KaReAGT.dvi


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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 9, No. 2, 2008

pp. 301-310

On some variations of multifunction continuity

A. Kanibir and I. L. Reilly

Abstract. This paper considers six classes of multifunctions bet-

ween topological spaces, namely almost ℓ−continuous multifunctions,

K-almost c−continuous multifunctions, nearly continuous multifunc-

tions, almost nearly continuous multifunctions, super continuous mul-

tifunctions, and δ−continuous multifunctions. We relate these classes

of multifunctions to others, and provide characterizations of related

concepts especially in terms of appropriate changes of topology.

2000 AMS Classification: 54A10, 54C05, 54C60, 54C10

Keywords: Almost ℓ−continuity, K-almost c−continuity, semi continuiy,
super continuity, δ−continuity, nearly continuity, change of topology, multi-
function.

1. Introduction and Preliminaries

There are several notions of generalized continuity for multifunctions in
the recent literature, for example almost continuity and ℓ−continuity [17], c-
continuity [10] and [17], almost c-continuity [14], nearly continuity [5], almost
nearly continuity [6], super continuity [1], δ−continuity [2] and [15]. In this
paper we consider each of the following continuity properties of multifunctions-
almost ℓ−continuity, K-almost c−continuity, nearly continuity, almost nearly
continuity, δ−continuity, and super continuity.

Throughout this paper, the closure (resp. interior) of a subset B in a topo-
logical space (Y, τ ) is denoted by clB (resp. intB). Then B is called regular
open if B = int(clB). The family of all regular open sets in (Y, τ ) which is
denoted by RO(Y, τ ), forms a base for a topology τS on Y , known as the
semiregularization of τ . In general τS ⊆ τ , and if τS = τ then (Y, τ ) is called
a semiregular space. A set A is said to be δ−closed [18] if for each point x /∈ A
there exists a regular open set containing x which has empty intersection with
A. A set B is δ−open [18] if and only if its complement is δ−closed. For any
topological space (Y, τ ) the collection of all δ−open sets forms a topology on X
which coincides with τS . For a topological space (Y, τ ), the cocompact topology



302 A. Kanibir and I. L. Reilly

of τ on Y is denoted by c(τ ) and defined by c(τ ) = {∅} ∪ {U ∈ τ : Y − U is
τ−compact}. The almost cocompact topology of τ on Y is denoted by e(τ )
and it has as a base e′(τ ) = {U ∈ RO(Y, τ ) : Y − U is τ−compact}. These
topologies are considered by Gauld [7] and [8]. If (Y, τ ) is a topological space
then the coLindelöf topology of τ on Y is denoted by ℓ(τ ) and defined by
ℓ(τ ) = {∅} ∪ {U ∈ τ : Y − U is τ−Lindelöf}, considered by Gauld, Mrsevic,
Reilly and Vamanamurthy [9]. Recall that a subset B of a topological space
(Y, τ ) is called N-closed in X if every regular open cover of B in (Y, τ ) has a
finite subcover. The concept of N-closed subsets was first considered by Carna-
han [3]. If (Y, τ ) is a topological space then the almost coN-closed topology of
τ on Y is denoted by p(τ ) and it has as a base p′(τ ) = {U ∈ RO(Y, τ ) : Y − U
is N-closed relative to τ}. The almost coLindelöf topology of τ on Y is denoted
by q(τ ) and it has as a base q′(τ ) = {U ∈ RO(Y, τ ) : Y − U is τ−Lindelöf}.
These topologies are considered by Konstadilaki-Savvopoulou and Reilly [12]
and [13].

By a multifunction F : (X, σ) → (Y, τ ), we mean a point-to-set corres-
pondence from (X, σ) into (Y, τ ), and we always assume that F (x) 6= ∅ for all
x ∈ X. For each B ⊆ Y , F +(B) = {x ∈ X : F (x) ⊆ B} and F −(B) = {x ∈
X : F (x) ∩ B 6= ∅}. For each A ⊆ X, F (A) = ∪x∈AF (x). As usual, F is said
to be a surjection if F (X) = Y . Moreover F : (X, σ) → (Y, τ ) is called upper
semicontinuous, abbreviated as u.s.c. (resp. lower semicontinuous, abbreviated
as l.s.c.) at a point x ∈ X if for each open set V in Y with F (x) ⊆ V ( resp.
F (x) ∩ V 6= ∅ ), there exists an open set U containing x such that F (U ) ⊆ V
(resp. F (z)∩V 6= ∅ for every z ∈ U ), or equivalently, if F +(V ) (resp. F −(V ))
is open in (X, σ) for every open set V of (Y, τ ).

A subset K ⊆ F (x0) is said to be a kernel [4] for F at x0, if the multifunction
FK : X → Y defined by:

FK (x) = K , if x = x0
FK (x) = F (x) ∩ (Y − F (x0)) , otherwise

is u.s.c. at x0.
For a multifunction F : X → Y , the graph multifunction GF : X → X × Y

is defined as GF (x) = {x} × F (x) for every x ∈ X.

This paper discusses how several versions of multifunction continuity
behave with respect to a change of topology approach. The results presented
here show that, in general, ”lower” multifunction continuities behave well from
this point of view. However, ”upper” multifunction continuities do not behave
at all well from this perspective. This behaviour of multifunctions can not be
predicted from a consideration of corresponding behaviour of (single-valued)
functions. It is unexpected, and a somewhat surprising situation.

2. Some properties

The following basic properties of almost ℓ−continuity and K-almost c−continuity
are useful in the sequel:



On some variations of multifunction continuity 303

Definition 2.1 ([11]). A multifunction F : X → Y is called

(a) upper almost ℓ−continuous (resp. upper K-almost c−continuous), or
u.a.ℓ−continuous (resp. u.K-a.c−continuous), at x ∈ X if for each
regular open subset V of Y with F (x) ⊆ V and having Lindelöf
(resp. compact) complement, there is an open neighbourhood U of x
such that F (U ) ⊆ V.

(b) lower almost ℓ−continuous (resp. lower K-almost c−continuous), or
l.a.ℓ−continuous (resp. l.K-a.c−continuous), at x ∈ X if for each
regular open subset V of Y with F (x) ∩ V 6= ∅ and having Lindelöf
(resp. compact) complement, there is an open neighbourhood U of x
such that F (z) ∩ V 6= ∅ for every point z ∈ U .

(c) almost ℓ−continuous (resp. K-almost c−continuous) at x ∈ X if it is
both u.a.ℓ−continuous (resp.u.K-a.c−continuous) and l.a.ℓ−continuous
(resp. l.K-a.c−continuous) at x ∈ X.

(d) almost ℓ−continuous (resp. u.a.ℓ−continuous ; l.a.ℓ−continuous, K-
a.c−continuous, u.K-a.c−continuous, l.K-a.c−continuous) if it is al-
most ℓ−continuous (resp. u.a.ℓ−continuous; l.a.ℓ−continuous, K-a.c−
continuous, u.K-a.c−continuous, l.K-a.c−continuous) at each point of
X.

Theorem 2.2 ([11]). Let F : (X, σ) → (Y, τ ) be a multifunction. Then

(a) F is upper (resp. lower) almost ℓ−continuous if and only if F +(V )
(resp. F −(V )) is open for each V ∈ q′(τ ).

(b) F is upper (resp. lower) K-almost c−continuous if and only if F +(V )
(resp. F −(V )) is open for each V ∈ e′(τ ).

Theorem 2.3. Let F : (X, σ) → (Y, τ ) be a multifunction and x0 ∈ X. Then
F is u.s.c. at x0 if and only if F (x0) is a kernel for F at x0.

Proof. (⇒) Let F : (X, σ) → (Y, τ ) be u.s.c. at x0 and K = F (x0). We will show
that FK : (X, σ) → (Y, τ ) is u.s.c. at x0. Let FK (x0) ⊆ V where V ∈ τ. Since
FK (x0) = F (x0) and F is u.s.c. at x0, there exists an open set U containing
x0 such that F (U ) ⊆ V. Therefore FK (U ) ⊆ V and so FK is u.s.c. at x0 and
hence F (x0) is a kernel for F at x0.

(⇐) Let K = F (x0) be a kernel for F at x0 and F (x0) ⊆ V where V ∈ τ.
Then FK (x0) = F (x0) ⊆ V. Since FK is u.s.c. at x0, there exists an open set
U containing x0 such that FK (U ) ⊆ V. Therefore F (U ) ⊆ V since F (x0) ⊆ V
and hence F is u.s.c. at x0. �

Theorem 2.4. Let F : (X, σ) → (Y, τ ) be a multifunction and F (x0) ∈ q
′(τ )

(resp. F (x0) ∈ e
′(τ )). Then F is upper almost ℓ−continuous (resp. upper

K-almost c−continuous ) at x0 ∈ X if and only if F (x0) is a kernel for F at
x0.

Proof. (⇒) Let K = F (x0) ∈ q
′(τ ) and FK (x0) = F (x0). We will show

that FK : (X, σ) → (Y, τ ) is u.s.c. at x0. Let FK (x0) ⊆ V where V ∈ τ.



304 A. Kanibir and I. L. Reilly

Since FK (x0) ∈ q
′(τ ) and F is u.a.ℓ−continuous multifunction, there exists an

open set U containing x0 such that F (U ) ⊆ FK (x0) = F (x0) ⊆ V. Therefore
FK (U ) ⊆ V and hence FK is u.s.c at x0 ∈ X. This shows that F (x0) is a kernel
for F at x0.

(⇐) It follows from Theorem 2.3 and the fact that every upper semi contin-
uous multifunction is upper almost ℓ−continuous [11].

The proof for upper K-almost c−continuity is similar. �

Theorem 2.5 ([16]). A subset A of (Y, τ ) is N-closed with respect to τ if
and only if A is compact in (Y, τS ).

Definition 2.6. A multifunction F : (X, σ) → (Y, τ ) is called

(a) upper (resp. lower) δ−continuous [2], or u.δ.c (resp. l.δ.c), if for
each x ∈ X and for each open subset V of Y with F (x) ⊆ V (resp.
F (x) ∩ V 6= ∅), there is an open neighbourhood U of x such that
F (int(clU )) ⊆ int(clV ) (resp. F (z) ∩ int(clV ) 6= ∅ for every point
z ∈ int(clU )).

(b) upper (resp. lower) super continuous [1], or u. sup .c (resp. l. sup .c),
if for each x ∈ X and for each open subset V of Y with F (x) ⊆ V
(resp. F (x) ∩ V 6= ∅), there is an open set U containing x such that
F (int(clU )) ⊆ V (resp. F (z) ∩ V 6= ∅ for every point z ∈ int(clU )).

(c) upper (resp. lower) c−continuous [10], or u.c.c (resp. l.c.c), if for
each x ∈ X and for each open subset V of Y with F (x) ⊆ V (resp.
F (x) ∩ V 6= ∅) and having compact complement, there is an open set
U containing x such that F (U ) ⊆ V (resp. F (z) ∩ V 6= ∅ for every
point z ∈ U ).

(d) upper (resp. lower) ℓ−continuous [17], or u.ℓ.c (resp. l.ℓ.c), if F +(V )
(resp. F −(V )) is open for each open subset V ⊆ Y having Lindelöf
complement.

(e) upper (resp. lower) nearly continuous [5] if for each x ∈ X and for
each open subset V of Y with F (x) ⊆ V (resp. F (x) ∩ V 6= ∅) and
having N-closed complement, there is an open set U containing x such
that F (U ) ⊆ V (resp. F (z) ∩ V 6= ∅ for every point z ∈ U ).

(f) upper (resp. lower) almost nearly continuous [6] if for each x ∈ X and
for each open subset V of Y with F (x) ⊆ V (resp. F (x) ∩ V 6= ∅)
and having N-closed complement, there is an open set U containing x
such that F (U ) ⊆ int(clV ) (resp. F (z) ∩ int(clV ) 6= ∅ for every point
z ∈ U ).

The following characterizations are important for our discussion.

Theorem 2.7. A multifunction F : (X, σ) → (Y, τ ) is

(a) upper (resp. lower) δ−continuous [2] if and only if F +(V ) (resp.
F −(V )) is δ−open for each regular open subset V ⊆ Y.

(b) upper (resp. lower) super continuous [1] if and only if F +(V ) (resp.
F −(V )) is δ−open for each open subset V ⊆ Y.



On some variations of multifunction continuity 305

(c) upper (resp. lower) c−continuous [10] and [17] if and only if F +(V )
(resp. F −(V )) is open for each open subset V ⊆ Y having compact
complement.

(d) upper (resp. lower) nearly continuous [5] if and only if F +(V ) (resp.
F −(V )) is an open set for any open subset V ⊆ Y having N-closed
complement.

(e) upper (resp. lower) almost nearly continuous [6] if and only if F +(V )
(resp. F −(V )) is an open set for any regular open subset V ⊆ Y having
N-closed complement.

The next six results indicate how change of topology is related to these
concepts of near continuity of multifunctions.

Theorem 2.8. Let F : (X, σ) → (Y, τ ) be a multifunction. Then F : (X, σ) →
(Y, τS ) is upper (resp. lower) c−continuous if and only if F : (X, σ) → (Y, τ )
is upper (resp. lower) nearly continuous.

Proof. (⇒) Let V be an open subset having N-closed complement. Then
Y − V is compact in (Y, τS ) by Theorem 2.5. Since F : (X, σ) → (Y, τS ) is
upper c−continuous, F +(V ) ∈ σ. Hence F : (X, σ) → (Y, τ ) is upper nearly
continuous. The proof for the case F lower nearly continuous is analogous.

(⇐) Let V ∈ τS and having compact complement. Then Y − V is N-closed
with respect to τ by Theorem 2.5. Since F : (X, σ) → (Y, τ ) is upper nearly
continuous, F +(V ) ∈ σ. Hence F : (X, σ) → (Y, τ ) is upper c−continuous. The
proof for the case F lower c−continuous is analogous. �

Corollary 2.9. Let F : (X, σ) → (Y, τ ) be a multifunction and Y be semi
regular. Then F : (X, σ) → (Y, τ ) is upper (resp. lower) c−continuous if and
only if F : (X, σ) → (Y, τ ) is upper (resp. lower) nearly continuous.

Theorem 2.10. Let F : (X, σ) → (Y, τ ) be a multifunction. Then F :
(X, σ) → (Y, τS ) is upper (resp. lower) K-almost c−continuous if and only
if F : (X, σ) → (Y, τ ) is upper (resp. lower) almost nearly continuous.

Proof. (⇒) Let V be a regular open subset having N-closed complement.
Then Y − V is compact in (Y, τS ) by Theorem 2.5. Since F : (X, σ) → (Y, τS )
is upper K-almost c−continuous, F +(V ) ∈ σ. Hence F : (X, σ) → (Y, τ ) is
upper almost nearly continuous. The proof for the case F lower almost nearly
continuous is analogous.

(⇐) Let V ∈ q′(τS ). Then Y − V is N-closed with respect to τ by Theorem
2.5. Since F : (X, σ) → (Y, τ ) is upper almost nearly continuous, F +(V ) ∈ σ.
Hence F : (X, σ) → (Y, τ ) is upper K-almost c−continuous. The proof for the
case F lower K-almost c−continuous is analogous. �

Corollary 2.11. Let F : (X, σ) → (Y, τ ) be a multifunction and Y be semi reg-
ular. Then F : (X, σ) → (Y, τ ) is upper (resp. lower) K-almost c−continuous
if and only if F : (X, σ) → (Y, τ ) is upper (resp. lower) almost nearly contin-
uous.



306 A. Kanibir and I. L. Reilly

We have the following corollary since upper (resp. lower) almost ℓ−continuous
multifunctions are upper (resp. lower) K-almost c−continuous [11].

Corollary 2.12. If F : (X, σ) → (Y, τS ) is upper (resp. lower) almost
ℓ−continuous, then F : (X, σ) → (Y, τ ) is upper (resp. lower) almost nearly
continuous.

Theorem 2.13. Let F : (X, σ) → (Y, τ ) be a multifunction. Then F :
(X, σ) → (Y, τ ) is lower almost nearly continuous if and only if F : (X, σ) →
(Y, p(τ )) is lower semi continuous.

Proof. (⇒) Let V ∈ p(τ ). We can write V = ∪α∈ΛVα where Vα is a regular
open set having N-closed complement for each α ∈ Λ. We have F −(∪α∈ΛVα) =
∪α∈ΛF

−(Vα). From Theorem 2.7(e) we have that F
−(Vα) is an open set for

each α ∈ Λ. So F −(V ) is an open set. Hence F : (X, σ) → (Y, p(τ )) is l.s.c.
(⇐) Obvious. �

A result analogous to Theorem 2.13 for upper almost nearly continuous
does not hold as the following example shows.

Example 2.14 ([11]). Let X = Y = {1, 2, 3, 4} and σ = {{1}, {1, 3, 4}, X, ∅}
be the topology on X and τ = {{1}, {2}, {1, 2}, Y, ∅} be the topology on Y.
Let F be defined as F (1) = {4}, F (2) = {1, 2}, F (3) = {3}, F (4) = {4}. The
family p′(τ ) = {{1}, {2}, Y, ∅} is a base consisting of regular open sets having
N-closed complement in Y for p(τ ) = τ. Then for any regular open set V having
N-closed complement we have F +(V ) ∈ σ. Therefore F : (X, σ) → (Y, τ ) is
upper almost nearly continuous. The topology p(τ ) contains the set {1, 2} but
F +({1, 2}) = {2} /∈ σ. Hence F : (X, σ) → (Y, p(τ )) is not u.s.c.

We have the following corollary by Theorem 2.13 and since lower semi con-
tinuous multifunctions are lower almost ℓ−continuous multifunctions [11].

Corollary 2.15. If the multifunction F : (X, σ) → (Y, τ ) is lower almost
nearly continuous, then F : (X, σ) → (Y, p(τ )) is l.a.ℓ−continuous.

The proof of each of the next three results is straightforward from Theorem
2.7 and Definition 2.6(d). Note that Propositions 2.16 and 2.23 provide change
of topology results for upper super continuity, and that this is an unusual
circumstance. These are exceptional results, as the counter-examples 2.14 and
2.20 show.

Proposition 2.16. Let F : (X, σ) → (Y, τ ) be a multifunction. Then F :
(X, σ) → (Y, ℓ(τ )) is u. sup .c.(resp. l. sup .c.) if and only if F : (X, σS ) →
(Y, τ ) is u.ℓ.c.(resp. l.ℓ.c.)

Proposition 2.17. Let F : (X, σ) → (Y, τ ) be a multifunction. Then F :
(X, σ) → (Y, τ ) is l.δ.c. if and only if F : (X, σ) → (Y, τS ) is l. sup .c.

Proposition 2.18. Let F : (X, σ) → (Y, τ ) be a multifunction. Then F :
(X, σS ) → (Y, τ ) is l.a.ℓ−continuous if and only if F : (X, σ) → (Y, q(τ )) is
l.δ.c.



On some variations of multifunction continuity 307

Theorem 2.19. Let F : (X, σ) → (Y, τ ) be a multifunction. Then the follow-
ing statements are equivalent:

(a) F : (X, σS ) → (Y, τ ) is l.a.ℓ−continuous.
(b) F : (X, σS ) → (Y, q(τ )) is l. sup .c.
(c) F : (X, σS ) → (Y, q(τ )) is l.δ.c.
(d) F : (X, σS ) → (Y, q(τ )) is l.s.c.
(e) F : (X, σS ) → (Y, q(τ )) is l.ℓ.c.

Proof. The proofs of (b)⇒ (c), (c)⇒ (d), (d)⇒ (e), (e)⇒ (a) are immediate.
We prove only (a)⇒ (b).

Let V ∈ q(τ ). Then we can write V = ∪α∈ΛVα where Vα is a regular open
set having Lindelöf complement, for each α ∈ Λ. We have F −(∪α∈ΛVα) =
∪α∈ΛF

−(Vα). Therefore F
−(Vα) ∈ σS by hypothesis. So F

−(V ) ∈ σS and
therefore is δ−open. Hence F : (X, σS ) → (Y, q(τ )) is l. sup .c. �

A result analogous to Theorem 2.19 for u.a.ℓ−continuous, u. sup .c., u.δ.c.,
u.s.c., u.ℓ.c. does not hold as the following example shows.

Example 2.20. Let us redefine the topology σ in Example 2.14 as follows
σ = {{1, 2}, {3, 4}, ∅, X}. No other changes are made to Example 2.14. It is
obvious that σS = σ and q(τ ) = τ. Then F is u.a.ℓ−continuous and u.δ.c but
not u. sup .c., u.s.c. and u.ℓ.c even if X is regular and Lindelöf and Y is semi
regular and Lindelöf.

Proposition 2.21 ([12]). If ( Y, τ ) is a Lindelöf and semi regular space, then
q(τ ) = τ.

We have the following corollary by Theorem 2.19 and Proposition 2.21.

Corollary 2.22. Let X be a semi regular topological space and let Y be a semi
regular and Lindelöf topological space and F : (X, σ) → (Y, τ ) be a multifunc-
tion. Then the following statements are equivalent:

(a) F : (X, σ) → (Y, τ ) is l.a.ℓ−continuous.
(b) F : (X, σ) → (Y, τ ) is l. sup .c.
(c) F : (X, σ) → (Y, τ ) is l.δ.c.
(d) F : (X, σ) → (Y, τ ) is l.s.c.
(e) F : (X, σ) → (Y, τ ) is l.ℓ.c.

A result analogous to Corollary 2.22 for u.a.ℓ−continuous, u. sup .c., u.δ.c.,
u.s.c., u.ℓ.c. does not hold as Example 2.20 shows.

The next two results are analogues of Propositions 2.16 and 2.18 respectively.
Their proofs are straightforward from Theorem 2.7.

Proposition 2.23. Let F : (X, σ) → (Y, τ ) be a multifunction. Then F :
(X, σ) → (Y, c(τ )) is u. sup .c.(resp. l. sup .c.) if and only if F : (X, σS ) →
(Y, τ ) is u.c.c.(resp. l.c.c.).



308 A. Kanibir and I. L. Reilly

Proposition 2.24. Let F : (X, σ) → (Y, τ ) be a multifunction. Then F :
(X, σS ) → (Y, τ ) is l.K-a.c−continuous if and only if F : (X, σ) → (Y, e(τ ))
is l.δ.c.

Similarly to Theorem 2.19, we can obtain the following characterizations.

Theorem 2.25. Let F : (X, σ) → (Y, τ ) be a multifunction. Then the follow-
ing statements are equivalent:

(a) F : (X, σS ) → (Y, τ ) is l.K-a.c−continuous.
(b) F : (X, σS ) → (Y, e(τ )) is l. sup .c.
(c) F : (X, σS ) → (Y, e(τ )) is l.δ.c.
(d) F : (X, σS ) → (Y, e(τ )) is l.s.c.
(e) F : (X, σS ) → (Y, e(τ )) is l.c.c.

A result analogous to Theorem 2.25 for u.K-a.c−continuous, u. sup .c., u.δ.c.,
u.s.c., u.c.c. does not hold as Example 2.20 shows.

Corresponding to Proposition 2.21 we have the following result, which seems
to be new.

Proposition 2.26. If ( Y, τ ) is a compact and semi regular space, then e(τ ) =
τ.

Corollary 2.27. Let X be a semi regular topological space and let Y be a semi
regular and compact topological space and F : (X, σ) → (Y, τ ) be a multifunc-
tion. Then the following statements are equivalent:

(a) F : (X, σ) → (Y, τ ) is l.K-a.c−continuous.
(b) F : (X, σ) → (Y, τ ) is l. sup .c.
(c) F : (X, σ) → (Y, τ )is l.δ.c.
(d) F : (X, σ) → (Y, τ )is l.s.c.
(e) F : (X, σ) → (Y, τ )is l.c.c.
(f) F : (X, σ) → (Y, τ ) is lower nearly continuous.
(g) F : (X, σ) → (Y, τ )is lower almost nearly continuous.

Proof. This is an immediate consequence of Theorem 2.25, Proposition 2.26,
and Corollaries 2.9 and 2.11. �

A result analogous to Corollary 2.27 for u.K-a.c−continuous, u. sup .c., u.δ.c.,
u.s.c., u.c.c., upper almost nearly continuous, upper nearly continuous does not
hold as Example 2.20 shows.

Several properties of upper and lower almost ℓ−continuous multifunctions
have been given in [11]. We now provide some more. We can also obtain
similar sets of results [which are parallel to the following results] for u.K-
a.c−continuous ( resp. l.K-a.c−continuous) multifunctions. We shall leave
them unstated.

The following result is immediate. The proof is left to the reader.



On some variations of multifunction continuity 309

Theorem 2.28. Let (X, σ), (Y, τ ), (Z, ϑ) be topological spaces and let F : X →
Y and G : Y → Z be multifunctions. If F : X → Y is upper (resp. lower)
semi continuous and G : Y → Z is upper (resp. lower) almost ℓ−continuous,
then F ◦ G : X → Z is upper(resp. lower) almost ℓ−continuous multifunction.

Theorem 2.29. Let F : (X, σ) → (Y, τ ) be a multifunction and let A ⊆
X be a nonempty subset. If F is u.a.ℓ−continuous (resp. l.a.ℓ−continuous)
then the restriction multifunction F |A : A → Y is u.a.ℓ−continuous (resp.
l.a.ℓ−continuous).

Proof. We prove only the assertion for F |A u.a.ℓ−continuous, the proof for
F |A l.a.ℓ−continuous being analogous. Let x ∈ A and V be a regular open
subset of (Y, τ ) having Lindelöf complement such that (F |A)(x) ⊆ V. Since
F is u.a.ℓ−continuous and (F |A)(x) = F (x), there exists U ∈ σ containing x
such that F (U ) ⊆ V. Then x ∈ A ∩ U and A ∩ U is open in A. Moreover
(F |A)(A ∩ U ) ⊆ V. This shows that F |A is u.a.ℓ−continuous. �

Theorem 2.30. Let F : (X, σ) → (Y, τ ) be a multifunction. Let {Vα : α ∈
Λ} be an open cover of X. If the restriction multifunction F α = F |Vα is
u.a.ℓ−continuous (resp. l.a.ℓ−continuous) multifunction for each α ∈ Λ, then
F is u.a.ℓ−continuous (resp. l.a.ℓ−continuous).

Proof. Let V ∈ q′(τ ). Since Fα is u.a.ℓ−continuous F
+
α

(V ) ⊆ intVα (F
+(V ))

and since Vα is open, we have F
+(V ) ∩ Vα ⊆ intVα (F

+(V ) ∩ Vα) and F
+(V ) ∩

Vα ⊆ int(F
+(V )) ∩ Vα. Since {Vα : α ∈ Λ} is an open cover of X, F

+(V ) =
int(F +(V )). Hence F is u.a.ℓ−continuous.

The proof for l.a.ℓ−continuous is similar. �

Corollary 2.31. Let {Vα : α ∈ Λ} be an open cover of X. A multifuntion
F : X → Y is u.a.ℓ−continuous (resp. l.a.ℓ−continuous) if and only if the
restriction F |Vα : Vα → Y is u.a.ℓ−continuous (resp. l.a.ℓ−continuous) for
each α ∈ Λ.

Proof. This is an immediate consequence of Theorems 2.29 and 2.30. �

Theorem 2.32. Let F : (X, σ) → (Y, τ ) be multifunction and let X × Y
be a Lindelöf space. If the graph multifunction of F is lower (resp. up-
per) almost ℓ−continuous multifunction, then F is lower (resp. upper) almost
ℓ−continuous multifuction.

Proof. Let x ∈ X and let V ∈ q′(τ ) with F (x)∩V 6= ∅. Then GF (x)∩(X×V ) 6=
∅ and X × V is a regular open set having Lindelöf complement. Since the
graph multifunction GF is lower almost ℓ−continuous, there exists an open
neighbourhood U of x such that GF (z) ∩ (X × V ) 6= ∅ for every point z ∈ U.
Thus F (z) ∩ V 6= ∅. Hence F is lower almost ℓ−continuous. The proof of the
upper almost ℓ−continuity of F is similar. �



310 A. Kanibir and I. L. Reilly

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Received April 2007

Accepted June 2008

A. Kanibir (kanibir@hacettepe.edu.tr)
Department of Mathematics, Hacettepe University, 06532 Beytepe, Ankara,
Turkey

I. L. Reilly (i.reilly@auckland.ac.nz)
Department of Mathematics, University of Auckland, P.B. 92019, Auckland,
New Zealand