LITMIR.BIZ

      is upper semicontinuous.

      (b) Let multimaps Fj : X → C(Y) be closed and . then the intersection is closed.

      Proof. (a) At first, let us prove the assertion for the case of two multimaps F₀ and F₁. For x ∈ X, let V be a neighborhood of the set F(x) = (F₀ ∩ F₁)(x). If at least one of the sets F₀(x) or F₁(x) is contained in V then the existence of such a neighborhood U(x) of x that F(U(x)) ⊂ V is evident. Otherwise F₀(x) \ V and F₁(x) \ V are nonempty disjoint closed sets. By virtue of the normality of the space Y there exist disjoint open sets W₀ and W₁ such that (F_j(x) \ V) ⊂ Wj, j = 0, 1. Then for every j = 0, 1 we have

      From the upper semicontinuity of the multimaps Fj it follows that for each j = 0, 1 there exists a neighborhood Uj(x) of x such that

      But if U(x) = U₀(x) ∩ U₁(x) then for every x′ ∈ U(x) we have

      proving the upper semicontinuity of F at x.

      The validity of the statement in the general case now follows from the mathematical induction principle.

      (b) The statement follows from the fact that the graph of the multimap is the intersection of the graphs of the multimaps F_j.

      Let us mention also the following assertion.

Theorem 1.3.3. Let a multimap F₀ : X → C(Y) be closed, a multimap F₁ : X → K(Y) upper semicontinuous and

      Then the intersection F = F₀ ∩ F₁ : X → K(Y) is upper semicontinuous.

      Proof. For an arbitrary x ∈ X let V ⊂ Y be any open neighborhood of the set (F₀ ∩ F₁)(x). We will show that there exists an open neighborhood U(x) of x such that (F₀ ∩ F₁)(U(x)) ⊂ V.

      When F₁(x) ⊂ V the existence of such a neighborhood follows from the upper semicontinuity of F₁. If K = F₁(x) \ V ≠

then the set K is compact and K ∩ F₀(x) =

. As F₀ is a closed multimap, for each point y ∈ K there exist neighborhoods V(y) ⊂ Y of y and Uy(x) ⊂ X of x such that F₀(U_y(x)) ∩ V(y) =

(see Theorem 1.2.24(b)).

      Let { be a finite cover of K formed by such neighborhoods V(y), and . The open set V ∪ V(K) contains F₁(x), hence there exists a neighborhood U₁(x) of x such that F₁(U₁(x)) ⊂ (V ∪ V(K)). Then the neighborhood

      is the required one. In fact, F₀(U(x)) ∩ V(K) =

and F₁(U(x)) ⊂ (V ∪ V(K)), therefore (F₀ ∩ F₁)(U(x)) ⊂ V.

      Corollary 1.3.4. Let a multimap F : X → K(Y) be upper semicontinuous, C ⊂ Y a closed set and F(x) ∩ C ≠

, ∀x ∈ X. Then the multimap F : X → K(Y),

      is upper semicontinuous.

      Proof. It is clear that the multimap F₀ : X → C(Y),

      is closed. Take F₁ = F and apply the previous theorem.

      Corollary 1.3.5. Let Y be a Hausdorff topological space, multimaps upper semicontinuous and . Then the intersection is upper semicontinuous.

      Proof. Let F_j0 be one of the multimaps from the family. Since al the multimaps Fj are closed (see Remark 1.2.30) the multimap

      is also closed ( Скачать книгу