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Let X, Y be topological spaces, F : X → P(Y) multimaps.

Definition 1.2.13. A multimap F is said to be upper semicontinuous at a point x ∈ X if for every open set V ⊂ Y such that F(x) ⊂ V there exists a neighborhood U(x) of x such that

Definition 1.2.14. A multimap F is called upper semicontinuous (u.s.c.) if it is upper semicontinuous at every point x ∈ X.

Consider some tantamount formulations.

Theorem 1.2.15. The following conditions are equivalent:

(a)the multimap F is u.s.c.;

(b)for every open set V ⊂ Y, the set is open in X;

(c)for every closed set W ⊂ Y, the set is closed in X;

(d)if D ⊂ Y then .

Proof. 1) The equivalence (a) ⇔ (b) is evident;

2) the equivalence (b) ⇔ (c) follows from Lemma 1.2.3(c) and Lemma 1.2.4(c);

3) is a closed set which contains ;

4) (d) ⇒ (c): if D is closed then , i.e., is closed.

Example 1.2.16. The multimaps from Examples 1.1.4 (a), (b); 1.1.5; 1.1.7 are u.s.c. The subdifferential multimap from Example 1.1.17 is also u.s.c. (see, e.g., [104]).

Definition 1.2.17. A multimap F is called lower semicontinuous at a point x ∈ X if for every open set V ⊂ Y such that F(x) ∩ V ≠ there exists a neighborhood U(x) of x such that F(x′) ∩ V ≠ for all x′ ∈ U(x).

Definition 1.2.18. A multimap F is said to be lower semicontinuous (l.s.c.) if it is lower semicontinuous at every point x ∈ X.

The lower semicontinuity also admits tantamount definitions.

Theorem 1.2.19. The following conditions are equivalent:

(a)the multimap F is l.s.c.;

(b)for every open set V ⊂ Y, the set is open in X;

(c)for every closed set W ⊂ Y, the set is closed in X;

(d)if a system of open sets {Vj}_j∈J forms a base for the topology of the space Y then for each Vj, the set is open in X;

(e)if D ⊂ Y is an arbitrary set then ;

(f)if A ⊂ X is an arbitrary set then .

Proof. 1) the equivalence (a) ⇔ (b) is evident;

2) the equivalences (b) ⇔ (c) and (c) ⇔ (e) can be proved similarly to the corresponding statements of Theorem 1.2.15;

3) the equivalence (b) ⇔ (d) follows from the fact that each set Vj is open and from Lemma 1.2.4 (d);

4) , but by virtue of Lemma 1.2.3(a) , hence . From Lemma 1.2.3(b) it follows: , therefore ;

5) but by virtue of Lemma 1.2.3 (b): , yielding . Applying to both sides of the last inclusion and using Lemma 1.2.3 (a) we get .

In the case of metric spaces we may obtain the following convenient sequential characterization of the lower and upper semicontinuity.

Theorem 1.2.20. Let X and Y be metric spaces.

(a)For the lower semicontinuity of a multimap F : X → P(Y) at a point x₀ ∈ X it is necessary and sufficient that:

(*) for every sequence , x_n → x₀ and each y₀ ∈ F(x₀) there exists a sequence , y_n ∈ F(x_n) such that y_n → y₀.

(b)For the upper semicontinuity of a multimap F : X → P(Y) at a point x₀ ∈ X it is necessary, and in the case of the compactness of the set F(x₀) it is also sufficient that:

(**) for every sequences , x_n → x₀ and , y_n ∈ F(x_n) the following relation holds: ϱY(y_n, F(x₀)) → 0.

Proof. (a)(i) Let condition (*) holds. If the multimap F is not l.s.c. at the point x₀ then there exist an open set V ⊂ Y such that F(x₀) ∩ V ≠ and a sequence , x_n → x₀ such that F(x_n) ∩ V = for all n = 1, 2, ... But these relations are in contradiction to the fact that we can, choosing a point y₀ ∈ F(x₀) ∩ V, to find a sequence y_n ∈ F(x_n) which converges to it.

(a)(ii) Let a multimap F be l.s.c. at a point x₀ and a certain sequence , x_n → x₀ and a point y₀ ∈ F(x₀) be given. Consider the sequence of open balls , m = 1, 2, ... centered at the point y₀. Let a number n₁ be such that for all n ≥ n₁. For every n < n₁ choose y_n ∈ F(x_n) arbitrarily. Further, let us find a number n₂ ≥ n₁ such that for all n ≥ n₂. For every n, n₁ ≤ n < n₂ choose y_n ∈ F(x_n) ∩ B₁(y₀). Continuing this process, we construct the desirable sequence yn converging to y₀.

(b)(i) Let a multimap F be u.s.c. at a point x₀. For an arbitrary ε > 0, consider an ε-neighborhood of the set F(x₀):

Since U_ε(F(x₀)) is the open set it follows that there exists a number n_ε such that for all n ≥ n_ε we get F(x_n) ⊂ U_ε(F(x₀)) implying ϱY(y_n, F(x₀)) ≤ ε.

(b)(ii) Let V ⊂ Y be an arbitrary open set containing F(x₀). Then, by virtue of compactness of the set F(x₀) there exists (see Chapter 0) ε > 0 such that U_ε(F(x₀)) ⊂ V. But then condition (**) yields the existence of such a neighborhood U(x₀) of the point x₀ such that F(U(x₀)) ⊂ U_ε(F(x₀)) ⊂ V.

Definition 1.2.21. If a multimap F is upper and lower semicontinuous it is called continuous.

It is clear that in the case of a single-valued map, both upper, as well as lower semicontinuity mean usual continuity. Notice also that the constant multimap F(x) ≡ Y₁ ⊂ Y is obviously continuous.

Example 1.2.22. (a) The multimaps from Examples 1.1.4 (a), (c); 1.1.5; 1.1.7; 1.1.8 are u.s.c. It can be verified by application of Definition 1.2.14 (do it!). Whence the multimaps from Examples 1.1.4 (a)(; 1.1.5; 1.1.7 are continuous. The multimap from Example 1.1.4 (b) is u.s.c., but not l.s.c., whereas the multimaps from Examples 1.1.4 (c)); 1.1.8 are l.s.c., but not u.s.c. In particular, for the multimap F from Example 1.1.8 we have , where

(b) Let T be a compact space; X a metric space, C(T; X) denote the space of continuous functions endowed with the usual sup-norm. For an arbitrary nonempty subset Ω ⊂ C(T, X), the multimap Q : T → P(x) defined as

is l.s.c. It can be checked up by using Theorem 1.2.20. Verify that if the set Ω is compact then the multimap Q is u.s.c. and hence continuous.

One more important class consists of closed multimaps.

Definition 1.2.23. A multimap F is called closed if its graph Γ_F (see Definition 1.1.2) is a closed subset of the space X × Y.

Consider some tantamount formulations.

Theorem 1.2.24. The following conditions are equivalent:

(a)the multimap F is closed;

(b)for each pair x ∈ X, y ∈ Y such that y ∉ F(x) there exist neighborhoods U(x) of x and V(y) of y such that F(U(x)) ∩ V(y) = ;

(c)for every nets {x_a} ⊂ X, {y_α} ⊂ Y such that x_α → x, y_α ∈ F(x_α), y_α → y, we have y ∈ F(x).

Proof. 1) (a) ⇔ (b): condition (b) means that a point (x, y) ∈ X × Y belongs to the complement of the graph Γ_F with a certain neighborhood;

2) (a) ⇔ (c): condition (c) means that if a net {(x_α, y_α)} ⊂ Γ_F converges to a point (x, y) ∈ X × Y then (x, y) ∈ Γ_F.

Notice that in the case when X and Y are metric spaces, it is sufficient to consider in condition (c) usual sequences.

Example 1.2.25. The multimaps from Examples 1.1.4 (a), (b); 1.1.5 - 1.1.8 are closed.

Example 1.2.26. Consider Example 1.1.9. If X, Y are topological spaces, the space Y is Hausdorff and f : X → Y is a continuous surjective map, then the inverse multimap F = f⁻¹ : Y → P(x) is closed.

Example 1.2.27. Consider Example 1.1.10. If X, Y, Z are topological spaces and the maps f and g are continuous then the implicit multimap F is closed.

Example 1.2.28. Consider Example1.1.11. If X, Y are topological spaces and function f is continuous then the multimap F_r is closed.

The validity of assertions in Examples 1.2.26 – 1.2.28 may be verified by applying Theorem 1.2.24(c) (do it!).

Introduce some notation which we will use in the sequel.

Let Y be a topological space.

Denote by C(Y), K(Y) the collections consisting of all nonempty closed, or respectively, compact subsets of Y. If the topological space Y is linear then Pv(Y) denotes the collection of all nonempty convex subsets of Y. Introduce also the following symbols:

When a multimap F maps into the collections C(Y), K(Y) or Pv(Y) we will say that F has closed, compact or convex values respectively.

From the definition of a closed multimap it follows that it has closed values.

The consideration of examples shows that closed and upper semicontinuous multimaps are a short distance apart. The relation between them is clarified by the following assertions.

Theorem 1.2.29. Let X and Y be topological spaces. If the space Y is regular and a multimap F : X → C(Y) is u.s.c. then F is closed.

Proof. Let y ∈ Y, y ∉ F(x). Since Y is regular there exist an open neighborhood V(y) of the point y and an open set W ⊃ F(x) such that V(y) ∩ W = . Let U(x) be a neighborhood of x such that F(U(x)) ⊂ W. Then F(U(x)) ∩ V(y) = and the statement follows from Theorem1.2.24(b).

Remark 1.2.30. It is clear from the proof that when F has compact values the condition of regularity of Y can be replaced with the weaker condition that Y is a Hausdorff space.

To formulate a sufficient condition for a closed multimap to be u.s.c. we need the following definitions.

Definition 1.2.31. A multimap F : X → P(Y) is called:

(a)compact if its range F(x) is relatively compact in Y, i.e., is compact in Y;

(b)locally compact if every point x ∈ X has a neighborhood U(x) such that the restriction F to U(x) is

(c)quasicompact if the restriction of F to each compact subset A ⊂ X is compact.

It is clear that (a) ⇒ (b) ⇒ (c).

Theorem 1.2.32. Let F : X → K(Y) be a closed multimap. If it is locally compact then it is u.s.c.

Proof. Let x ∈ X, V an open set in Y such that F(x) ⊂ V. Let U(x) be a neighborhood of x such that the restriction of F to it is compact and let . If y ∈ W then since F is closed, there exist neighborhoods V(y) of y and U_y(x) of x such that F(U_y(x)) ∩ V(y) = . By the compactness of W we can extract a finite subcover V(y₁), . . . , V(y_n). Consider the following open neighborhood of x:

Notice now that implies F(x′) ∩ V(y_j) = for all j = 1, 2, . . . , n and hence F(x′) ∩ W = . From the other side, . Therefore, .

The difference between closed and u.s.c. multimaps is illustrated by Examples 1.1.6–1.1.8. As it was mentioned already, the multimaps in these examples are closed, but they are not u.s.c. Notice that the multimap from Example 1.1.6 has compact values and the condition of its upper semicontinuity is violated at the same point x = π/2 in which the condition of the local compactness is not satisfied.

Let us consider some properties of closed and u.s.c. multimaps.

Theorem 1.2.33. Let F : X → C(Y) be a closed multimap. If A ⊂ X is a compact set then its image F(A) is a closed subset of Y.

Proof. The case F(A) = Y is trivial. Let y ∈ Y \ F(A). For any x ∈ A, let U(x) and V_x(y) be neighborhoods of x and y such that

If U(x₁), . . . , U(x_n) are neighborhoods forming a finite cover of A then is a neighborhood of y such that V(y) ∩ F(A) = .

Remark 1.2.34. The condition of compactness of the set A is essential: the image of a closed set under the action of a closed multimap can be a non-closed set. In fact, in Example 1.1.7: .

In the sequel an important role will be played by the following property of u.s.c. multimaps.

Theorem 1.2.35. Let F : X → K(Y) be a u.s.c. multimap. If A ⊂ X is a compact set then its image F(A) is a compact subset of A.

Proof. Let {Vj}_j∈J be an open cover of the set F(A). For each point x ∈ A, the value F(x) can be covered by a finite collection of sets V_j1, ...., V_jn(x). We denote . The sets , x ∈ A form an open cover of A. If we select a finite subcover

then the sets V_x1, ..., V_xm form an open cover of the set F(A).

Remark 1.2.36. The condition of a upper semicontinuity is essential in this theorem. In fact, for a closed multimap F with compact values in Example 1.1.6 we have F([0, π]) = .

Let us mention also the following property.

Theorem 1.2.37. Let X and Y be topological spaces; A ⊂ X a connected set and F : X → P(Y) a multimap. If one of the following conditions holds true:

(i)F is upper or lower semicontinuous and the values F(x) are connected for each x ∈ A;

(ii)F is continuous and a value F(x₀) is connected for some x₀ ∈ A

then F(A) is a connected subset of Y.

Proof. (i) Consider the case of a upper semicontinuous multimap F. Suppose the contrary, then there exist open sets V₀ and V₁ in the space Y such that:

a)F(A) ⊂ (V₀ ∪ V₁);

b)F(A) ∩ V_i ≠ , i = 0, 1;

c)(F(A) ∩ V₀) ∩ (F(A) ∩ V₁) = .

Consider the sets . These sets are open by virtue of the upper semicontinuity of F. Notice that the value of each point x ∈ A is contained only in one of the sets V₀ or V₁ since otherwise the set F(x) would be disconnected. Hence A ⊂ (U₀ ∪ U₁); A ∩ U_i ≠ for each i = 0, 1 and (A ∩ U₀) ∩ (A ∩ U₁) = that contradicts to the connectedness of the set A.

In the case when the multimap F is lower semicontinuous, it is sufficient to note that open sets arising in the definition of a connected set may be replaced with closed ones and to carry out the same reasonings as above, by using Theorem 1.2.19 (c).

(ii) Also suppose the contrary. Then, by virtue of its connectedness, the set F(x₀) must lie either in V₀ or V₁. Suppose for determinacy that F(x₀) ⊂ V₀ and hence . Then we get

and moreover, by the continuity of the multimap F, both last sets are non-empty, disjoint and open. But this contradicts to the connectedness of A.