4.5 Basic Operations on Subsets
4.5.6 The Union of the Elements
Whenever we have a set of subsets of some space, we can take the union of its elements. For any open U ⊂ X we have S
K∈UUK = U because for each x∈ U, {x}
is compact. Since the setsUU are a base for the topology of K(X), it follows that the union of all elements of an open subset of K(X) is open. If U and V1, . . . , Vk
are open, then UU ∩ VV1 ∩ · · · ∩ VVk = ∅ if there is some j with U ∩ Vj = ∅, and otherwise
{x, y1, . . . , yk} ∈ UU ∩ VV1 ∩ · · · ∩ VVk
whenever x ∈ U and y1 ∈ V1 ∩ U, . . . , yk ∈ Vk ∩ U, so the union of all K ∈ UU∩ VV1 ∩ · · · ∩ VVk is again U. Therefore the union of all the elements of an open
1Proof: an open cover of the subset, together with its complement, is an open cover of the space, any finite subcover of which yields a finite subcover of the subset.
2Proof: fixing a point y in the complement of the compact set K, for each x ∈ K there are disjoint neighborhoods of Uxof x and Vxof y,{Ux} is an open cover of K, and if Ux1, . . . , Uxn is a finite subcover, then Vx1∩ . . . ∩ Vxn is a neighborhood of y that does not intersect K.
4.5. BASIC OPERATIONS ON SUBSETS 75 subset of H(X) is open. If X is either T1 or regular, then similar logic shows that for either T ∈ {K0,H0} the union of the elements of an open subset of T (X) is open.
If a subset C ofH(X) or H0(X) is compact, then it is automatically compact in the coarser topology ofK(X) or K0(X). Therefore the following two results imply the analogous claims for the H(X) and H0(X), which are already interesting.
Lemma 4.5.14. If S ⊂ K(X) is compact, then L :=S
K∈SK is compact.
Proof. Let {Uα : α ∈ A} be an open cover of L. For each K ∈ S let VK be the union of the elements of some finite subcover. Then K ∈ UVK, so { UVK : K ∈ S } is an open cover of S; let UVK1, . . . ,UVKr be a finite subcover. Then L⊂ Sr
i=1VKi, and the various sets from {Uα} that were united to form the VKi are the desired finite subcover of L.
Lemma 4.5.15. If X is regular and S ⊂ K0(X) is compact, then D :=S
C∈SC is closed.
Proof. We will show that X\ D is open; let x be a point in this set. Each element of S is a closed set that does not contain x, so (since X is regular) it is an element of UX0\N for some closed neighborhood N of x. Since S is compact we have S ⊂ UX\N0 1 ∪ . . . ∪ UX0\Nk for some N1, . . . , Nk. Then N1 ∩ . . . ∩ Nk is a neighborhood of x that does not intersect any element of S, so x is in the interior of X \ D as desired.
Topologies on Functions and Correspondences
In order to study of robustness of fixed points, or sets of fixed points, with respect to perturbations of the function or correspondence, one must specify topologies on the relevant spaces of functions and correspondences. We do this by identifying a function or correspondence with its graph, so that the topologies from the last chapter can be invoked. The definitions of upper and lower semicontinuity, and their basic properties, are given in Section 5.1. There are two topologies on the space of upper semicontinuous correspondences from X to Y . The strong upper topology, which is defined and discussed in Section 5.2, turns out to be rather poorly behaved, and the weak upper topology, which is usually at least as coarse, is presented in Section 5.3. When X is compact the strong upper topology coincides with the weak upper topology.
We will frequently appeal to a perspective in which a homotopy h : X× [0, 1] → Y is understood as a continuous function t7→ ht from [0, 1] to the space of contin-uous functions from X to Y . Section 5.4 presents the underlying principle in full generality for correspondences. The specializations to functions of the strong and weak upper topologies are known as the strong topology and the weak topology respectively. If X is regular, then the weak topology coincides with the compact-open topology, and when X is compact the strong and weak topologies coincide.
Section 5.5 discusses these matters, and presents some results for functions that are not consequences of more general results pertaining to correspondences.
The strong upper topology plays an important role in the development of the topic, and its definition provides an important characterization of the weak upper topology when the domain is compact, but it does not have any independent signif-icance. Throughout the rest of the book, barring an explicit counterindication, the space of upper semicontinuous correspondences from X to Y will be endowed with the weak upper topology, and the space of continuous functions from X to Y will be endowed with the weak topology.
76
5.1. UPPER AND LOWER SEMICONTINUITY 77
5.1 Upper and Lower Semicontinuity
Let X and Y be topological spaces. Recall that a correspondence F : X → Y maps each x∈ X to a nonempty F (x) ⊂ Y . The graph of F is
Gr(F ) ={ (x, y) ∈ X × Y : y ∈ F (x) }.
If each F (x) is compact (closed, convex, etc.) then F is compact valued (closed valued, convex valued, etc.). We say that F is upper semicontinuous if it is compact valued and, for any x∈ X and open set V ⊂ Y containing F (x), there is a neighborhood U of x such that F (x′)⊂ V for all x′ ∈ U. When F is compact valued, it is upper semi-continuous if and only if F−1(UV) is a open whenever V ⊂ Y is open. Thus:
Lemma 5.1.1. A compact valued correspondence F : X → Y is upper semi-continuous if and only if it is semi-continuous when regarded as a function from X to K(Y ).
In economics literature the graph being closed in X× Y is sometimes presented as the definition of upper semicontinuity. Useful intuitions and simple arguments flow from this point of view, so we should understand precisely when it is justified.
Proposition 5.1.2. If F is upper semicontinuous and Y is a Hausdorff space, then Gr(F ) is closed.
Proof. We show that the complement of the graph is open. Suppose (x, y) /∈ Gr(F ).
Since Y is Hausdorff, y and each point z ∈ F (x) have disjoint neighborhoods Vz and Wz. Since F (x) is compact, F (x)⊂ Wz1 ∪ · · · ∪ Wzk for some z1, . . . , zk. Then V := Vz1 ∩ · · · ∩ Vzk and W := Wz1 ∪ · · · ∪ Wzk are disjoint neighborhoods of y and F (x) respectively. If U is a neighborhood of x with F (x′)⊂ W for all x′ ∈ U, then U× V is a neighborhood (x, y) that does not intersect Gr(F ).
If Y is not compact, then a compact valued correspondence F : X → Y with a closed graph need not be upper semicontinuous. For example, suppose X = Y = R, F (0) ={0}, and F (t) = {1/t} when t 6= 0.
Proposition 5.1.3. If Y is compact and Gr(F ) is closed, then F is upper semi-continuous.
Proof. Fix x∈ X. Since (X × Y ) \ Gr(F ) is open, for each y ∈ Y \ V we can choose neighborhoods Uy of x and Vy of y such that (Uy× Vy)∩ Gr(F ) = ∅. In particular, Y \ F (x) =S
y∈Y \F (x)Vy is open, so F (x) is closed and therefore compact. Thus F is compact valued.
Now fix an open neighborhood V of F (x). Since Y \ V is a closed subset of a compact space, hence compact, there are y1, . . . , yksuch that Y \V ⊂ Vy1∪. . .∪Vyk. Then F (x′)⊂ V for all x′ ∈ Uy1 ∩ . . . ∩ Uyk.
Proposition 5.1.4. If F is upper semicontinuous and X is compact, then Gr(F ) is compact.
Proof. We have the following implications of earlier results:
• Lemma 4.5.7 implies that the function x 7→ {x} ∈ K(X) is continuous;
• Lemma 5.1.1 implies that F is continuous, as a function from X to K(Y );
• Proposition 4.5.9 states that (K, L) 7→ K × L is a continuous function from K(X) × K(Y ) to K(X × Y ).
Together these imply that ˜F : x 7→ {x} × F (x) is continuous, as a function from X toK(X × Y ). Since X is compact, it follows that ˜F (X) is a compact subset of K(X × Y ), so Lemma 4.5.14 implies that Gr(F ) =S
x∈XF (x) is compact.˜
We say that F is lower semicontinuous if, for each x ∈ X, y ∈ F (x), and neighborhood V of y, there is a neighborhood U of x such that F (x′)∩ V 6= ∅ for all x′ ∈ U. If F is both upper and lower semi-continuous, then it is said to be continuous. When F is compact valued, it is lower semicontinuous if and only if F−1(VV) is open whenever V ⊂ Y is open. Combining this with Lemma 5.1.1 gives:
Lemma 5.1.5. A compact valued correspondence F : X → Y is continuous if and only if it is continuous when regarded as a function from X to H(Y ).