Download Chapter 6 Convergences Preserving Continuity

Chapter 6
Convergences Preserving Continuity
In Chapter 3 we introduced the space C(X, (Y, d) and we learned that continuity is preserved under
the uniform convergence. Let us consider again a topological space (X, T ), a metric space (Y, d)
and the hyperspace C(C(Y), H) of all nonempty closed subsets of Y equipped with the Hausdorff
metric (derived from d). Then it makes sense to speak about the space C(X, (C(Y), H)), actually,
this space can be also perceived as a space of all Hausdorff continuous multifunctions from X to Y
with nonempty closed values. The following theorem is the consequence of theorem 13.13:
THEOREM 6.1. Let (X, T ) be a topological space, (Y, d) a metric space. Let {Fγ }γ∈Γ be a net
of Hausdorff continuous multifunctions Fγ : X −−≺ Y with closed nonempty values, converging
uniformly to a multifunction F : X −−≺ Y with closed nonempty values (uniformly with respect to
the metric ρ, where ρ(F1 , F2 ) = sup x∈X Hd (F1 (x), F2 (x))). Then the multifunction F is Hausdorff
continuous, too.
Remark 6.2. Later on we are going to speak about continuous selections of multifunctions. In
1984 Skoczylas showed that for a "typical" space Z = C(X, (C(Y), H)) the set S ⊂ Z of all multifunctions from Z having a continuous selection is nowhere dense in Z (int(S̄ ) = ∅).
The procedure used at the beginning of this chapter (perceiving multifunction spaces as known
function spaces and obtaining continuity preserving theorems in that way) could be repeated in
various contexts and with many different notions of continuity. But this kind of approach would
require a deeper insight into the interplay between various properties of Y and C(Y). For this
reason, in what follows, we prefer to concentrate on a more direct approach to this subject.
6.1 Strong convergence and continuity of multifunctions
We remind the reader that if (Y, T ) is a topological space, r an open cover of Y and A ⊂ Y, then the
set
[
r? (A) =
{V ∈ T : A ∩ V , ∅}
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is called the r−star of A. Intuitively, r−stars can sometimes serve to measure the distance of two
sets, of course, in a purely relative way, dependent on a concrete open cover r.
Now, we are going to define a strong type of convergence for multifunctions. The use of open
covers will enable us to obtain some continuity preserving theorems in topological setting.
Definition 6.3. Let X and Y be two topological spaces. Let n = {Fγ }γ∈Γ be a net of multifunctions
from X to Y. Let F be a multifunction from X to Y. We say that n converges strongly to F if
for every open cover r of Y there exists an index γ0 such that for all γ ≥ γ0 and for all x ∈ X,
F(x) ⊆ r? (Fγ (x)) and Fγ (x) ⊆ r? (F(x)).
Definition 6.4. Let X and Y be two topological spaces. Let n = {Fγ }γ∈Γ be a net of multifunctions
from X to Y. It is called frequently upper (lower) semicontinuous at a point a ∈ X if for each
γ0 ∈ Γ there exists a γ ≥ γ0 such that the multifunction Fγ is upper (lower) semicontinuous at a.
The following two theorems show that under quite general conditions, the strong convergence
preserves upper-semicontinuity and lower-semicontinuity of multifunctions.
THEOREM 6.5. Let X be a topological space, let Y be a normal topological space. Let n =
{Fγ }γ∈Γ be a net of multifunctions from X to Y. Let F be a multifunction from X to Y. Let a be an
element of X, let the set F(a) be closed. If the net n converges strongly to F and if it is frequently
upper semicontinuous at a, then F is upper-semicontinuous at a.
Proof. Let V be an open subset of Y such that F(a) ⊂ V. Since Y is normal and F(a) closed, there
exist two open subsets O, W of Y such that
F(a) ⊂ O ⊆ Ō ⊆ W ⊆ W̄ ⊆ V.
It is clear that
r = {V, Y \ W̄}, c = {W, Y \ Ō}
are open covers of Y.
Since the net n converges to F strongly, there exists and index γ0 such that
F(x) ⊆ r? (Fγ (x))
and
Fγ (x) ⊆ c? (F(x))
(†)
for all γ ≥ γ0 and x ∈ X.
Moreover, since the net n is frequently upper semicontinuous at the point a, there exists a γ ∈ Γ,
with γ ≥ γ0 , such that the multifunction Fγ is upper semicontinuous at the point a.
Since
Fγ (a) ⊆ c? (F(a)) = W,
and W is open, there exists an open neighbourhood U of a such that ∀x ∈ U, Fγ (x) ⊆ W. Because
of (†) we obtain ∀x ∈ U F(x) ⊆ r? (Fγ (x)) = V.
F is proved to be upper semicontinuous at a.
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THEOREM 6.6. Let X be a topological space, let Y be a regular one. Let n = {Fγ }γ∈Γ be a net
of multifunctions from X to Y. Let F be a multifunction from X to Y. Let a be an element of X. If
the net n converges strongly to F and if it is frequently lower semicontinuous at a, then F is lower
semicontinuous at a.
Proof. Let U ⊂ Y be an open set such that F(a) ∩ U , ∅. Let us consider a point b ∈ F(a) ∩ U.
Since Y is regular, there exist two open subsets V, W of U such that
b ∈ V ⊆ V̄ ⊆ W ⊆ W̄ ⊆ U.
Denote
r = {W, Y \ V̄},
c = {U, Y \ W̄}.
Since the net n converges to F strongly, there exists an index α such that
∀x ∈ X, F(x) ⊆ r? (Fα (x)) and Fα (x) ⊆ c? (F(x)).
(‡)
Moreover, we can consider an index α such that Fα is lower semicontinuous at a.
First we prove by contradiction Fα (a) ∩ W = ∅ is not true.
Since by (‡), F(a) ⊆ r? (Fα (a)) = Y \ V̄, this is a contradiction with the fact that F(a) ∩ V ⊃ {b} , ∅.
The multifunction Fα is lower semicontinuous at a, so there exists an open neighbourhood O of a
such that ∀t ∈ O, Fα (t) ∩ W , ∅.
Hence, because of (‡) we obtain
∀t ∈ O, Fα (t) ⊆ c? (F(t)).
Therefore ∀t ∈ O, F(t) ∩ U , ∅, because if for a t0 from O the set F(t0 ) ∩ U were an empty set,
then Fα (t0 ) ⊆ c? (F(t0 )) = Y \ W̄ would lead to a contradiction.
F is proved to be lower semicontinuous at a.
The following example shows that the regularity conditions of the two preceding theorems
cannot be omitted.
Example 6.7. Let Y = {1, 2} with the topology T = {∅, {1}, {1, 2}}. Let us define functions f :
R → Y and g : R → Y as follows
∀x ∈ R \ {0} : f (x) = 2, f (0) = 1;
∀x ∈ R : g(x) = 2.
Put
∀n ∈ N : fn = g.
Then the sequence { fn }n∈N converges strongly to f . But it does not converge pointwise to f . Of
course, f is not continuous.
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Next we show that under certain condition a uniform (Hausdorff like) convergence of multifunctions coincides with the strong one. Remark that the strong convergence always implies the
uniform one.
What do we mean by a uniform convergence of multifunctions?
Definition 6.8. Let X be a set, let (Y, U) be a uniform space. We say that a net n of multifunctions
converges uniformly to a multifunction F : X −−≺ Y if for each V ∈ U there exists an index γ0 such
that for every γ ≥ γ0 and for every x ∈ X
F(x) ⊆ V(Fγ (x))
and
Fγ (x) ⊆ V(F(x))
hold.
The following theorem shows when the uniform convergence implies the strong one.
THEOREM 6.9. Let X be a set, let (Y, U) be a compact uniform space. Let n be a net of multifunctions which converges uniformly to a multifunction F : X −−≺ Y. Then n converges to F
strongly.
Proof. Let r be an open cover of Y. Then by the Lebesgue Covering Theorem (see [20]), there
exists a W ∈ U such that for each y ∈ Y there exists an Oy ∈ r such that W(y) ⊆ Oy . Moreover, by
the uniform convergence of the net n = {Fγ }γ∈Γ to F, there exists a γ ∈ Γ such that, for all α ≥ γ
and x ∈ X, we have
F(x) ⊆ W(Fα (x)) and Fα (x) ⊆ W(F(x)).
So
F(x) ⊆ W(Fα (x)) ⊆ r? (Fα (x))
hold.
and
Fα (x) ⊆ W(F(x)) ⊆ r? (F(x))
Exercises
Exercise 1 Find a T 1 topological space (X, T ), a function f : R → X and a net of continuous
functions { fγ }γ∈Γ from R to X such that { fγ }γ∈Γ converges to f strongly and f is not continuous.
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