Applied Mathematics E-Notes, 1(2001), 47-55 \begin{matrix}c \\ ◯ \end{matrix} Available free at mirror sites of http://math2.math.nthu.edu.tw/∼amen/

Super and Strongly Faintly Continuous Multifunctions ^{∗}

İdris Zorlutuna^{†}, Yalçın Küçük^{‡}

Received 13 December 2000

Abstract

In this paper, we introduce and discuss some strong forms of faintly continuity for multifunctions. Basic properties and characterizations of such multifunctions are established.

1 Introduction

In a series of papers, Long and Herrington [6] introduced and discussed θ-topology and faintly continuous functions. In 1990, Yalva\begin{matrix}c \\ . \end{matrix} [17] introduced the concepts of super and strongly faintly continuous functions. The main purpose of this paper is to define super and strongly faintly continuous multifunctions and to obtain several characterizations and basic properties of such multifunctions.

Let A be a subset of a topological space (X, τ). intA and clA denote the interior and closure of A respectively. A subset A of X is called regular open (regular closed) iff A = int(cl(A)) (A = cl(int(A))). The family of all regular open subsets of (X, τ) form a base for a smaller topology τ_{s} on X, called semi regularizations of τ (see [6]). A point x ∈ X said to be a δ-cluster point of the subset A of (X, τ) if U ∩ A ≠ ∅ for every τ-regular open set U containing x. The set of all δ-cluster points of A is called the δ-closure of A [16] and is denoted by δ-clA. If A = δ-clA then A is called δ-closed and the complement of a δ-closed set is called δ-open. A point x ∈ X is said to be a θ-cluster point of A if clU ∩ A ≠ ∅ for each open neighborhood U of x. The set of all θ-cluster points of A is called the θ-closure of A [16] and is denoted by θ-clA. If A = θclA then A is called θ-closed and the complement of a θ-closed set is called θ-open. In a similar manner, the θ-interior of a set A is defined to be the set of all x ∈ A for which there exists a closed neighborhood of x contained in A. In a topological space (X, τ), θ-open sets form a topology τ_{θ} on X and (X, τ) is regular iff τ = τ_{θ} [6].

A space (X, τ) is said to be almost regular [14] if for every regular closed set F and each point x not belonging to F, there exist disjoint open sets U and V containing F and x respectively. A subset A of a topological space (X, τ) is called N-closed [1] (H-set [16]) if every open cover of A by open sets in X has a finite subfamily whose interior of closures (resp. closures) cover A. A space (X, τ) is called nearly compact [15] (quasi H-closed [12]) if it is N-closed subset (resp. H-set) of it.

∗Mathematics Subject Classifications: 54C10, 54C60.

†Department of Mathematics, Cumhuriyet University, 58140, Sivas-Turkey

‡Department of Mathematics, Anadolu University, Eskişehir-Turkey

The net (x_{α})_{α∈I} is θ-convergent (δ-convergent [9]) to x if for each θ-open (resp. regular open) set U containing x, there exists a α_{0} ∈ I such that α ≥ α_{0} implies x_{α} ∈ U. The net (x_{α})_{α∈I} is r-convergent [4] to x if for each open set U containing x, there exists a α_{0} ∈ I such that α ≥ α_{0} implies x_{α} ∈ clU.

For a given topological space (X, τ), the collection of all sets of the form U^{+} = {T ⊆ X : T ⊆ U} (U^{−} = {T ⊆ X : T ∩ U ≠ ∅}) with U in τ form a basis (subbasis) for a topology on 2^{X}, where 2^{X} is the set of all nonempty subsets of X (see [7]). This topology is called upper (lower) Vietoris topology and denoted by τ_{V}^{+} (τ_{V}^{−}). A multifunction F of a set X into Y is a correspondence such that F (x) is a nonempty subset of Y, for each x ∈ X. We will denote such a multifunction by F : X ↪ Y. For a multifunction F, the upper and lower inverse set of a set B of Y will be denoted by F^{+}(B) and F^{−}(B) respectively, that is F^{+}(B) = {x ∈ X : F (x) ⊆ B} and F^{−}(B) = {x ∈ X : F (x) ∩ B ≠ ∅}. Given a multifunction F : X ↪ Y where F (x) is nonempty, we define the induced function f on X into 2^{Y} by setting f(x) = F (x) for each x ∈ X. Note that f is single valued and f will always denote the function induced by F unless otherwise stated. A function f : X → Y is said to be super continuous [8] (strongly θ-continuous [5]) if for each x ∈ X and each open set V containing f(x), there is an open set U containing x such that f(int(clU)) ⊆ V (resp. f(clU) ⊆ V). A multifunction F : X ↪ Y is called upper semi continuous or u.s.c. [11] (lower semi continuous or l.s.c. [11]) at a point x ∈ X if for each open set V ⊆ Y with F (x) ⊆ V (resp. F (x) ∩ V ≠ ∅), there is an open set U containing x such that F (U) ⊆ V (resp. F (z) ∩ V ≠ ∅ for each z ∈ U). A multifunction F : X ↪ Y is called upper strongly θ-continuous or u.s.θ-c. [2] (lower strongly θ-continuous or l.s.θ-c. [2]) at a point x ∈ X if for each open set V ⊆ Y with F (x) ⊆ V (resp. F (x) ∩ V ≠ ∅), there is an open set U containing x such that F (clU) ⊆ V (F (z) ∩ V ≠ ∅ for each z ∈ clU). A multifunction F : X ↪ Y is called upper δ-continuous or u.δ-c. [3] (lower δ-continuous or l.δ-c. [3]) at a point x ∈ X if for each open set V ⊆ Y with F (x) ⊆ int(clV) (resp. F (x) ∩ int(clV) ≠ ∅), there is an open set U containing x such that F (int(clU)) ⊆ int(clV) (F (z) ∩ int(clV) ≠ ∅ for each z ∈ int(clU)).

2 Super Faintly Continuous Multifunctions

In this section, we define upper (lower) super faintly continuous multifunctions and we obtain many characterizations and basic properties of these multifunctions.

DEFINITION 1. A multifunction F : X ↪ Y is said to be (a) upper super faintly continuous (briefly u.s.f.c.) at a point x ∈ X if for each θ-open set V in Y with F (x) ⊆ V, there exists an open set U containing x such that F (int(clU)) ⊆ V; (b) lower super faintly continuous (briefly l.s.f.c.) at a point x ∈ X if for each θ-open set V in Y with F (x)∩V ≠ ∅, there exists an open set U containing x such that F (z)∩V ≠ ∅ for every z ∈ int(clU); and (c) upper (lower) super faintly continuous on X if it has the property at each point x ∈ X.

EXAMPLE 1. Let X = {0, 1, 2} and Y = {a, b, c, d, e}. Let τ and υ be respectively topologies on X and on Y given by τ = {∅, X, {0}, {1}, {0, 1}} and υ = {∅, Y, {a, b}, {c, d}, {c, d, e}, {a, b, c, d}}. Define the multifunction F : X ↪ Y by F (0) = {a, b, c}, F (1) = {d}, F (2) = {b, c, e}. Then F is u.s.f.c.

EXAMPLE 2. Let X = {0, 1, 2} and Y = {a, b, c, d, e}. Let τ and υ be respectively topologies on X and on Y given by τ = {∅, X, {0}, {1}, {0, 1}} and υ = {∅, Y, {a, b}, {c, d}, {c, d, e}, {a, b, c, d}}. Define the multifunction F : X ↪ Y by F (0) = {a, b, c}, F (1) = {d}, F (2) = {c, d}. Then F is l.s.f.c.

THEOREM 1. Let F : (X, τ) ↪ (Y, υ) be a multifunction, then the following statements are equivalent.

(1) F is l.s.f.c.

(2) For any θ-open set V ⊆ Y and for each x of X with F (x) ∩ V ≠ ∅, there is a regular open set U containing x such that z ∈ U implies F (x) ∩ V ≠ ∅.

(3) F : (X, τ_{s}) ↪ (Y, υ_{θ}) is l.s.c.

(4) F^{−}(V) ⊆ X is δ-open in X for every θ-open set V of Y.

(5) F^{+}(K) ⊆ X is δ-closed in X for every θ-closed set K of Y.

(6) The induced mapping f : (X, τ) → (2^{Y} , (υ_{θ})_{V}^{−}) is super continuous.

(7) The induced mapping f : (X, τ_{s}) → (2^{Y} , (υ_{θ})_{V}^{−}) is continuous.

(8) For each x ∈ X and for each net (x_{α})_{α∈I} which is δ-converging to x and any θ-open set V with F (x) ∩ V ≠ ∅, there exists α_{0} ∈ I such that α ≥ α_{0} implies F (x_{α}) ∩ V ≠ ∅.

(9) For each y ∈ F (x) and for every net (x_{α})_{α∈I} which is δ-converging to x, there exists a subnet (z_{β})_{β∈ξ} of the net (x_{α})_{α∈I} and a net (y_{β})_{(β,V)∈ξ} in Y with y_{β} ∈ F (z_{β}) is θ-convergent to y.

PROOF. (1)⇒(2). Let x ∈ X and let V be a θ-open set in Y with F (x) ∩ V ≠ ∅. Since f is l.s.f.c. at x, there exists an open set W of X containing x such that z ∈ int(clW) implies F (z) ∩ V ≠ ∅. Put int(clW) = U. Then U is a regular open set in X and z ∈ U implies F (z) ∩ V ≠ ∅.

(2)⇒(3) and (3)⇒(4). These are immediate.

(4)⇒(5). Let K ⊆ Y be any θ-closed set. Then Y − K ⊆ Y is a θ-open set, by (4), F^{−}(Y − K) ⊆ X is a δ-open set. Since we can write F^{+}(K) = X − F^{−}(Y − K), F^{+}(K) is a δ-closed set in X.

(5)⇒(1). Let x ∈ X and let V ⊆ Y be a θ-open set with x ∈ F^{−}(V). Then Y − V ⊆ Y is a θ-closed set, by (5), F^{+}(Y − V) ⊆ X is a δ-closed set. Since we can write F^{−}(V) = X − F^{+}(Y − V), F^{−}(V) is a δ-open set in X and x ∈ F^{−}(V). Therefore, there is an open set U containing x such that x ∈ int(clU) ⊆ F^{−}(V).

(1)⇒(6). For any x ∈ X, let f(x) ∈ ∩_{i=1}^{n}V_{i}^{−} and ∩_{i=1}^{n}V_{i}^{−} be a (υ_{θ})_{V}^{−}−open set in 2^{Y}. Then V_{1}, ..., V_{n} are θ-open sets in Y and since f(x) ∈ V_{i}^{−} for i = 1, 2, ..., n, we have F (x) ∩ V_{i} ≠ ∅. Since F is l.s.f.c. at x ∈ X, there exists an open set U_{i} containing x such that z ∈ int(clU_{i}) implies F (z) ∩ V_{i} ≠ ∅ for all i = 1, 2, ..., n. Put U = ∩_{i=1}^{n}U_{i}. Then, we obtain f(z) ∈ ∩_{i=1}^{n}V_{i}^{−} for all z ∈ int(clU) i.e. f(int(clU)) ⊆ ∩_{i=1}^{n}V_{i}^{−}.

(6)⇒(1). Let x ∈ X and V be a θ-open set with x ∈ F^{−}(V). Then f(x) = F (x) ∈ V^{−} and V^{−} is a (υ_{θ})_{V}^{−}-open set in 2^{Y}. Since f is super continuous at x ∈ X, there exists an open set U such that F (int(clU)) = f(int(clU)) ⊆ V^{−}. Thus we obtain an open set U containing x such that z ∈ int(clU) implies F (z) ∩ V ≠ ∅.

(6)⇔(7). This is obvious from [13].

(4)⇒(8). Let x ∈ X and (x_{α})_{α∈I} be a net δ-converging to x. By (4), for any θ-open set V in Y with F (x) ∩ V ≠ ∅, F^{−}(V) is an δ-open set in X and x ∈ F^{−}(V). Hence there exists an open set U such that x ∈ int(clU) ⊆ F^{−}(V). For this U, since (x_{α})_{α∈I} is δ-convergent to x, there is a α_{0} ∈ I such that x_{α} ∈ F^{−}(V) for all α ≥ α_{0} and F (x_{α}) ∩ V ≠ ∅ for all α ≥ α_{0} and (8) follows.

(8)⇒(4). Suppose (4) is not true. Then there is a θ-open set V in Y with x ∈ F^{−}(V) such that for each open set U of X containing x, x ∈ int(clU) ⊈ F^{−}(V) i.e. there is a x_{U} ∈ int(clU) such that x_{U} ∉ F^{−}(V). Define D = {(x_{U}, U) : U ∈ τ(x), x_{U} ∈ int(clU), x_{U} ∉ F^{−}(V)}. Now the ordering ≤ defined by (x_{U_{1}}, U_{1}) ≤ (x_{U}, U) ⇔ U ⊆ U_{1} is a direction on D and g defined by g : D → X, g((x_{U}, U)) = x_{U} is a net on X. The net (x_{U})_{(x_{U} ,U)∈D} is δ-converging to x. But F (x_{U}) ∩ V ≠ ∅ for all (x_{U}, U) ∈ D. This is a contradiction.

(1)⇒(9). Suppose F is l.s.f.c. at x_{0}. Let (x_{α})_{α∈I} be a net δ-converging to x_{0}. Let y ∈ F (x_{0}) and V be any θ-open set containing y. So we have F (x_{0}) ∩ V ≠ ∅. Since F is l.s.f.c. at x_{0}, there exists an open set U such that x ∈ int(clU) ⊆ F^{−}(V). Since the net (x_{α})_{α∈I} is δ-convergent to x_{0}, for this U, there exists α_{0} ∈ I such that α ≥ α_{0} ⇒ x_{α} ∈ int(clU). Therefore, we have the implication α ≥ α_{0} ⇒ x_{α} ∈ F^{−}(V). For each θ-open set V ⊆ Y containing y, define the sets I_{V} = {α_{0} ∈ I : α ≥ α_{0} ⇒ x_{α} ∈ F^{−}(V)} and ξ = {(α, V) : α ∈ I_{V} , y ∈ V and V is θ-open} and an ordering ≥ on ξ as follows: “(ά, V́) ≥ (α, V) ⇔ V́ ⊆ V and ά ≥ α”. Define φ : ξ → I by φ((β, V)) = β. Then φ is increasing and cofinal in I, so φ defines a subnet of (x_{α})_{α∈I}. We denote the subnet (z_{β})_{(β,V)∈ξ}. On the other hand, for any (β, V) ∈ ξ, if β ≥ β_{0} then x_{β} ∈ F^{−}(V) and we have F (z_{β}) ∩ V = F (x_{β}) ∩ V ≠ ϕ. Pick y_{β} ∈ F (z_{β}) ∩ V ≠ ϕ. Then the net (y_{β})_{(β,V)∈ξ} is θ-convergent to y. To see this, let V_{0} be a θ-open set containing y. Then there exists β_{0} ∈ I such that φ((β_{0}, V_{0})) = β_{0} and y_{β_{0}} ∈ V. If (β, V) ≥ (β_{0}, V_{0}) this means that β ≥ β_{0} and V ⊆ V_{0}. Therefore, y_{β} ∈ F (z_{β}) ∩ V = F (x_{β}) ∩ V ⊆ F (x_{β}) ∩ V_{0}, so y_{β} ∈ V_{0}. Thus (y_{β})_{(β,V)∈ξ} is θ-convergent to y.

(9)⇒(1). Suppose (1) is not true, i.e. F is not l.s.f.c. at x_{0}. Then there exists a θ-open set V ⊆ Y so that x_{0} ∈ F^{−}(V) and for each open set U ⊆ X containing x_{0}, there is a point x_{U} ∈ int(clU) for which x_{U} ∉ F^{−}(V). Let us consider the net (x_{U})_{U∈τ(x_{0})} where τ(x_{0}) is the system of τ-neighborhoods of x_{0}. Obviously (x_{U})_{U∈τ(x_{0})} is δ-convergent to x_{0}. Let y_{0} ∈ F (x_{0}) ∩ V. By (9), there is a subnet (z_{w})_{w∈W} of (x_{U})_{U∈τ(x_{0})} and y_{w} ∈ F (z_{w}) like (y_{w})_{w∈W} is θ-convergent to y_{0}. As y_{0} ∈ V and V ⊆ Y is a θ-open set, there is ẃ_{0} ∈ W so that w ≥ ẃ_{0} implies y_{w} ∈ V. On the other hand, (z_{w})_{w∈W} is a subnet of the net (x_{U})_{U∈τ(x_{0})} and so there is a function h : W → τ(x_{0}) such that z_{w} = x_{h(w)} and for each U ∈ τ(x_{0}), there is w̃_{0} ∈ W such that h(w̃_{0}) ≥ U. If w ≥ w̃_{0}, then h(w) ≥ h(w̃_{0}) ≥ U. Let us consider w_{0} ∈ W so that w_{0} ≥ ẃ_{0} and w_{0} ≥ w̃_{0}. Therefore, y_{w} ∈ V for each w ≥ w_{0}. By the definition of the net (x_{U})_{U∈τ(x_{0})}, we have F (z_{w}) ∩ V = F (x_{h(w)}) ∩ V = ∅ and y_{w} ∉ V. This is a contradiction and so F is l.s.f.c. at x_{0}.

THEOREM 2. For a multifunction F : (X, τ) ↪ (Y, υ), the following statements are equivalent.

(1) F is u.s.f.c.

(2) For any θ-open set V ⊆ Y and for each x of X with F (x) ⊆ V, there is a regular open set U containing x such that F (U) ⊆ V.

(3) F : (X, τ_{s}) ↪ (Y, υ_{θ}) is u.s.c.

(4) F^{+}(V) ⊆ X is δ-open in X for every θ-open set V of Y.

(5) F^{−}(K) ⊆ X is δ-closed in X for every θ-closed set K of Y.

(6) The induced mapping f : (X, τ) → (2^{Y} , (υ_{θ})_{V}^{+}) is super continuous.

(7) The induced mapping f : (X, τ_{s}) → (2^{Y} , (υ_{θ})_{V}^{+}) is continuous.

(8) For each x ∈ X and for each net (x_{α})_{α∈I} which is δ-converging to x and any θ-open set V with F (x) ⊆ V, there exists α_{0} ∈ I such that α ≥ α_{0} implies F (x_{α}) ⊆ V.

The proof is similar to that of Theorem 1, and is omitted.

For a given multifunction F : X ↪ Y, the graph multifunction G_{F} : X ↪ X × Y is defined as G_{F} (x) = {x} × F (x) for every x ∈ X. In [10], it was shown that for a multifunction F : X ↪ Y, G_{F}^{+}(A × B) = A ∩ F^{+}(B) and G_{F}^{−}(A × B) = A ∩ F^{−}(B) where A ⊆ X and B ⊆ Y.

THEOREM 3. If the graph multifunction of F : X ↪ Y is u(l).s.f.c., then F is u(l).s.f.c.

PROOF. We shall prove only the case where F is l.s.f.c. Let x ∈ X and V be a θopen set in Y such that x ∈ F^{−}(V). Then G_{F} (x)∩(X ×V) = ({x}×F (x))∩(X ×V) = {x} × (F (x) ∩ V) ≠ ∅ and X × V is θ-open in X × Y by Theorem 5 in [6]. Since the graph multifunction G_{F} is l.s.f.c., there exists an open set U containing x such that z ∈ int(clU) implies G_{F} (z) ∩ (X × V) ≠ ∅. Therefore, we obtain int(clU) ⊆ G_{F}^{−}(X × V) = F^{−}(V) from above equalities. Consequently, F is l.s.f.c.

PROPOSITION 4. Let (X, τ) be a topological space, A ⊂ Y an open set and U ⊂ X a regular open set. Then W = A ∩ U is regular open set in A [3].

THEOREM 5. For a multifunction F : X ↪ Y, the following statements are true.

a) If F is u(l).s.f.c. and A is an open set of X, then F |_{A}: A ↪ Y is u(l).s.f.c.

b) Let {A_{α} : α ∈ I} be a regular open cover of X. Then a multifunction F : X ↪ Y is u(l).s.f.c. iff the restrictions F |_{A_{α}}: A_{α} ↪ Y are u(l).s.f.c. for every α ∈ I.

The proof is obvious from the above proposition and we omit it.

THEOREM 6. If F : X ↪ Y is a l.δ-c. multifunction and G : Y ↪ Z is a l.s.f.c. multifunction, then G ∘ F : X ↪ Z is a l.s.f.c. multifunction.

PROOF. Let V be a θ-open set of Z. We know that (G ∘ F)^{−}(V) = F^{−}(G^{−}(V)). Since G is l.s.f.c., G^{−}(V) is a δ-open set in Y and since F is l.δ-c., F^{−}(G^{−}(V)) is an δ-open set in X by Theorem 2.2 in [3]. Thus we obtain that (G ∘ F)^{−}(V) is δ-open in X, and so G ∘ F is l.s.f.c.

A multifunction F : X ↪ Y is said to be point closed (resp. point compact) iff for each x ∈ X, F (x) is closed (resp. compact) in Y.

THEOREM 7. Let F : (X, τ) ↪ (Y, υ) be a point compact and u.s.f.c. multifunction. If A is N-closed in X, then F (A) is υ_{θ}-compact in Y.

PROOF. Let A be a N-closed set in X, and Σ be υ_{θ}-open cover of F (A). If a ∈ A, then F (a) ⊆ ∪Σ. Since υ_{θ} ⊆ υ and F (a) is compact, there exists a finite subfamily Σ_{n(a)} of Σ such that F (a) ⊆ ∪Σ_{n(a)}. Let ∪Σ_{n(a)} be V_{a}. V_{a} is a θ-open set in Y. Since F is u.s.f.c. at a, there exists an open set U_{a} of X such that a ∈ int(clU_{a}) ⊆ F^{+}(V_{a}). Therefore, Ψ = {U_{a} : a ∈ A} is an open cover of A. Since A is N-closed set in X, there exist a_{1}, a_{2}, ..., a_{k} ∈ A such that A ⊆ ∪{int(clU_{a_{i}}) : a_{i} ∈ A, i = 1, 2, ..., k}. So we obtain

F (A) ⊆ F (∪{int(clU_{a_{i}}) : a_{i} ∈ A, i = 1, 2, ..., k})
⊆ ∪{V_{a_{i}} : a_{i} ∈ A, i = 1, 2, ..., k}
⊆ ∪{∪Σ_{n(a_{i})} : a_{i} ∈ A, i = 1, 2, ..., k}.

Thus F (A) is υ_{θ}-compact in Y.

We know that in almost regular space (Y, υ), quasi H-closedness and υ_{θ}-compactness are the same [18]. Therefore, we have the following corollary.

COROLLARY 8. Let F : (X, τ) ↪ (Y, υ) be a point compact and u.s.f.c. multifunction. If X is nearly compact and F is surjective, then Y is υ_{θ}-compact. In addition, if (Y, υ) is almost regular, then space (Y, υ) is quasi H-closed.

3 Strongly Faintly Continuous Multifunctions

In this section, we define upper (lower) strongly faintly continuous multifunctions and we obtain many characterizations and basic properties of these multifunctions.

DEFINITION 2. A multifunction F : X ↪ Y is said to be (a) upper strongly faintly continuous (briefly u.str.f.c.) at a point x ∈ X if for each θ-open set V in Y with F (x) ⊆ V, there exists an open set U containing x such that F (clU) ⊆ V; (b) lower strongly faintly continuous (briefly l.str.f.c.) at a point x ∈ X if for each θ−open set V in Y with F (x) ∩ V ≠ ∅, there exists an open set U containing x such that F (z) ∩ V ≠ ∅ for every z ∈ clU; and (c) upper (lower) strongly faintly continuous on X if it has the property at each point x ∈ X.

As an example, let X = {0, 1} with topology τ = {∅, X, {0}}, let Y = {a, b, c} with topology υ = {∅, Y, {a}, {b}, {a, b}}, and let F : X ↪ Y be defined as F (0) = {a} and F (1) = {a, b}. Then F is u(l).str.f.c. since the only θ−open set in Y is Y itself.

COROLLARY 9. If a multifunction F is u(l).str.f.c., then F is u(l).s.f.c.

Note that the converse of the above corollary is false in general. Indeed, in Example 1, F is u.s.f.c., but F is not u.str.f.c at 1 ∈ X. Also, in Example 2, F is l.s.f.c., but F is not l.str.f.c at 0 ∈ X.

THEOREM 10. For a multifunction F : (X, τ) ↪ (Y, υ), the following statements are equivalent.

(1) F is l.str.f.c.

(2) F : (X, τ_{θ}) ↪ (Y, υ_{θ}) is l.s.c.

(3) F : (X, τ) ↪ (Y, υ_{θ}) is l.s.θ−c.

(4) F^{−}(V) ⊆ X is θ-open in X for every θ-open set V of Y.

(5) F^{+}(K) ⊆ X is θ-closed in X for every θ-closed set K of Y.

(6) The induced mapping f : (X, τ) → (2^{Y} , (υ_{θ})_{V}^{−}) is strongly θ-continuous.

(7) The induced mapping f : (X, τ_{θ}) → (2^{Y} , (υ_{θ})_{V}^{−}) is continuous.

(8) For each x ∈ X and for each net (x_{α})_{α∈I} which is r-converging to x and any θ-open set V with F (x) ∩ V ≠ ∅, there exists α_{0} ∈ I such that α ≥ α_{0} implies F (x_{α}) ∩ V ≠ ∅.

(9) For each y ∈ F (x) and for every net (x_{α})_{α∈I} which is r-converging to x, there exists a subnet (z_{β})_{β∈ξ} of the net (x_{α})_{α∈I} and a net (y_{β})_{(β,V)∈ξ} in Y with y_{β} ∈ F (z_{β}) is θ-convergent to y.

THEOREM 11. For a multifunction F : (X, τ) ↪ (Y, υ), the following statements are equivalent.

(1) F is u.str.f.c.

(2) F : (X, τ_{θ}) ↪ (Y, υ_{θ}) is u.s.c.

(3) F : (X, τ) ↪ (Y, υ_{θ}) is u.s.θ-c.

(4) F^{+}(V) ⊆ X is θ-open in X for every θ-open set V of Y.

(5) F^{−}(K) ⊆ X is θ-closed in X for every θ-closed set K of Y.

(6) Induced mapping f : (X, τ) → (2^{Y} , (υ_{θ})_{V}^{+}) is strongly θ-continuous.

(7) Induced mapping f : (X, τ_{θ}) → (2^{Y} , (υ_{θ})_{V}^{+}) is continuous.

(8) For each x ∈ X and for each net (x_{α})_{α∈I} which is r-converging to x and any θ-open set V with F (x) ⊆ V, there exists α_{0} ∈ I such that α ≥ α_{0} implies F (x_{α}) ⊆ V.

THEOREM 12. If the graph multifunction of F : X ↪ Y is u(l).str.f.c., then F is u(l).str.f.c.

PROOF. We shall only prove the case where F is u.str.f.c. Let x ∈ X and V be a θ-open set in Y such that x ∈ F^{+}(V). Then G_{F} (x) ⊆ X × V and X × V is θ-open in X × Y by Theorem 5 in [6]. Since the graph multifunction G_{F} is u.str.f.c., there exists an open set U containing x such that G_{F} (clU) ⊆ X × V. Therefore, we obtain clU ⊆ G_{F}^{+}(X × V) = F^{+}(V). Consequently, F is u.str.f.c.

THEOREM 13. If F : X ↪ Y and G : Y ↪ Z are u(l).str.f.c. multifunctions, then G ∘ F : X ↪ Z u(l).str.f.c. multifunction.

The proof is similar to that of Theorem 5 by Theorem 7 and Theorem 8.

The graph G(F) of the multifunction F : X ↪ Y is θ-closed with respect to X if for each (x, y) ∉ G(F), there exist an open set U containing x and an open set V containing y such that (clU × V) ∩ G(F) = ∅.

THEOREM 14. Let F : (X, τ) ↪ (Y, υ) be a point closed multifunction. If F is u.str.f.c. and assume that Y is regular, then G(F) is θ-closed with respect to X.

PROOF. Suppose (x, y) ∉ G(F). Then we have y ∉ F (x). Since Y is regular, there exist disjoint open sets V_{1}, V_{2} of Y such that y ∈ V_{1} and F (x) ⊆ V_{2}. By regularity of Y, V_{2} is also θ-open in Y. Since F is u.str.f.c. at x, there exists an open set U in X containing x such that F (clU) ⊆ V_{2}. Therefore, we obtain x ∈ U, y ∈ V_{1} and (x, y) ∈ clU × V_{1} ⊆ (X × Y) − G(F). So G(F) is θ-closed with respect to X.

THEOREM 15. Let F : (X, τ) ↪ (Y, υ) be a point compact and u.str.f.c. multifunction. If A is H-set, then F (A) is υ_{θ}-compact in Y.

PROOF. Let A be a H-set and Σ be υ_{θ}-open cover of F (A). If a ∈ A, then F (a) ⊆ ∪Σ. Since υ_{θ} ⊆ υ and F (a) is compact, there exists a finite subfamily Σ_{n(a)} of Σ such that F (a) ⊆ ∪Σ_{n(a)}. Let ∪Σ_{n(a)} be V_{a}. V_{a} is a θ-open set in Y. Since F is u.str.f.c. at a, there exists an open set U_{a} of X such that a ∈ clU_{a} ⊆ F^{+}(V_{a}). Therefore, Ψ = {U_{a} : a ∈ A} is an open cover of A. Since A is H-set, there exist a_{1}, a_{2}, ..., a_{k} ∈ A such that A ⊆ ∪{clU_{a_{i}} : a_{i} ∈ A, i = 1, 2, ..., k}. So we obtain

F (A) ⊆ F (∪{clU_{a_{i}} : a_{i} ∈ A, i = 1, 2, ..., k})
⊆ ∪{V_{a_{i}} : a_{i} ∈ A, i = 1, 2, ..., k}
⊆ ∪{∪Σ_{n(a_{i})} : a_{i} ∈ A, i = 1, 2, ..., k}.

Thus F (A) is υ_{θ}−compact in Y.

COROLLARY 16. Let F : (X, τ) ↪ (Y, υ) be a point compact and u.str.f.c. multifunction. If X is quasi H-closed and F is surjective, then Y is υ_{θ}-compact. In addition, if (Y, υ) is almost regular, then space (Y, υ) is quasi H-closed.

THEOREM 17. Let F : (X, τ) ↪ (Y, υ) be a point closed and u.str.f.c. multifunction. If F satisfies x_{1} ≠ x_{2} ⇒ F (x_{1}) ≠ F (x_{2}) and Y is a regular space, then X will be Hausdorff.

PROOF. Let distinct x_{1}, x_{2} belong to X. Then F (x_{1}) ≠ F (x_{2}). Since F is point closed and Y is regular, for all y ∈ F (x_{1}) with y ∉ F (x_{2}), there exist θ-open sets V_{1}, V_{2} containing y and F (x_{2}) respectively such that V_{1} ∩ V_{2} = ∅. Since F is u.str.f.c. and F (x_{2}) ⊂ V_{2}, there exists an open set U containing x_{2} such that F (clU) ⊆ V_{2}. Thus x_{1} ∉ clU. Therefore, U and X − clU are disjoint open sets separating x_{1} and x_{2}.

The following example shows that if upper strongly faintly continuity is replaced by upper semi continuity, Theorem 17 will be false.

EXAMPLE 3. Let X = {a, b, c} with the topology τ = {∅, X, {a}{c}{a, c}} and Y = [0, 1] with the usual topology. Define the multifunction F : X ↪ Y, by F (x) = [0, 1/4] for x = a, F (x) = Y for x = b and F (x) = {1/3} for x = c. Then F is point closed, u.s.c. and Y is regular, but X is not Hausdorff.

We know that a space (Y, υ) is regular iff υ = υ_{θ}. Therefore, for a multifunction which is defined on a regular space, strongly θ-continuousness and strongly faintly continuousness are equivalent. Hence, the proofs of the following corollaries are similar to those of [2]. First, if F : (X, τ) ↪ (Y, υ) is a one-to-one point compact, u.str.f.c. multifunction and Y is a T_{3}-space, then X is Urysohn. Next, if F : (X, τ) ↪ (Y, υ) is a multifunction and Y is a regular space, and if G_{F} is u.str.f.c., then X is a regular space. Here a multifunction F : (X, τ) ↪ (Y, υ) is one-to-one in case x_{1} ≠ x_{2} ⇒ F (x_{1}) ∩ F (x_{2}) = ∅ for all x_{1},x_{2} ∈ X. However, if upper strongly faintly continuity is replaced by upper semi continuity, the last corollary will be false in general. Indeed, let X = {a, b, c} with the topology τ = {∅, X, {a}, {a, b}} and Y = [0, 1] with the usual topology. Define the multifunction F : X ↪ Y, by F (x) = (1/3, 2/3) for x = a, F (x) = (1/4, 3/4) for x = b and F (x) = Y for x = c. Then the graph of F is u.s.c. and Y is regular, but X is not regular.

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