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Let X, Y be topological spaces; {Fj}_j∈J, Fj : X → P(Y) a family of multimaps.

Theorem 1.3.1. (a) Let multimaps Fj be upper semicontinuous. If the set of indices J is finite then the union of multimaps

is upper semicontinuous;

(b) Let the multimaps Fj be lower semicontinuous. Then their union is lower semicontinuous;

(c) Let multimaps Fj : X → C(Y) be closed. If the set of indices J is finite then the union is closed.

Proof. (a) Let V ⊂ Y be open, then in accordance with Lemma 1.2.7 (a)

and hence this set is open and by Theorem 1.2.15 (b) the multimap is u.s.c.

(b) The assertion similarly follows from Lemma 1.2.8 (a) and Theorem 1.2.19 (b).

(c) It is easy to verify (do it!) that the graph of the multimap is the union of the graphs from where the asserion follows.

Theorem 1.3.2. (a) Let multimaps Fj : X → C(Y) be upper semicontinuous. If the set of indices J is finite, the space Y is normal and then the intersection of multimaps ,

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 (Theorem 1.3.2(b)). Take F₀ = F_j0 and apply the theorem.

The continuity properties of the intersection of lower semicontinuous multimaps are more complicated. The following example demonstrates that in general case such an intersection is not lower semicontinuous.

Example 1.3.6. Consider the multimaps F₀, F₁ : [0, π] → Kv(²) defined in the following way. The multimap Fo is constant:

whereas the multimap F₁ is defined as

(See fig. 8).

Fig. 8

The multimaps F₀ and F₁ are l.s.c. (they are even continuous) but their intersection F₀ ∩ F₁ defined on the whole interval [0, π] loses the lower semicontinuity property at the points 0 and π (why?)

To clarify the conditions under which we can guarantee the lower semi-continuity of the intersection of multimaps the following notion is useful.

Definition 1.3.7. A multimap F : X → P(Y) is called quasi-open at a point x ∈ X if

and for every y ∈ intF (x) there exist neighborhoods V(y) ⊂ Y of y and U(x) ⊂ X of x such that V(y) ⊂ F(x′) for all x′ ∈ U(x). A multimap F is said to be quasi-open provided it is quasi-open at every point x ∈ X.

It is easy to see that a multimap F : X → P(Y) such that intF(x) ≠ for all x ∈ X is quasi-open if and only if the multimap intF : X → P(Y),

has the open graph Γ_intF ⊂ X × Y.

We have the following important characterization of a quasi-open multimap.

Theorem 1.3.8. Let Y be a finite-dimensional linear topological space. A multimap F : X → Cv(Y) is quasi-open at a point x ∈ X if and only if intF(x) ≠ and F is lower semicontinuous at x.

Proof. 1) Let F be quasi-open at x ∈ X then intF(x) ≠ . If V ⊂ Y is an open set such that V ∩ F(x) ≠ then it is easy to see that V ∩ intF(x) ≠ . For an arbitrary y ∈ Y ∩ intF(x) let V(y) ⊂ Y and U(x) ⊂ X be neighborhoods such that V(y) ⊂ F(x′) for all x′ ∈ U(x). But V(y) ∩ V ≠ implies that V ∩ F(x′) ≠ for all x′ ∈ U(x), giving the lower semicontinuity of F at x.

2) Conversely, let intF(x) ≠ and F be l.s.c. at x ∈ X. Let y ∈ intF(x) and B_δ(y) ⊂ F(x) for some δ > 0. Take δ₁, 0 < δ₁ < δ. Since the space Y is finite-dimensional, the ball B_δ(y) is relatively compact. By applying the reasonings similar to those that were used while proving the necessity part of Theorem 1.2.40 we get that there exists a neighborhood U(x) of x such that for each point x′ ∈ U(x) we have B_δ(y) ⊂ F_η(x′), where η = δ → δ₁.

Let now y′ ∈ B_δ1(y) but y′ ∉ F(x′) for some x′ ∈ U(x). Then from the convexity of the set F(x′) it follows that the ball (y′) will contain points whose distance from F(x′) is greater than η. But this contradicts to the fact that B_η(y′) ⊂ B_δ(y) ⊂ F_η(x′). Therefore, B_δ1(y) ⊂ F(x′) for all x′ ∈ U(x).

We now formulate a condition that guarantees the lower semicontinuity of the intersection of multimaps.

Theorem 1.3.9. Let X, Y be topological spaces; a multimap F₀ : X → P(Y) be lower semicontinuous at x₀ ∈ X and a multimap F₁ : X → P(Y) be quasi-open at x₀, and

for all x ∈ X. If

then the intersection F₀ ∩ f₁ is lower semicontinuous at x₀.

Proof. Let V ⊂ Y be an open set such that V ∩ (F₀ ∩ F₁)(x₀) ≠ . From the assumptions it follows that there exists a point y ∈ V ∩ (F₀ ∩ F₁)(x₀) which is an interior point of the set F₁(x₀). Let V(y) be a neighborhood of y such that V(y) ⊂ (V ∩ F₁(x₀)). By using the quasi-openness of the multimap F₁ we can assume, without loss of generality, that there exists a neighborhood U₁(x₀) of x₀ such that V(y) ⊂ F₁(x′) for all x′ ∈ U₁(x₀).

Since y ∈ F₀(x₀) and the multimap F₀ is l.s.c. there exists a neighborhood U₀(x₀) of x₀ such that F₀(x″) ∩ V(y) ≠ for all x″ ∈ U₀(x₀). But then for every we have , i.e., , that means the lower semicontinuity of the multimap F₀ ∩ F₁ at x₀.

As a corollary, we can obtain now a sufficient condition for the lower semicontinuity of the intersection of l.s.c. multimaps.

Theorem 1.3.10. Let X be a topological space, Y a finite-dimensional linear topological space, F₀, F₁ : X → Cv(Y) l.s.c. multimaps. Assume that F₀(x) ∩ F₁(x) ≠ for all x ∈ X and

for some x₀ ∈ X. Then the intersection F₀ ∩ F₁ : X → Cv(Y) is l.s.c. at x₀.

Proof. From Theorem 1.3.8 it follows that the multimap F₁ is quasi-open at x₀. Let y ∈ (F₀ ∩ F₁) be an arbitrary point and .

It is clear that and . This means that the multimaps F₀ and F₁ satisfy the assumptions of Theorem 1.3.9.

It is worth noting that the loss of the lower semicontinuity for the intersection of multimaps in Example 1.3.6 occurs exactly at the points where the above condition is violated.

Now consider some continuity properties of the composition of multimaps (see Definition 1.2.9).

Let X, Y, and Z be topological spaces.

Theorem 1.3.11. If the multimaps F₀ : X → P(Y) and F₁ : Y → P(Z) are u.s.c. (l.s.c.) then their composition F₁ ○ F₀ : X → P(Z) is u.s.c. (respectively, l.s.c.).

Proof. The assertion follows immediately from Theorems 1.2.15(b), 1.2.19(b) and Lemma 1.2.10.

Theorem 1.3.12. Let F₀ : X → K(Y) be a u.s.c. multimap and F₁ : Y → C(Z) a closed multimap. Then the composition F₁ ○ F₀ : X → C(Z) is a closed multimap.

Proof. Let z ∈ Z be such that z ∉ F₁ ○ F₀(x), x ∈ X. Applying Theorem 1.2.24(b) to the closed multimap F₁ we can find for each point y ∈ F₀(x), neighborhoods W_y(z) of z and V(y) of y such that

Let be a finite cover of the set F₀(x). If now U(x) is a neighborhood of x such that

then

and the application of Theorem 1.2.24(b) concludes the proof.

Remark 1.3.13. The condition of upper semicontinuity of the multimap F₀ is essential. The following example shows that the composition of closed multimaps is not necessarily a closed multimap.

Example 1.3.14. The multimaps F₀ : → K(),

and F₁ : → K(),

are closed but not u.s.c. Their composition F₁ ○ F₀ : → K(),

is not closed.

We consider now the Cartesian product of multimaps (see Definition 1.2.11).

Theorem 1.3.15. If multimaps F₀ : X → P(Y), F₁ : X → P(Z) are lower semicontinuous then their Cartesian product F₀ × F₁ : X → P(Y × Z) is lower semicontinuous.

Proof. Notice that the sets V₀ × V₁, where V₀ ⊂ Y, V₁ ⊂ Z are open sets form a base for the topology of the space Y × Z and apply Theorem 1.2.19(d) and Lemma 1.2.12(b).

Theorem 1.3.16. If multimaps F₀ : X → C(Y), F₁ : X → C(Z) are closed then their Cartesian product F₀ × F₁ : X → C(Y × Z) is closed.

Proof. Consider nets {x_α} ⊂ X, {υ_α} ⊂ Y × Z such that x_α → x, υ_α ∈ (F₀ × F₁)(x_α), υ_α → υ. Then υ_α = y_α × z_α, y_α ∈ F₀(F₁(x_a). from the definition of the topology in Y × Z, the convergence υ_α → υ = (y, z) implies the convergences y_α → y and z_α → z. From the closedness of the multimaps F₀ and F₁ it follows that y ∈ F₀(x), z ∈ F₁(x) (Theorem 1.2.24(c)) but it means that υ ∈ (F₀ × F₁)(x) concluding the proof.

To consider the upper semicontinuity of the Cartesian product of multimaps we need the compactness of their values.

Theorem 1.3.17. If multimaps F₀ : X → K(Y), F₁ : X → K(Z) are upper semicontinuous then their Cartesian product F₀ × F₁ : X → K(Y × Z) is upper semicontinuous.

Proof. The Tychonoff theorem (see Chapter 0) implies that the multimap F₀ × F₁ has compact values. For an arbitrary point x ∈ X, let G ⊃ (F₀ × F₁)(x) be an open subset of Y × Z. From the definition of the product topology in Y × Z it follows that for every (y, z)(F₀ × F₁)(x) there exist open sets G₀(y, z) ⊂ Y and G₁(y, z) ⊂ Z such that (y, z) ∈ G₀(y, z) × G₁(y, z) ⊂ G. For each y ∈ F₀(x) consider the cover of the set F₁(x). Since the set F₁(x) is compact we can select a finite subcover . The set is a neighborhood of y in Y and the set is a neighborhood of F₁(x) in Z and moreover (V(Y) × W_y) ⊂ G. The sets V(Y), y ∈ F₀(x) form an open cover of the compact set F₀(x). Choose a finite subcover and set and . Then V is a neighborhood of F₀(x) in Y whereas W is a neighborhood of F₁(x) in Z and V × W ⊂ G.

Then from Lemma 1.2.12(a) and the upper semicontinuity of the multimaps F₀ and F₁ it follows that there exists a neighborhood U(x) of x such that (F₀ × F₁)(U(x)) ⊂ V × W ⊂ G proving the upper semicontinuity of F₀ × F₁.