Download ON UPPER AND LOWER CONTRA-CONTINUOUS

ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII “AL.I. CUZA” DIN IAŞI (S.N.)
MATEMATICĂ, Tomul LIV, 2008, f.1
ON UPPER AND LOWER CONTRA-CONTINUOUS
MULTIFUNCTIONS
BY
ERDAL EKICI, SAEID JAFARI and TAKASHI NOIRI
Abstract. In 1996, Dontchev [4] introduced and investigated the notion of contracontinuity. In this paper we introduce and study the basic properties of upper (lower)
contra-continuous multifunctions.
Mathematics Subject Classification 2000: 54C60.
Key words: strongly S-closed space, multifunction, contra-continuity.
1. Introduction. Throughout this paper, spaces X and Y mean
topological spaces. For a subset A of X, cl(A) and int(A) represent the
closure of A and the interior of A, respectively.
In this paper, F : X → Y presents a multifunction. For a multifunction
F : X → Y , we shall denote the upper and lower inverse of a set A of Y
by F + (A) and F − (A), respectively, that is, F + (A) = {x ∈ X : F (x) ⊂ A}
and F − (A) = {x ∈ X : F (x) ∩ A 6= ∅} [3].
The graph multifunction GF : X → X ×Y of a multifunction F : X → Y
is defined as follows GF (x) = {x} × F (x) for every x ∈ X.
Definition 1. ([10]) The set ∩{A ∈ τ : B ⊂ A} is called the kernel of
a subset B of a space (X, τ ) and is denoted by ker(B).
A multifunction F : X → Y is called upper semi-continuous (resp. lower
semi-continuous) [14] if F + (V ) (resp. F − (V )) is open in X for every open
set V of Y .
Lemma 2. ([12]) Let X and Y be topological spaces and let A ⊂ X and
B ⊂ Y . The following properties hold for a multifunction F : X → Y :
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ERDAL EKICI, SAEID JAFARI and TAKASHI NOIRI
(1)
(2)
G+
F (A
G−
F (A
2
F + (B),
× B) = A ∩
× B) = A ∩ F − (B).
Definition 3. A subset A of a space X is said to be
(1) α-open [11] if A ⊂ int(cl(int(A))).
(2) semi-open [8] if A ⊂ cl(int(A)).
(3) preopen [9] if A ⊂ int(cl(A)).
(4) β-open [1] if A ⊂ cl(int(cl(A))).
The intersection of all α-closed (resp. semi-closed, preclosed, β-closed)
sets of X containing A is called the α-closure (resp. semi-closure, preclosure,
β-closure) of A and is denoted by α-cl(A) (resp. s-cl(A), p-cl(A) and βcl(A)).
2. Contra-continuous multifunctions
Definition 4. A multifunction F : (X, τ ) → (Y, σ) is called
(1) lower contra-continuous at x ∈ X if for each closed set A such that
x ∈ F − (A), there exists an open set U containing x such that U ⊂ F − (A),
(2) upper contra-continuous at x ∈ X if for each closed set A such that
x ∈ F + (A), there exists an open set U containing x such that U ⊂ F + (A).
(3) lower (upper) contra-continuous if F has this property at each point
of X.
Theorem 5. The following are equivalent for a multifunction F : (X, τ )
→ (Y, σ):
(1) F is upper contra-continuous,
(2) F + (A) is an open set for any closed set A ⊂ Y ,
(3) F − (U ) is a closed set for any open set U ⊂ Y ,
(4) for each x ∈ X and each closed set A containing F (x), there exists
an open set U containing x such that if y ∈ U , then F (y) ⊂ A.
Proof. (1)⇔(2): Let A be a closed set in Y and x ∈ F + (A). Since F is
upper contra-continuous, there exists an open set U containing x such that
U ⊂ F + (A). Thus, F + (A) is open.
The converse of the proof is similar.
(2)⇔(3): This follows from the fact that F + (Y \A) = X\F − (A) for
every subset A of Y .
(1)⇔(4): Obvious.
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Lemma 6. ([6]) Let A, B be subsets of a space (X, τ ). The following
properties hold:
(1) x ∈ ker(A) if and only if A ∩ B 6= ∅ for any closed set B containing
x.
(2) If A ∈ τ , then A = ker(A).
Theorem 7. Let F : (X, τ ) → (Y, σ) be a multifunction. If cl(F − (A)) ⊂
for every subset A of Y , then F is upper contra-continuous.
F − (ker(A))
Proof. Suppose that cl(F − (A)) ⊂ F − (ker(A)) for every subset A of
Y . Let A ∈ τ . By Lemma 6, cl(F − (A) ⊂ F − (ker(A)) = F − (A). Thus,
cl((F − (A)) = F − (A) and hence F − (A) is closed in X. Consequently, by
Theorem 5, F is upper contra-continuous.
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Definition 8. ([5]) A multifunction F : X → Y is called
(1) lower clopen continuous if for each x ∈ X and each
such that x ∈ F − (V ), there exists a clopen set U containing
U ⊂ F − (V ).
(2) upper clopen continuous if for each x ∈ X and each
such that x ∈ F + (V ), there exists a clopen set U containing
U ⊂ F + (V ).
open set V
x such that
open set V
x such that
Definition 9. ([15, 16]) A multifunction F : X → Y is said to be:
(1) lower weakly continuous if for each x ∈ X and each open set V of
Y such that x ∈ F − (V ), there exists an open set U in X containing x such
that U ⊂ F − (cl(V )).
(2) upper weakly continuous if for each x ∈ X and each open set V of
Y such that x ∈ F + (V ), there exists an open set U in X containing x such
that U ⊂ F + (cl(V )).
Theorem 10. If F : X → Y is upper/lower contra-continuous, then F
is upper/lower weakly continuous.
Proof. Let F be upper contra-continuous, x ∈ X and V any open set
of Y contining F (x). Then cl(V ) is a closed set contining F (x). Since F is
upper contra-continuous by Theorem 5 there exists an open set U containing
x such that U ⊂ F + (cl(V )). Hence F is upper weakly continuous.
The proof for lower contra-coninuous is similar.
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ERDAL EKICI, SAEID JAFARI and TAKASHI NOIRI
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Remark 11. The following diagram hold for a multifunction F : X →
Y:
upper/lower semi-continuous ⇒ upper/lower weakly continuous
⇑
⇑
upper/lower clopen continuous ⇒ upper/lower contra-continuous
None of these implications is reversible as shown in the following examples.
Example 12. Let X = {a, b, c, d} and τ = {∅, X, {a}, {a, b}, {a, b, c}}.
Define a multifunction F : X → X by F (a) = {b, c}, F (b) = {a}, F (c) =
{a, d}, F (d) = {a}. Then F is upper contra-continuous but it is not upper
semi-continuous. Define a multifunction F : X → X by F (a) = {a, b},
F (b) = {b}, F (c) = {a, b}, F (d) = {d}. Then F is upper semi-continuous
but it is not upper contra-continuous.
Example 13. Let X = {a, b, c} and τ = {∅, X, {a}, {c}, {a, c}, {b, c}}.
Define a multifunction F : X → X by F (a) = {b, c}, F (b) = {a, c},
F (c) = {a, b}. Then F is upper contra-continuous but it is not upper
clopen continuous. Define a multifunction G : X → X by G(a) = {b, c},
G(b) = {a, b}, G(c) = {a, c}. Then G is upper semi-continuous but it is not
upper contra-continuous.
Theorem 14. The following are equivalent for a multifunction F : X →
Y :
(1) F is lower contra-continuous multifunction,
(2) F − (A) is an open set for any closed set A ⊂ Y ,
(3) F + (U ) is a closed set for any open set U ⊂ Y ,
(4) for each x ∈ X and for each closed set A such that F (x) ∩ A 6= ∅,
there exists an open set U containing x such that if y ∈ U , then F (y)∩A 6= ∅.
Proof. The proof is similar to that of Theorem 5.
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Theorem 15. Suppose that one of the following properties holds for a
multifunction F : (X, τ ) → (Y, σ):
(1) F (cl(A)) ⊂ ker(F (A)) for every subset A of X,
(2) cl(F + (A)) ⊂ F + (ker(A)) for every subset A of Y .
Then F is lower contra-continuous.
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Proof. Suppose that F (cl(A)) ⊂ ker(F (A)) for every subset A of
X. Let A ⊂ Y . Then, F (cl(F + (A))) ⊂ ker(A) and thus cl(F + (A)) ⊂
F + (ker(A)). Therefore, the implication (1)⇒(2) holds.
Suppose that cl(F + (A)) ⊂ F + (ker(A)) for every subset A of Y . Let A ∈
τ . By Lemma 6, cl(F + (A) ⊂ F + (ker(A)) = F + (A). Thus, cl((F + (A)) =
F + (A) and hence F + (A) is closed in X. Consequently, by Theorem 14, F
is lower contra-continuous.
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Corollary 16. ([4]) For a function f : (X, τ ) → (Y, σ), the following
are equivalent:
(1) f is contra-continuous,
(2) f −1 (A) is closed for any open set A in Y ,
(3) for each x ∈ X and for each closed set A containing f (x), there
exists an open set U containing x such that f (U ) ⊂ A.
Corollary 17. Let f : (X, τ ) → (Y, σ) be a function. Suppose that one
of the following properties hold:
(1) f (cl(A)) ⊂ ker(f (A)) for every subset A of X,
(2) cl(f −1 (A)) ⊂ f −1 (ker(A)) for every subset A of Y .
Then f is contra-continuous.
Definition 18. A topological space X is called strongly S-closed [4] if
every closed cover of X has a finite subcover.
Theorem 19. Let F : X → Y be an upper contra-continuous surjective
multifunction. Suppose that F (x) is strongly S-closed for each x ∈ X. If X
is compact, then Y is strongly S-closed.
Proof. Let {Ak }k∈I be a closed cover of Y . Since F (x) is strongly
S-closedSfor each x ∈ X, there exists a finite subset Ix of I such that
F (x) ⊂ k∈Ix Ak (= Ax ). Since F is upper contra-continuous, there exists
an open set Ux of X containing x such that F (Ux ) ⊂ Ax . The family
{Ux }x∈X is an open cover of
S X. Since X is compact, there exist x1 , x2 , x3 ,
...,xn in X such that X = ni=1 Uxi . Thus,
Y = F (X) = F (
n
[
Uxi ) =
i=1
and hence Y is strongly S-closed.
n
[
F (Uxi ) ⊂
i=1
n
[
Axi =
i=1
n [
[
Ak
i=1k∈Ixi
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ERDAL EKICI, SAEID JAFARI and TAKASHI NOIRI
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Theorem 20. If F : X → Y is an upper/lower contra-continuous punctually connected surjective multifunction and X is connected, then Y is
connected.
Proof. Since F is upper/lower contra-continuous, then by Theorem
10, F is upper/lower weakly continuous. Then, the conclusion follows from
Theorem 11 of [16].
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Corollary 21. If f : X → Y is contra-continuous surjection and X is
connected, then Y is connected.
Theorem 22. Let F : X → Y and G : Y → Z be multifunctions. If F
is upper (lower) semi-continuous and G is upper (lower) contra-continuous,
then G ◦ F : X → Z is upper (lower) contra-continuous.
Proof. Let A ⊂ Z be a closed set. We have
(G ◦ F )+ (A) = F + (G+ (A)) ((G ◦ F )− (A) = F − (G− (A))).
Since G is upper (lower) contra-continuous, then G+ (A) (G− (A)) is an
open set. Since F is upper (lower) semi-continuous, then F + (G+ (A))
(F − (G− (A))) is an open set. Thus, G ◦ F is an upper (lower) contracontinuous multifunction.
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Theorem 23. Let F : X → Y be a multifunction and let A ⊂ X. If
F is a lower (upper) contra-continuous multifunction, then the restriction
multifunction F |A : A → Y is lower (upper) contra-continuous.
Proof. Let B ⊂ Y be a closed set and x ∈ A and let x ∈ (F |A )− (B).
Since F is lower contra-continuous multifunction, then there exists an open
set U in X containing x such that U ⊂ F − (B). This implies that x ∈ U ∩ A
is open in A and hence U ∩ A ⊂ (F |A )− (B). Thus, F |A is lower contracontinuous.
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Theorem 24. The following are equivalent for an open cover {Ai }i∈I
of a space X:
(1) A multifunction F : X → Y is upper contra-continuous (resp. lower
contra-continuous),
(2) The restriction F |Ai : Ai → Y is upper contra-continuous (resp.
lower contra-continuous) for each i ∈ I.
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ON UPPER AND LOWER CONTRA-CONTINUOUS MULTIFUNCTIONS
81
Proof. (1)⇒(2): Let i ∈ I and B be any closed set of Y . Since F
is upper contra-continuous, F + (B) is open in X. Then (F |Ai )+ (B) =
F + (B) ∩ Ai is open in Ai . Thus, F |Ai is upper contra-continuous.
(2)⇒(1): Let B be a closed set in Y . Since F |Ai is upper contracontinuous for each i ∈ I, (F |Ai )+ (B) = F + (B) ∩ Ai is open in Ai . Since
+
+
A
Si is open in+X, (F |Ai ) (B) is open in X for each i ∈ I and hence F (B) =
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i∈I (F |Ai ) (B) is open in X. Thus, F is upper contra-continuous.
Theorem 25. Let F : X → Y be a multifunction. Suppose that F (X)
is endowed with the subspace topology. If F is upper contra-continuous, then
F : X → F (X) is upper contra-continuous.
Proof. Let F be an upper contra-continuous multifunction. Then
F + (V ∩ F (X)) = F + (V ) ∩ F + (F (X)) = F + (V )
is open for each closed subset V of Y . Thus, F : X → F (X) is upper
contra-continuous.
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Definition 26. A subset A of a space X is called:
(1) α-paracompact [17] if every open cover of A is refined by a cover of
A which consists of open sets of X and locally finite in X,
(2) α-regular [7] if for each x ∈ A and each open set U of X containing
x, there exists an open set V of X such that x ∈ V ⊂ cl(V ) ⊂ U .
Lemma 27. ([7]) If A is an α-regular α-paracompact set of a space X
and U is an open neighbourhood of A, then there exists an open set V of X
such that A ⊂ V ⊂ cl(V ) ⊂ U .
Definition 28. ([2]) For a multifunction F : X → Y , a multifunction
cl(F ) : X → Y is defined by cl(F )(x) = cl(F (x)) for each point x ∈ X.
Similarly, we denote s-cl(F ), p-cl(F ), α-cl(F ), β-cl(F ).
Lemma 29. If F : X → Y is a multifunction such that F (x) is αregular α-paracompact for each x ∈ X, then
(1) G+ (U ) = F + (U ) for each open set U of Y ,
(2) G− (K) = F − (K) for each closed set K of Y ,
where G denotes cl(F ), s-cl(F ), p-cl(F ), α-cl(F ), β-cl(F ).
Proof. The proof follows from Lemma 3.6 of [13].
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ERDAL EKICI, SAEID JAFARI and TAKASHI NOIRI
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Lemma 30. For a multifunction F : X → Y the following properties
hold:
(1) G− (U ) = F − (U ) for each open set U of Y ,
(2) G+ (K) = F + (K) for each closed set K of Y ,
where G denotes cl(F ), s-cl(F ), p-cl(F ), α-cl(F ), β-cl(F ).
Proof. The proof follows from Lemma 3.7 of [13].
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Theorem 31. Let F : X → Y be a multifunction. The following are
equivalent:
(1) F is upper contra-continuous,
(2) G is upper contra-continuous.
Proof. (1)⇒(2): Let K be a closed set of Y . Then by Theorem 5
and Lemma 30, G+ (K) = F + (K) is an open set of X. Hence G is upper
contra-continuous.
(2)⇒(1): Let K be a closed set of Y . Then by Theorem 5 and Lemma
30, F + (K) = G+ (K) is an open set of X. Hence F is upper contracontinuous.
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Theorem 32. Let F : X → Y be a multifunction such that F (x) is
α-regular α-paracompact for each x ∈ X. The following are equivalent:
(1) F is lower contra-continuous,
(2) G is lower contra-continuous.
Proof. (1)⇒(2): Let K be a closed set of X. Then by Lemma 29 and
Theorem 14, G− (K) = F − (K) is open in X. Hence G is lower contracontinuous.
(2)⇒(1): Let K be a closed set of Y . Then by Lemma 29 and Theorem l4, F − (K) = G− (K) is an open set of X. Hence F is lower contracontinuous.
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3. The graph multifunction and the product spaces
Theorem 33. Let F : X → Y be a multifunction. If the graph multifunction of F is upper contra-continuous, then F is upper contra-continuous.
Proof. Let GF : X → X × Y be upper contra-continuous and x ∈ X.
Let A be any closed set of Y containing F (x). Since X × A is closed in
X × Y and GF (x) ⊂ X × A, there exists an open set U containing x such
+
that GF (U ) ⊂ X × A. By Lemma 2, U ⊂ G+
F (X × A) = F (A) and
F (U ) ⊂ A. Thus, F is upper contra-continuous.
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ON UPPER AND LOWER CONTRA-CONTINUOUS MULTIFUNCTIONS
83
Theorem 34. Let F : X → Y be a multifunction. If GF : X → X × Y
is lower contra-continuous, then F is lower contra-continuous.
Proof. Let GF be lower contra-continuous and x ∈ X. Let A be any
closed set in Y such that x ∈ F − (A). This implies that X × A is closed in
X × Y and
GF (x) ∩ (X × A) = ({x} × F (x)) ∩ (X × A) = {x} × (F (x) ∩ A) 6= ∅.
Since GF is lower contra-continuous, there exists an open set U containing
−
x such that U ⊂ G−
F (X × A). By Lemma 2, U ⊂ F (A). Thus, F is lower
contra-continuous.
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Corollary 35. Let f : X → Y be a function. If the graph function
g : X → X × Y , defined by g(x) = (x, f (x)) for each x ∈ X, is contracontinuous, then f is contra-continuous
Theorem 36. Let (X, τQ) and (Xi , τi ) be topological spaces (i ∈ I). If
a multifunction F : X → i∈I Xi is an upper (lower) contra-continuous
multifunction, then Pi ◦ F is an upper
Q (resp. lower) contra-continuous multifunction for each i ∈ I, where Pi : i∈I Xi → Xi is the projection for each
i ∈ I.
Proof. Let Ai0 be a closed set in (Xi0 , τi0 ). We have
Q
(Pi0 ◦ F )+ (Ai0 ) = F + (Pi+0 (Ai0 )) = F + (Ai0 ×
Xi ).
i6=i0
Q
Since F is an upper contra-continuous multifunction, then F + (Ai0 × i6=i0 Xi )
is open in (X, τ ). This implies that Pi0 ◦ F is an upper contra-continuous
multifunction. Thus, Pi ◦ F is upper contra-continuous for each i ∈ I.
The proof for lower contra-continuity is similar.
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Theorem 37. Let (Xi , τi ), (Yi , υi ) be topological spaces
and F
Q
Qi : Xi →
Yi be a multifunction Q
for each i ∈ I. Suppose that F : i∈I Xi → i∈I Yi is
defined by F ((xi )) = i∈I Fi (xi ). If F is upper (lower) contra-continuous,
then Fi is upper (lower) contra-continuous for each i ∈ I.
Proof. Let Ai ⊂ Yi be a closed set. Since F is upper contra-continuous,
then
Y
Y
F + (Ai × Yj ) = Fi+ (Ai ) × Xj
i6=j
i6=j
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ERDAL EKICI, SAEID JAFARI and TAKASHI NOIRI
10
is an open set. Thus, Fi+ (Ai ) is an open set and hence Fi is upper contracontinuous.
The proof for lower contra-continuity is similar.
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Acknowledgment. We would like to express our sincere gratitude
to the Referee for valuable suggestions and comments which improved the
paper.
REFERENCES
1. Abd El-Monsef, M.E.; El-Deeb, S.N.; Mahmoud, R.A. – β-open sets and βcontinuous mappings, Bull. Fac. Sci. Assiut Univ., 12 (1983), 77-90.
2. Banzaru, T. – Multifunctions and M-product spaces (Romanian), Bull. Stiin. Teh.
Inst. Politeh Timisoara Ser. Mat. Fiz Mec. Teor. Apl., 17 (31) (1972), 17-23.
3. Berge, C. – Espaces Topologiques, Fonctions Multivoques, Dunod, Paris, 1959.
4. Dontchev, J. – Contra-continuous functions and strongly S-closed spaces, Internat
J. Math. Math. Sci., 19 (1996), 303-310.
5. Ekici, E.; Popa, V. – Some properties of upper and lower clopen continuous multifunctions, Bul. Şt. Univ. Politehnica Timisoara, Seria Mat.-Fiz., 50 (64), (2005),
1-11.
6. Jafari, S.; Noiri, T. – Contra-super-continuous functions, Anal. Univ. Sci. Budapest, 42 (1999), 27-34.
7. Kovačević, I. – Subsets and paracompactness, Univ. u Novom Sadu, Zb. Rad. Prirod.
Mat. Fak. Ser. Mat., 14 (1984), 79-87.
8. Levine, N. – Semi-open sets and semi-continuity in topological spaces, Amer. Math.
Monthly, 70 (1963), 36-41.
9. Mashhour, A.S.; Abd El-Monsef, M.E.; El-Deeb, S.N. – On precontinuous and
weak precontinuous mappings, Proc. Phys. Soc. Egypt, 53 (1982), 47-53.
10. Mrsevic, M. – On pairwise R0 and pairwise R1 bitopological spaces, Bull. Math.
Soc. Sci. Math. R. S. Roumanie 30 (1986), 141-148.
11. Njåstad, O. – On some classes of nearly open sets, Pacific J. Math., 15 (1965),
961-970.
12. Noiri, T.; Popa, V. – Almost weakly continuous multifunctions, Demonstratio
Math., 26 (1993), 363-380.
13. Noiri, T.; Popa, V. – A unified theory of weak continuity for multifunctions, Stud.
Cerc. St. Ser. Mat. Univ. Bacau, 16 (2006), 167-200.
11
ON UPPER AND LOWER CONTRA-CONTINUOUS MULTIFUNCTIONS
85
14. Ponomarev, V.I. – Properties of topological spaces preserved under multivalued continuous mappings on compacta, Amer. Math. Soc. Translations, 38 (2) (1964), 119140.
15. Popa, V. – Weakly continuous multifunctions, Boll. Un. Mat. Ital. (5), 15-A (1978),
379-388.
16. Smithson, R.E. – Almost and weak continuity for multifunctions, Bull. Calcutta
Math. Soc., 70 (1978), 383-390.
17. Wine, D. – Locally paracompact spaces, Glasnik Mat., 10 (30) (1975), 351-357.
Received: 18.VI.2007
Revised: 14.IX.2007
Department of Mathematics,
Canakkale Onsekiz Mart University,
Terzioglu Campus, 17020 Canakkale,
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College of Vestsjaelland South,
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