1
+ (Vα1 × ... × VαN × (

∏
α6=α1,α2,...,αN

Yα)) =

(ϕα1)
−1
+ (Vα1 ) × ... × (ϕαN )

−1
+ (VαN ) × (

∏
α6=α1,α2,...,αN

Xα). Since each ϕα is

lower cl-supercontinuous, ϕ−1+ (B) cl-open in
∏
α∈Λ Xα and so ϕ is lower cl-

supercontinuous. �

Theorem 4.11. Let ϕ : X ⊸ Y be multifunction and let g : X ⊸ X × Y
defined by g(x) = {(x,y) ∈ X × Y |y ∈ ϕ(x)} for each x ∈ X be the graph
multifunction. Then g is lower cl-supercontinuous if and only if ϕ is lower
cl-supercontinuous and the space X is zero dimensional.

Proof. Suppose that g is lower cl-supercontinuous. By Theorem 4.5 the multi-
function ϕ = pyog is lower cl-supercontinuous. Next we shall show that X is
zero dimensional. Let U be an open set in X and let x ∈ U. Then U × Y is an
open set in X ×Y and g(x)∩(U ×Y ) 6= ∅. Since g is lower cl-supercontinuous,
there exists a clopen set W containing x such that g(z) ∩ (U × Y ) 6= ∅ for
every z ∈ W and so W ⊂ g−1+ (U × Y ) = U. Hence x ∈ W ⊂ U and X is zero
dimensional.
Conversely Suppose that ϕ is lower cl-supercontinuous. Let x ∈ X and let
W be an open set with g(x) ∩ W 6= ∅. Then there exist open sets U in X
and V in Y such that g(x) ∩ (U × V ) 6= ∅ and so x ∈ U and ϕ(x) ∩ V 6= ∅.
Since X is zero dimensional, there exists a clopen set G1 containing x such that
x ∈ G1 ⊂ U. Again since ϕ is lower cl- super continuous, there exists a clopen
set G2 containing x such that ϕ(h)∩V 6= ∅ for each h ∈ G2. Let G = G1 ∩G2.
Then G is a clopen set containing x and it is easily verified that g(h) ∩ W 6= ∅
for each h ∈ G. This proves that g is lower cl-supercontinuous. �

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(Received October 2009 – Accepted April 2010)

J.K. Kohli (jk kohli@yahoo.co.in)
Dept.of Mathematics, Hindu College, University of Delhi, Delhi 110007, India.

C. P. Arya (carya28@gmail.com)
Dept.of Mathematics, University of Delhi, Delhi 110007, India.


	Upper and lower cl-supercontinuous[8pt] multifunctions. By J. K. Kohli and C. P. Arya