Volume 4, 1979
Pages 149–158
http://topology.auburn.edu/tp/
CONVERGENCE OF SEQUENCES OF
CLOSED SETS
by
Gabriella Salinetti and Roger J-B. Wets
Topology Proceedings
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Mail:
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Volume 4
TOPOLOGY PROCEEDINGS
149
1979
CONVERGENCE OF SEQUENCES OF CLOSED SETS
Gabriella Salinetti and Roger J-D. Wets
The theorem proved in this short note yields new cri­
teria for the convergence of sequences of closed subsets of
a locally compact, separable metric space (E,d).
This work
was motivated by the study of convergence of measurable
multifunctions [10] and optimization problems, e.g.
We say that a sequence of closed sets {F
n
c
[2].
E, n E N}
converges to the closed set F in E, in which case we write
F
=
limnENFn' if
lim sup F
n
=
F
=
lim inf F •
n
This is the classical definition (due to Borel).
Here
lim inf F n = {x = lim
x n Ix n E F for all n E N}
.
n
and
lim sup F
n
= {x = lim x
m
Ix m
E F
m
for all m E MeN}.
By N we always denote a countable ordered index set, typi­
cally the natural numbers; M denotes generically a naturally
ordered infinite subset of N, i.e. {x ' m E M} is a subse­
m
quence of {x ' n EN}.
n
Let J
=
subsets of E.
{F c
ElF closed} be the hyperspace of closed
There are many ways to topologize
[8] for example.
We equip
J
J, cf. [6],
with the topology J generated
by the subbase consisting of all families of sets of the form
{j<} and {J G}
Supported in part by grants of Consiglio Nazionale
delle Ricerche (GNAFA) and the National Science Foundation
(MCS 78-02864).
Salinetti and Wets
150
where
J
K = {F E
J
IF
n
K =
~} and J G
=
{F E J IF
n
G
':I ~},
and, as in the rest of this note, K,K ,K , ••• are compact,
l
2
G,G ,G , •.• are open and F,F ,F , ••• are closed sets in E.
l
2
l
2
The resulting base consists of all sets of the type:
p
This is a variant of the Vietoris topology.
F
=
1,2, ...
We write
J - lim F n
if the elements {F , n E N} of J converge--in this topology-­
n
to F E
J.
It is easy to verify that the topological space
(J,J) is Hausdorff, regular, second countable and thus met­
rizable.
To see that it is compact we can either rely on
the fact that the collection
C=
{J K}
n
{J G } satisfies the
finite intersection property or use the equivalence between
the notions of J-convergence in
J
and convergence for
sequences of sets in E (cf. part (i) of the Theorem), this
implies that every infinite sequence of elements of the
metrizable space (J,]) contains a convergent subsequence
since every infinite sequence of sets contains a convergent
subsequence [12, Theorem (7.11)].
The space (J,]) being
metrizable, convergence could be defined in terms of a met­
ric, for example the metric constructed in the proof of
Urysohn's metrization Lenuna, cf.
[3, p. 24].
Although the
Theorem does not identify a new metric compatible with
J,
it yields criteria for ]-convergence that quantify the dis­
tance between the limit set and the elements in the sequence.
Notations.
An arbitrary point of E is designated as
the origin and denoted
o.
Let d(x,F)
=
inf [d(x,y) Iy E F]
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151
1979
be the distance between a closed set F and a point x in E,
d(x,~)
with the usual convention that
=
In E, the
+00.
closed ball of radius rand center x is denoted by Br(x)
and the open ball by BO(x).
r
simply Dr for D
n
Given any set D c E we write
B (0) , i.e. the intersection of 0 with
r
the closed ball of radius r centered at the origin.
E
> 0,
For
E-neighborhood ED of a nonempty set DeE
the (open)
is the set {yld(y,D) < E}i and for the empty set ~, E~
[B -1 (0)]
c
=
•
E
Note that we have not excluded the possibility that the
limit set of a sequence is empty.
The Lemma below yields a
characterization of convergence to the empty set that is
exploited repeatedly in the proof of the Theorem.
Lemma.
Suppose that
closed sets in E.
{F
n, n
Then lim F
E N}
~
n
there corresponds n K such that
F n K = ~ for all n
n
~
is a sequence of
if and only if to each K
n K•
Equivalently, if and only if to each r
n(r) such that F r
=
n
~ 0,
there corresponds
~ for all n ~ n(r).
The equivalence between these two assertions
Ppoof.
follows directly from the nature of E.
First, suppose that lim F
that x
{x }
m
c
m
E F
m
n K
~ ~
n
=
~
but there exists K such
for all m E MEN.
The sequence
K admits a cluster point which, by construction, is
also in lim sup F , contradicting lim F =
n
m
Since lim inf F
n
c
lim sup F , to prove the only if
n
part it suffices to show that lim sup F
Take x E lim sup F
~.
n
=~.
Suppose not.
n with x = 1im{xm Ixm E Fm , m E M} and let
Salinetti and Wets
152
K be a neighborhood of x.
n Fm
Then x E K
for all m suffi­
ciently large, contradicting the hypothesis.
We are now prepared to state the main theorem.
The
equivalence between the four first statements was already
known or could be deduced from existing results, see for
example [1],
[6] and [5].
Short (new) elementary proofs
are included to make the paper self-contained.
Theorem.
Suppose that {F;Fn,n E N} is a collection
of closed sets in E.
Then F
=
J-lim F
if and only if, in
n
anyone of the following statements, both parts a and bare
satisfied:
i
i
a
b
ii
ii
a
b
iii
a
F
lim inf F
c
lim sup F
n
c
if F
n G
if F
n K =
if F
n BO (x)
~ ~
~
£
n'
F;
then F
then F
~ ~
n
n
n G
~ ~
n K
~
then F
n
for all n
~
n ,
G
for all n 2: n K;
n BO(x)
~ ~
for all
n B£ (x)
~
foT' all
£
n 2: n(£,x) ,
iii b
if F n B£(X) =
n
iv
iV
a
b
~
~
then
F
n
n'(£,x);
foT' all x in E, lim sup d(x,F n ) ~ d(x,F),
for all x in E, d(x,F) 5 lim inf d (x,F) ;
va
lim (F'£Fn)
~
for all £ > 0,
vb
lim (Fn'£F)
~
foT' all £ > 0;
vi a
vi
b
Fr
c
£F n for all £ > 0, r >
Fr
n
c:
£F for all £ > 0, r > 0 and n
o and n
~
~
n(£,r),
n' (£,r);
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limrtoolim inf F~,
vii a
F
vii b
limrtoolim sup F~
C
153
1979
F;
C
The equivalence between ]-convergenc~ and (ii)
Proof.
is an immediate consequence of the base structure of
J.
Thus the Theorem will be proved if we establish the equiva­
lence between (ii) and the other statements.
sider the case F nonempty; if F
=
~
We only con­
the implicatiOns are
either trivial or require an elementary appeal to the pre­
vious Lemma.
The rest is proved in two parts, we show first
that all a statements are equivalent; this is done by estab­
lishing the following sequence of implications
i
i a :;> viia.
n E Nlx
n
a
:;> vii
a
:::>
vi :;> v
a
a
E F }.
n
a
a
~
ii
lim inf F , i.e. x
n
For n sufficiently large and r
Take any x E F
x n
E F~ and thus x E lim inf F~
V11 :;> vi
iv :;> iii
a
a
:::>
If x E F r then x
c
a
=
~
:::>
i .
a
lim{x ,
n
s > d(O,x),
limrtoolim inf F~.
S
S
S
limsoon
t lim x with x E F .
n
n
n
r + £, there exists n'(£,s)
=
Hence for £ > 0 and with s =
such that x S E F S c F for all n ~ n' (£,s) or equivalently
n
n
n
x E EF for all n ~ n' (E,S) •
n
via:;> va.
Apply the Lemma to the sequence {F\£F n , n EN}.
va:;> iVa·
Suppose not.
Then there exists x E E, £ > 0 and
MeN such that d(x,F m) > d(x,F) + 2£ for all m E M or
equivalently d(x,£Fm) > d(x,F) + £. It follows that
{Yld(x,y)
= d(x,F)}
c
lim SUPn(F\£Fn) which contradicts va.
iv :;> i i i . Note that lim sup d(x,F n ) ~ d(x,F) holds only
a
a
if for all £ > 0 such that d(x,F) < £, d(x,F n ) < £ for n
Salinetti and Wets
154
sufficiently large or equivalently if F
that F
iii
a
n B;(X)
n
~
lla.
n BO(x)
~ ~ implies
~ ~ for n sufficiently large.
Simply note that the properties of E allow us
to write every open set as the countable union of open balls.
ii
a
i
~
Take x E F and {G , i
i
a
E I} a fundamental (nested)
system of open neighborhoods of x with G
I
G. n F ~ ~ for all i.
1
By ii
a
Clearly
l.
1
C
F.
this implies that G. n F
for n :2: n. and thus there exists x
i.e. F
~
n
n ~ ~
E F n such that x
limx
n'
lim inf F .
n
Next we prove that the b statements are equivalent.
But this time we derive a string of implications in the
opposite order, i.e.
ib~
ii ·
b
that F
n K
Suppose not.
~
but for F
sequence {x E F
m
m
to x.
n
Then there exists a compact K such
m
n
K
~ ~
for m E MeN.
Every
K} admits a convergent subsequence, say
By construction x E K n lim sup F
n
c
K n F, a con­
tradiction.
ii
b
~
iiib~
iii •
b
iv .
b
Evident.
Since d(x,F) > E if and only if F n Bs(X)
which in view of iii
b
implies that F
n
n BE(X) =
lently d(x,F ) > E, for n sufficiently large.
n
iV
iV
b
b
~,
=
~
or equiva­
From this
follows directly.
~
vb-
Suppose not; then lim sup (Fn\EF)
exists {xm E Fm'EF, m E M} such that lim x m
~ ~,
=
i.e. there
x E lim sup
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Volume 4
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1979
On one hand we have that for all m E M
~
E < d(xm,F)
d(x,F) + d(x,x )
m
and thus d(x,F) > E-d(x,x ), on the other hand d(x,F ) ~
m
m
d(xm,F ) + d(x,x ).
m
m
o
< E-lim d(x,x ),
m
vb~
contradicting iv .
b
Apply the Lemma to the sequence {Fn\EF), n EN}.
vi ·
b
vib~
This implies that lim inf d(x,F )
m
vii .
b
Since for all E > 0, r > 0 there exists n(E,r)
such that for all n ~ n(E,r) it follows that lim sup
Since this holds for all E > 0 and lim sup
also have that lim sUPn F~ C F.
n
F
r
n
n
F
r
n
CEF.
is closed we
From this the assertion
follows directly.
vii
b
~
i
b
.
If x E lim sUPn F
m E M C N.
n
then x
=
Take s > d(O,x), then the x
ciently large.
lim x ' x m E F ,
m
m
m
S
E Fm
for m suffi­
Thus x E lim sup F~ C limrtoolim sup F~
=
F.
In the following Corollary we give an abbreviated
version of a number of statements appearing in the Theorem.
The proofs follow directly from the inclusion lim inf C lim
sup for any sequence of subsets of E and the inequality
lim inf
~
lim sup for any sequence of numbers.
Coro l lary 1.
Suppose that
of closed subsets of E.
(i)
F
lim F
(ii)
F
J-lim F
(iii)
(iv)
(v)
{F; F n'
Then the following are equivalent
n
n
for all x in E, d(x,F)
limn[(F'EF n )
F
=
n E N} is a co l lection
U (Fn'EF)]
limrtoolim inf F~
=
=
limnd(x,F )
n
= ¢
limrtoolim sup F~.
Salinetti and Wets
156
The last one of the above statements suggest the defi­
nition of an r-distance between subsets of E.
This has been
done for linear subspaces [4], convex cones [11] and for
convex subsets of RP = E [7],
[9].
Let C,D be two closed
convex subsets of RP , then the r-distance between C and D
is by definition
o if Cr = Dr = ~
hr(C,D)
+00 if Cr
=
~ or Dr
h(Cr,D r ) otherwise.
Corollary 2.
[7]
Suppose that
{FiFn,
collection of closed convex subsets of RP .
n E N} is a
Then F
lim F
if and only if there exists r O > 0 such that for all r
lim hr(F,F ) =
n
~
n
r O'
o.
In [9] it was conjectured that a statement similar to
that of Corollary 2 holds for closed sets if the r-distance
is defined as follows Pr(C,D) = lims~rhs(C,D).
It can be
shown that if limp (F,F ) =0 for all r ~ r O > 0 then F
r
n
=
lim F ·
n
Unfortunately the converse does not hold as can be seen from
the following example:
Consider E
= R2
tance, i.e., Bl(O) is the unit square.
F
=
with
K =
F
{[(-I, k
n
Clearly F
{[(-l,k),
with d the loo-dis­
Let
(l,k)], k E K},
{1,2, ••• } and
=
+ n- l ), (O,k)]
lim F
n
U
[(O,k),
(-1, k
+ n- l )], k E K}.
but for all k in K and all n in N,
Pk(F,F n ) = 1.
Finally note that the main theorem could be proved
with E locally compact, Hausdorff and second countable.
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1979
But then E is metrizab1e and separable.
After defining a
metric on E compatible with the underlying topology, we
would be in the setting considered in this paper.
Acknowledgement:
The presentation of the paper has profited
from friendly conversations with Sam Nadler.
References
[1]
G. Choquet, Convergences, Anna1es de l'Univ. de
Grenoble 23 (1947-1948), 55-112.
[2]
G. Dantzig, J. Folkman and N. Shapiro, On the con­
tinuity of the minimum set of a continuous function,
J. Math. Anal. App1. 17 (1967), 519-548.
[3]
N. Dunford and J. Schwartz, Linear Operators, Part 1:
General Theory, Interscience Pub1. Inc., New York,
1957.
[4]
T. Kato, Perturbation Theory for Linear Operators,
Springer Verlag, Berlin, 1966.
[5]
G. Matheron, Random Sets and Integral Geometry, J.
Wiley, New York, 1975.
[6]
E. Michael, Topologies on space of subsets, Trans.
Amer. Math. Soc. 71 (1951), 151-182.
[7]
U. Mosco, Convergence of convex sets and the solutions
of variational inequalities, Adv. Mathern 3 (1969),
510-584.
[8]
S. Nadler, Hyperspaces of Sets, M. Dekker Inc., New
York, 1978.
[9]
G. Sa1inetti and R. Wets, On the convergence of se­
quences of convex sets in finite dimensions, SIAM
Review 21 (1979).
[10]
, Convergence of Closed-Valued Measurable Multifunctions, Tech. Report 1979 (to appear).
[11]
D. Walkup and R. Wets, Continuity of some convex-
cone valued mappings, Proc. Amer. Math. Soc. 18 (1967),
229-235.
[12]
G. Whyburn, Analytic topology, Amer. Math. Soc.,
Providence, R. I., 1942.
158
Universita' di Roma
and
University of Kentucky
Salinetti and Wets