Applied Mathematics E-Notes, 1(2001), 104-110 \begin{matrix}c \\ ◯ \end{matrix} Available free at mirror sites of http://math2.math.nthu.edu.tw/∼amen/

On Upper and Lower D-Continuous Multifunctions ^{∗}

Metin Akdağ^{†}

Received 12 May 2001

Abstract In this paper, we define upper and lower D-continuous multifunctions and obtain some of their characterizations and basic properties. Also some relationships between D-continuity and other types of continuity are given.

In 1968, Singal and Singal [9] introduced and investigated the concept of almost continuous functions. In 1981, Heldermann [2] introduced some new regularity axioms and studied the class of D-regular spaces. In 1990, Kohli [3] introduced the concept of D-continuous functions and some properties of D-continuous functions are given by him. The purpose of this paper is to extend this concept and to provide some properties of multifunctions.

A multifunction F : X ↪ Y is a correspondence from X to 2^{Y} with F (x) a nonempty subset of Y, for each x ∈ X. Let A be a subset of a topological space (X, τ). A^{∘} and \overline{A} denote the interior and closure of A respectively. A subset A of X is called regular open (regular closed) [12] if, and only if, A = (\overline{A})^{∘} (respectively A = \overline{(A^{∘})}). A space (X, τ) is said to be almost regular [8] 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. 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, forms a basis (respectively subbasis) for a topology on 2^{X} (see [5]). This topology is called upper (respectively lower) Vietoris topology and denoted by τ_{V}^{+} (respectively τ_{V}^{−}). 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 ≠ ∅}. The graph G(F) of the multifunction F : X ↪ Y is strongly closed [4] if for each (x, y) ∉ G(F), there exist open sets U and V containing x and containing y respectively such that (U × \overline{V}) ∩ G(F) = ∅.

In [7], a multifunction F : X ↪ Y is said to be (i) upper semi continuous (or u.s.c.) at a point x ∈ X if for each open set V in Y with F (x) ⊆ V, there exists an open set U containing x such that F (U) ⊆ V; and (ii) lower semi continuous (or l.s.c.) at a point x ∈ X if for each 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 ∈ U.

In [10], a multifunction F : X ↪ Y is said to be (i) upper weakly continuous (or u.w.c.) at a point x ∈ X if for each open set V in Y with F (x) ⊆ V, there exists an open set U containing x such that F (U) ⊆ \overline{V}; and (ii) lower weakly continuous (or l.w.c.) at a point x ∈ X if for each open set V in Y with F (x) ∩ V ≠ ∅, there exists an open set U containing x such that F (z) ∩ \overline{V} ≠ ∅ for every z ∈ U.

∗Mathematics Subject Classifications: 54C10, 54C60

†Department of Mathematics, Cumhuriyet Univeristy, Sivas 58140, Turkey

Let F : X ↪ Y be a multi function. F is said to be upper D-continuous (briefly u.D.c.) at x_{0} ∈ X, if for each open F_{σ}-set V with F (x_{0}) ⊂ V, there exists an open neighborhood U_{x_{0}} of x_{0} such that the implication x ∈ U_{x_{0}} ⇒ F (x) ⊂ V holds. F is said to be lower D-continuous (briefly l.D.c.) at x_{0} ∈ X, if for each open F_{σ}-set V with F (x_{0}) ∩ V ≠ ∅ there exists an open neighborhood U_{x_{0}} of x_{0} such that the implication x ∈ U_{x_{0}} ⇒ F (x_{0})∩V ≠ ∅ holds. F is said to be D-continuous (briefly D.c.) at x_{0} ∈ X, if it is both u.D.c. and l.D.c. at x_{0} ∈ X. Finally, F is said to be u.D.c. (l.D.c. or D.c.) on X, if it has this property at each point x ∈ X.

THEOREM 1. Let X and Y be topological spaces. For a multifunction F : X ↪ Y, the following statements are equivalent: (a) F is u.D.c. (l.D.c.). (b) For every open F_{σ}-set V, F^{+}(V) (F^{−}(V)) is an open set in X. (c) For every closed G_{δ}-set K, F^{−}(K) (F^{+}(K)) is closed in X. (d) For each x ∈ X and each net {x_{α}}_{α∈△} which converges to x, if V is an open F_{σ}-set with F (x) ⊂ V (F (x) ∩ V ≠ ∅), then there is an α_{o} ∈ △ such that for every α ≥ α_{o}, F (x_{α}) ⊂ V (respectively F (x_{α}) ∩ V ≠ ∅).

PROOF. (a)⇒(b): If V is an open F_{σ}-set of Y, then for each x ∈ F^{+}(V), F (x) ⊂ V and hence there is an open neighborhood U of x such that ⋃_{x∈U} F (x) ⊂ V. Thus F^{+}(V), being a neighborhood of each of its points, is open.

(b)⇒(c): Let K be a closed G_{δ}-set of Y. Then Y \K is an open F_{σ}-set and F^{+}(Y \K) = X\F^{−}(K) is open. Thus F^{−}(K) is closed in X.

(c)⇒(b): Let V be an open F_{σ}-set. Then Y \V is a closed G_{δ}-set and F^{−}(Y \V) = X\F^{+}(V) is closed in X. Thus F^{+}(V) is an open set in X.

(b)⇒(a): Let x ∈ X and let V be an open F_{σ}-set containing F (x). Then F^{+}(V) is an open set containing x and F (F^{+}(V)) ⊂ V. Thus F is u.D.c. at x.

(b)⇒(d): Let {x_{α}}_{α∈△} be a net in X which converges to x and let V be an open F_{σ}-set containing F (x). Then F^{+}(V) is an open set containing x. Since {x_{α}} converges to x, there is an α_{o} ∈ △ such that for every α ≥ α_{o}, x_{α} ∈ F^{+}(V). Thus for every α ≥ α_{o}, F (x_{α}) ⊂ V.

(d)⇒(b): Let V be an open F_{σ}-set of Y. To show that F^{+}(V) is open, assume to the contrary that there is x ∈ F^{+}(V) such that F^{+}(V) is not neighborhood of x. Then there is a net {x_{α}} in X which converges to x and misses F^{+}(V) frequently. Then {F (x_{α})} misses V frequently, which is a contradiction.

The proof for the case where F is l.D.c. is similarly proved. The proof is complete.

As an example, let X = {0, 1}, τ = {∅, X, {1}} and Y = {a, b, c}, ϑ = {∅, Y, {a}, {b}, {a, b}}. If we define F : (X, τ) ↪ (Y, ϑ) with F (0) = {a}, F (1) = {b}, then F is u.D.c. (l.D.c.) but not u.s.c. (respectively l.s.c.) at x_{0} = 0.

THEOREM 2. Let F : (X, τ) ↪ (Y, ϑ) be a multifunction. If F is u.s.c. (l.s.c.), then F is u.D.c. (respectively l.D.c.).

PROOF. Suppose that F is u.s.c. (l.s.c.) at x_{0} ∈ X. If V is an open F_{σ}-set in Y with F (x_{0}) ⊂ V (respectively F (x_{0}) ∩ V ≠ ∅) then F^{+}(V) (respectively F^{−}(V)) is an open set in X. Thus F is u.D.c. (respectively l.D.c.) at x_{0} ∈ X. The proof is complete.

THEOREM 3. Let X be a topological space and let Y be a D-regular space [2]. If F is point compact and u.D.c. (l.D.c.), then F is u.s.c. (respectively l.s.c.).

PROOF. Suppose that V is an open set in Y with F (x_{0}) ⊂ V. Since Y is D-regular for every y ∈ F (x_{0}), there is an open F_{σ}-set G_{y} such that y ∈ G_{y} and G_{y} ⊂ V. If we define the family Σ = {G_{y} : y ∈ F (x_{0})}, then it is an open cover of F (x_{0}) and F (x_{0}) ⊂ ⋃ G_{y} ⊂ V. Since F is point compact and for each y ∈ F (x_{0}), G_{y} is an open F_{σ}-set, there is a finite subcover of F (x_{0}) such that F (x_{0}) ⊂ ⋃ _{i=1}^{n}G_{y_{i}} ⊂ V, and if we take ⋃ G_{y_{i}} = G, then it is an open F_{σ}-set. Also since F is u.D.c., for F (x_{0}) ⊂ G, there is an open set U_{x_{0}} such that x_{0} ∈ U_{x_{0}} and the implication x ∈ U_{x_{0}} ⇒ F (x) ⊂ G ⊂ V holds. Thus F is u.s.c. at x_{0} ∈ X. The other case is similarly proved. The proof is complete.

THEOREM 4. Let X and Y be topological spaces and let F : X ↪ Y be a multifunction. If the graph function G_{F} : X → X × Y is u.D.c. (l.D.c.), then F is u.D.c. (respectively l.D.c.).

PROOF. Suppose G_{F} is u.D.c. at x_{0} ∈ X. Let V be an open F_{σ}-set with F (x_{0}) ⊂ V. Then G_{F} (x) ⊂ X × V and X × V is an open F_{σ}-set in X × Y. Since G_{F} is u.D.c., there is an open set U with x_{0} ∈ U such that G_{F} (U) ⊂ X × V. From [6], U ⊂ G_{F}^{+}(X × V) = X ∩ F^{+}(V) = F^{+}(V) and so F is u.D.c. at x_{0} ∈ X. Suppose G_{F} is l.D.c. at x_{0} ∈ X. Let V be an open F_{σ}-set with F (x_{0}) ∩ V ≠ ∅. Then

G_{F} (x_{0}) ∩ (X × V) = ({x_{0}} × F (x_{0})) ∩ (X × V) = {x_{0}} × (F (x_{0}) ∩ V) ≠ ∅

and X × V is an open F_{σ}-set in X × Y. Since G_{F} is l.D.c., there is an open set U with x_{0} ∈ U such that U ⊂ G_{F}^{−}(X × V). From [6], U ⊂ G_{F}^{−}(X × V) = X ∩ F^{−}(V) = F^{−}(V) and so F is l.D.c. at x_{0} ∈ X. The proof is complete.

Let (X, τ) be a topological space and let {K_{β} : β ∈ Δ} be a closed cover of X. If for any subset F of X and for the collection \{K_{β} : β ∈ Δ\} the equation ⋃ \overline{K_{β} ⋂ F} = \overline{⋃ (K_{β} ⋂ F)} holds, then the collection is called a hereditarily closure preserving closed cover of X [3].

THEOREM 5. Let X and Y be topological spaces. Then the following statements are true: (a) If F : X ↪ Y is u.D.c.(l.D.c.), then the restriction multifunction F |_{A} : A ↪ Y is u.D.c. (l.D.c.). (b) Let F : X ↪ Y be a multifunction. If {U_{α} : α ∈ △} is an open cover of X and for each α, F_{α} = F |_{U_{α}} is u.D.c.(l.D.c.), then F is u.D.c. (l.D.c.). (c) Let F : X ↪ Y be a multifunction. If {K_{β} : β ∈ △} is a hereditarily closure preserving closed cover of X and for each β ∈ Δ, F_{β} = F |_{K_{β}} is u.D.c. (l.D.c.), then F is u.D.c. (respectively l.D.c.).

PROOF. (a) Let V be an open F_{σ}-set in A with F |_{A}(x_{0}) ⊂ V (F |_{A}(x_{0}) ∩ V ≠ ∅). Since F is u.D.c. (respectively l.D.c.) and F |_{A}(x_{0}) = F (x_{0}) ⊂ V (respectively F |_{A}(x_{0}) = F (x_{0}) ∩ V ≠ ∅), there is an open neighborhood of x_{0} such that the implication x ∈ U ⇒ F (x) ⊂ V (respectively F (x)∩V ≠ ∅) holds. If we take U_{1} = U ∩A, then U_{1} is an open neighborhood of x_{0} in A and F |_{A} (U_{1}) ⊂ V (respectively U_{1} ⊂ F^{−}(V)). Thus F |_{A} is u.D.c. (respectively l.D.c.) at x_{0} ∈ X.

(b) Let V be an open F_{σ}-set of Y. Then F^{+}(V) = ⋃{F_{α}^{+}(V) : α ∈ △} (F^{−}(V) = ⋃{F_{α}^{−}(V) : α ∈ △}) and since for each α ∈ △, F_{α} is u.D.c.(l.D.c.) and F_{α}^{+}(V)

(respectively F_{α}^{−}(V)) is an open set in U_{α} and hence in X. Thus F^{+}(V) (respectively F^{−}(V)) being the union of open sets is open.

(c) Let K be a closed G_{δ}-set of Y. Then F^{+}(K) = ∪{F_{β}^{+}(K) : β ∈ △} (F^{−}(K) = ∪{F_{β}^{−}(K) : β ∈ △}) and since for each α ∈ △, F_{β} is u.D.c. (respectively l.D.c.) and F_{β}^{+}(K) (respectively F_{β}^{−}(K)) is closed in K_{β} and hence in X. Also since {K_{β} : β ∈ △} is a hereditarily closure preserving closed cover of X, the collection {F_{β}^{+}(K) : β ∈ △} (respectively {F_{β}^{−}(K) : β ∈ △}) is a closure preserving collection of closed sets. Thus F^{+}(K) (respectively F^{−}(K)) is closed.

The proof is complete.

THEOREM 6. Let F : X ↪ Y and G : Y ↪ Z be two multifunctions. If F is u.s.c. (l.s.c.) and G : Y ↪ Z is u.D.c. (respectively l.D.c.), then G ∘ F : X ↪ Z is a u.D.c. (respectively l.D.c.)

PROOF. Let V be an open F_{σ}-set in Z. Since G is u.D.c. (l.D.c.), G^{+}(V) (respectively G^{−}(V)) is an open set in Y. Also since F is u.s.c. (respectively l.s.c.), F^{+}(G^{+}(V)) = (G ∘ F)^{+}(V) (respectively F^{−}(G^{−}(V)) = (G ∘ F)^{−}(V)) is an open set in X. Thus G ∘ F is u.D.c. (respectively l.D.c.) The proof is complete.

THEOREM 7. Let F : (X, τ) ↪ (Y, ϑ) be a multifunction and let Y be extremally disconnected space. If F is l.D.c. (u.D.c.), then F is l.w.c. (respectively u.w.c.).

PROOF. Let V be an open set of Y. Since Y is extremally disconnected, \overline{V} is an open set of Y and so \overline{V} is an open F_{σ}-set of Y. Also since F is u.D.c. (l.D.c.), F^{+}(\overline{V}) (respectively F^{−}(\overline{V})) is open in X. Thus F is u.w.c. (respectively l.w.c.). The proof is complete.

THEOREM 8. Let F : (X, τ) ↪ (Y, ϑ) be a multifunction and let Y be a regular space. If F is l.w.c., then F is l.D.c.

PROOF. Let F be l.w.c. at x_{0} ∈ X and let V be an open F_{σ}-set in Y with F (x_{0}) ∩ V ≠ ∅. Since Y is a regular space, for each y ∈ F (x_{0}) ∩ V, there is an open set G_{y} such that y ∈ G_{y} ⊂ \overline{G_{y}} ⊂ V. Thus F (x_{0}) ∩ G_{y} ≠ ∅. Also since F is l.w.c., there is an open neighborhood U of x_{0} such that the implication x ∈ U ⇒ F (x) ∩ \overline{G_{y}} ≠ ∅ holds. Hence F (U) ∩ \overline{G_{y}} ⊂ F (U) ∩ V ≠ ∅ and F is l.D.c. at x_{0} ∈ X. The proof is complete.

THEOREM 9. Let F : (X, τ) ↪ (Y, ϑ) be a multifunction and let Y be a regular space. If the family \overline{Σ} = \{\overline{T} : T ∈ ϑ\} has the local finite property and F is u.w.c., then F is u.D.c.

PROOF. Let V be an open F_{σ}-set in Y with F (x_{0}) ⊂ V. Since Y is regular, for each y ∈ F (x_{0}), there is an open set G_{y} such that y ∈ G_{y} ⊂ \overline{G_{y}} ⊂ V. So F (x_{0}) ⊂ ⋃_{y∈F (x_{0})} G_{y} ⊂ ⋃ \overline{G_{y}} ⊂ V. If we take V_{1} = ⋃_{y∈F (x_{0})} G_{y}, then since F is u.w.c. at x_{0} ∈ X, for F (x_{0}) ⊂ V_{1}, there is an open neighborhood U of x_{0} such that F (U) ⊂ \overline{V_{1}}. Also since \overline{Σ} = \{\overline{G_{y}}|G_{y} ∈ ϑ\} has the local finite property \overline{V_{1}} = \overline{∪G_{y}} ⊂ \overline{∪\overline{G_{y}}} = ∪\overline{G_{y}} ⊂ V, F^{+}(V) is open in X. Thus F is u.D.c. at x_{0} ∈ X. The proof is complete.

Now we give a multifunction F which is u.D.c. (l.D.c.) but not u.w.c. (respectively l.w.c.). Let X = {0, 1}, τ = {∅, X, {1}} and Y = {a, b, c}, ϑ = {∅, Y, {a}, {b}, {a, b}}. If we define F : (X, τ) ↪ (Y, ϑ) with F (0) = {a}, F (1) = {b}, then F is u.D.c. (l.D.c.) but not u.w.c. (respectively l.w.c.) at x_{0} = 0.

THEOREM 10. Let F : X ↪ Y be a quotient multifunction. Then a multifunction G : Y ↪ Z is u.D.c. if, and only if, G ∘ F is u.D.c.

PROOF. Since quotient map is u.D.c., from Theorem 6, G∘F is u.D.c. Conversely, let V be an open F_{σ}-set of Z. Then (G ∘ F)^{+}(V) = F^{+}(G^{+}(V)) is open in X. Since F is a quotient map, G^{+}(V) is open in Y, and so G is u.D.c. The proof is complete.

THEOREM 11. Suppose for each α ∈ △, F_{α} : X_{α} ↪ Y_{α} is a multifunction and let F : ΠX_{α} ↪ ΠY_{α} be a multifunction defined by F ((x_{α})) = (F_{α}(x_{α})) for each point (x_{α}) in ΠX_{α}. If F is u.D.c. (l.D.c.), then for each α ∈ △, F_{α} is u.D.c. (respectively l.D.c.).

PROOF. Let G_{α_{o}} be a closed G_{δ}-set of Y_{α_{o}}. Then G_{α_{o}} × ∏_{α≠α_{o}} Y_{α} is a closed G_{δ}- set of ΠY_{α}. Since F is u.D.c. (l.D.c.), F^{−}(G_{α_{o}} × ∏_{α≠α_{o}} Y_{α}) = F^{−}(G_{α_{o}}) × ∏_{α≠α_{o}} X_{α} (respectively F^{+}(G_{α_{o}} × ∏_{α≠α_{o}} Y_{α}) = F^{+}(G_{α_{o}}) × ∏_{α≠α_{o}} X_{α}) is closed in ΠX_{α}. Consequently F_{α_{o}}^{−}(G_{α_{o}}) (resepctively F^{+}(G_{α_{o}})) is closed in X_{α_{o}} and so G_{α_{o}} is u.D.c. (respectively l.D.c.). The proof is complete.

THEOREM 12. Let F : X → ΠX_{α} be a multifunction into a product space. If F is u.D.C. (l.D.c), then each α ∈ △, P_{α} ∘ F is u.D.c. (respectively l.D.c.).

PROOF. Let G_{α_{o}} be an open F_{σ}-set of X_{α_{o}}. Then, (P_{α_{o}}∘F)^{+}(G_{α_{o}}) = F^{+}(P_{α_{o}}^{+}(G_{α_{o}})) = F^{+}(G_{α_{o}} × ∏_{α≠α_{o}} X_{α}) (respectively (P_{α_{o}} ∘ F)^{−}(G_{α_{o}}) = F^{−}(P_{α_{o}}^{−}(G_{α_{o}})) = F^{−}(G_{α_{o}} × ∏_{α≠α_{o}} X_{α})). Since F is u.D.c. (respectively l.D.c.) and since G_{α_{o}} × ∏_{α≠α_{o}} X_{α} is an open F_{σ}-set, F^{+}(G_{α_{o}} × ∏_{α≠α_{o}} X_{α}) (respectively F^{−}(G_{α_{o}} × ∏_{α≠α_{o}} X_{α})) is open in X. Thus P_{α} ∘ F is u.D.c. (respectively l.D.c.). The proof is complete.

THEOREM 13. The set of all points of X for which F : X ↪ Y is not u.D.c. is identical to the union of the boundaries of the inverse image of open F_{σ}-sets of Y.

PROOF. Suppose F is not u.D.c. at a point x ∈ X. Then there exists an open F_{σ}-set V containing F (x) such that for every open set U containing x, F (U) ⊈V. Thus for every open set U containing x, U ∩ (X\F^{+}(V)) ≠ ∅. Therefore, x cannot be an interior point of F^{+}(V). Hence x is a boundary point of F^{+}(V). Now, let x belong to the boundary of F^{+}(V) for some open F_{σ}-set of Y (that is x ∈ F^{+}(V) but x ∉ [F^{+}(V)]^{o}). Then F (x) ⊂ V. If F is u.D.c. at x, then there is an open set U containing x such that F (U) ⊂ V. Thus x ∈ U ⊂ F^{+}(V), and so x is an interior point of F^{+}(V). This is contrary to the fact that x belongs to the boundary of F^{+}(V). Hence F is not u.D.c. at x. The proof is complete.

THEOREM 14. A u.D.c. image of a connected space is connected for a multifunction F.

PROOF. Let F : X ↪ Y be a u.D.c. multifunction from a connected space X onto a space Y. Suppose Y is not connected and let Y = A ∪ B be a partition of Y. Then both A and B are open and closed subsets of Y. Since F is u.D.c., F^{+}(A) and F^{+}(B) are open subsets of X. In view of the fact that F^{+}(A) and F^{+}(B) are disjoint, X = F^{+}(A) ∪ F^{+}(B) is a partition of X. This is contrary to the connectedness of X. The proof is complete.

THEOREM 15. Let F : X → Y be u.D.c. If every pair of distinct points of Y are contained in disjoint open sets such that one of them may be chosen to be an F_{σ}-set. Then F has strongly closed graph.

PROOF. Suppose (x, y) ∉ G(F). Then y ∉ F (x). By the hypothesis on Y, there are disjoint open sets V_{1} and V_{2} containing F (x) and y respectively, and V_{1} is an F_{σ}-set. Since F is u.D.c., F^{+}(V_{1}) is open. Thus U = F^{+}(V_{1}) is an open set containing x and F (U) ⊂ V_{1} ⊂ Y \\overline{V_{2}}. Consequently, U × \overline{V} does not contain any points of G(F), and so G(F) is strongly closed in X × Y. The proof is complete.

Let (X, τ) be a topological space. Then X is said to be a D-normal space if for every distinct closed subsets K and F of X, there are two open F_{σ}-sets U and V such that K ⊆ U, F ⊆ V and U ∩ V = ∅.

THEOREM 16. Let F and G be u.D.c. and point closed multifunctions from a space X to a D-normal space Y. Then the set A = {x|F (x) ∩ G(x) ≠ ∅} is closed in X.

PROOF. Let x ∈ X\A. Then F (x) ∩ G(x) = ∅ and so by the hypothesis on Y, there are disjoint open F_{σ}-sets U and V containing F (x) and G(x) respectively. Since F and G are u.D.c., the sets F^{+}(U) and G^{+}(V) are open and contain x. Let H = F^{+}(U) ∩ G^{+}(V). Then H is an open set containing x and H ∩ A = ∅. Thus A is closed in X. The proof is complete.

As a corollary, the set of fixed points of a u.D.c. self map of a D-normal space is closed.

THEOREM 17. Let F : X ↪ Y be u.D.c., F (x) ≠ F (y) for each distinct pair x, y ∈ X and point closed from a topological space X to a D-normal space Y. Then X is Hausdorff.

PROOF. Let x and y be any two distinct points in X. Then F (x) ∩ F (y) = ∅. Since Y is D-normal, there are disjoint open F_{σ}-sets U and V containing F (x) and F (y) respectively. Thus F^{+}(U) and F^{+}(V) are disjoint open sets containing x and y respectively. Thus X is Hausdorff. The proof is complete.

Let (X, τ) be a topological space. Since the intersection of two open F_{σ}-sets is an open F_{σ}-set, the collection of all open F_{σ}-subsets of (X, τ) is a base for a topology τ^{∗} on X. It is immediate that a space (X, τ) is D-regular if, and only if, τ^{∗} = τ [3]. The following example shows that a D-regular space may not be first countable.

EXAMPLE. Let X be the set of positive integers. Let N(n, E) denote the number of integers in a set E ⊂ X which are less than or equal to n. We describe the Appert’s topology on X by declaring open any set which excludes the integer 1, or any set E containing 1 for which lim_{n→∞}N(n, E) = 1. Then the Appert space is completely normal, completely regular and hence from [2] D-regular. However, it is not first countable.

THEOREM 18. Let (X, τ) be a topological space. Then the following statements are equivalent: (a) (X, τ) is a D-regular space. (b) Every u.D.c. and point compact multifunction F from a topological space Y into (X, τ) is u.s.c. (c) The identity mapping I_{X} from (X, τ^{∗}) onto (X, τ) is continuous.

PROOF. (a)⇒(b): Let F : (Y, ϑ) ↪ (X, τ) be a u.D.c. multifunction and let V be an open set in X with F (x) ⊂ V. Then since F is point compact and (X, τ) is D-regular, there is an open F_{σ}-set V_{1} such that F (x) ⊂ V_{1} ⊂ V. Since F is u.D.c.,there exits an open set U containing x such that F (U) ⊂ V_{1} ⊂ V. Thus F is u.s.c. at x.

(b)⇒(c): Let I_{X} : (X, τ^{∗}) ↪ (X, τ) be the identity mapping. Let F (x) ⊂ V and V be an open F_{σ}-set in X. Then I_{X}^{+}(V) = V is an open F_{σ}-set and I_{X}^{+}(V) ∈ τ^{∗}. Thus I_{X} is u.D.c. at x. From (b), I_{X} is u.s.c. at x.

(c)⇒(a): Let V an open set in (X, τ) with x ∈ V. From (c), I_{X} : (X, τ^{∗}) ↪ (X, τ) is u.s.c. and, for I_{X}(x) = x ⊂ V, there is an open F_{σ}-set U in (X, τ^{∗}) such that I_{X}(U) ⊂ V and x ∈ U = I_{X}(U) ⊂ V. Thus (X, τ) is D-regular space. The proof is complete.

In [1], a space X is said to be sequential if a subset U of X is open if, and only if, every sequence converging to a point in U is eventually in U.

THEOREM 19. Let F : X → Y be a u.D. continuous function from a sequential space X into a countably compact Hausdorff space Y. If Y has a neighborhood base of closed G_{δ}-sets then F is upper continuous.

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