A Generalization of a Theorem on Continuous Selections
E. Michael
Proceedings of the American Mathematical Society, Vol. 105, No. 1. (Jan., 1989), pp. 236-243.
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PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 105, Number I , January 1989
A GENERALIZATION OF A THEOREM ON CONTINUOUS SELECTIONS E. MICHAEL
(Communicated by Dennis Burke)
The author's selection theorem for functions with finite-dimensional
domain is sharpened in two directions.
ABSTRACT.
The purpose of this note is to sharpen the following theorem in two related
directions, and to give some applications. The relevant definitions, which have
all appeared in the literature, will be reviewed later in this introduction.
Theorem 1.1 [3, Theorem 1.21. Let X be paracompact, Y completely metrizable, and y, : X + 2' 1.s.c. with y,(x) E F ( Y ) for all x E X . Suppose that
A c X is closed, that dim, X\A 5 n + 1 , and that { y,(x) : x E X ) is equi- LCn .
Then every selection g for y, 1 A extends to a selection f for y, 1 U for some open
U 3 A , and one can take U = X if y,(x) is C n for all x E X\A .
We now introduce some additional terminology. Suppose that y,: X +
9 ( Y ) , where 9 ( Y ) = { E : E c Y ) . We define @ : X + 9 ( X x Y) by
@(x)= {x) x y,(x), and (following [3, p. 5741) we say that the function y,
is equi- L C n if {@(x): x E X ) is an equi- LC" collection of subsets of X x Y .
This property of a, is strictly weaker than {y,(x): x E X) being an equi- L C n
collection of subsets of Y (see Corollary 1.5), but it suffices for Theorem 1.1
(see Theorem 1.2) and it is conserved under a natural and useful operation
which does not conserve {y,(x): x E X) being equi- LCn (see Lemma 5.1).
The following result generalizes Theorem 1.1. Part (a) of this generalization
answers a question which was raised by the author on p. 575 of [3], and which
had originated in a conversation between the author and S. Eilenberg.
Theorem 1.2. Theorem 1.1 can be sharpened simultaneously in two directions:
(a) The assumption that {y,(x): x E X) is equi- L C n can be weakened to
assuming only that y, is equi- L C n .
Received by the editors October 8, 1987 and, in revised form, February 16, 1988 and May 12,
1988.
1980 Mathematics Subject Classification (1985 Revision). Primary 54C65, 54C99.
Key words and phrases. Continuous selections, equi- L C n , G6-set.
@ 1989 American Mathematical Society
0002-9939189 $1.00 + S.25 per page
A THEOREM ON CONTINUOUS SELECTIONS
237
(b) The assumption that a,(x) E F ( Y ) for every x E X can be weakened
to assuming only that there exists a G,-subset D of X x Y such that
@(x)E F ( D ) for every x E X .
As we shall see in $2, Theorem 1.2 follows almost immediately from Theorem
1.1 in the special case of metrizable X ; the difficulty consists in proving it for all
paracompact X , and thus obtaining a genuine generalization of Theorem 1.1.
This is an unusual situation, since paracompact spaces are the natural domains
for almost all selection theorems and, as a rule, it is no easier to prove a selection theorem (such as Theorem 1.1) for metrizable X than for paracompact
X . Theorem 1.2 presents a striking exception to this rule. The only other such
exception known to the author is [4, Theorem 1.21, whose proof required the development of a special technique which was of some independent interest. The
proof of Theorem 1.2 also depends on a-quite different-special technique,
and it is this technique, as much as the theorem itself and its applications, which
gives Theorem 1.2 its significance.
It is not easy to further weaken the conditions in Theorem 1.2. In particular,
(a) cannot be weakened to simply assume that every a,(x) is LC" , even when
n = 0 (see [5, Example 10.6]), and (b) cannot be weakened to simply assume
that every a,(x) is a G, in Y , even when n = - 1 (see [2, Example 6.11).
The following result, which is applied in 56, is proved with the aid of both
parts of Theorem 1.2. We adopt the convention that, if D c X x Y and
x E X , then D ( x ) = { y E Y: (x , y ) E D ) ; moreover, if a,: X 9 ( Y ) , then
a,, : X
9 ( Y ) is defined by a,, ( x ) = a,(x) n D ( x ) .
-+
-+
Corollary 1.3. Let X be paracompact with dim X 5 n + 1 , Y completely metrizable, and a,: X + 3 ( Y ) 1.s.c. and equi- LC" . Suppose that D c X x Y is open
and that q,(x) is nonempty and C" for all x E X . Then q, has a selection.
Let us now briefly define the concepts in Theorem 1.1. A space Y is completely metrizable if it can be metrized with a complete metric. Letting P ( Y ) =
{ E :E c Y ) , we define 2'= { E E P ( Y ) : E # 0 ) and F ( Y ) = { E E 2': E
.
closed in Y) . A function a,: X + 2 Y is
1.s.c. if {x E X : a,(x) n V # 0 )
2' is a conis open in X for every open V in Y . A selection for a,: X
tinuous f : X
Y such that f (x) E a,(x) for every x E X . The inequality
dim, X\A 5 n + 1 means that dim B 5 n + 1 for every closed (in X ) set
B c X\,4.
To define C" and equi- LC" , we introduce the following notation, which will
be used throughout this paper: If F c E , then F + E will mean that, for all
m 5 n , every continuous g : sm+ F extends to a continuous f : B ~ " + E
(where S m is the m-sphere and B ~ + t' he ( m + 1)-ball). Now E is called
C" if E + E , and $ c 2' is called equi- LC" if, for every y E U 8 ,
every
neighborhood W of y in Y contains a neighborhood V of y in Y such that
V n E + W n E forevery E E ~ .
-+
-+
238
E. M I C H A E L
We now pause for a simple characterization of equi- LC" functions y,: X
9 ( Y ) , as defined after Theorem 1.1.
Lemma 1.4. If y,: X
+
+
9 ( Y ) , then the following are equivalent.
(a) y, is equi- LC" .
(b) If x * E X , y* E y,(x) , and if W is a neighborhood of y* in Y , then
(x* , y") has a neighborhood U x V in X x Y such that V n y,(x) +
W n y,(x) for every x E U .
In contrast to Lemma 1.4, {y,(x): x E X ) is equi- LC" if and only if 1.4(b)
is satisfied with U = X . We thus obtain the following corollary.
Corollary 1.5. If a, : X
is equi- LCn . I
+9
( Y ) , and if { y,(x) : x E X ) is equi- LC" , then y,
The easy proof of Theorem 1.2 for metrizable X is given in 52. The technique needed to prove Theorem 1.2 in full generality is developed in $3, and
Theorem 1.2 is then proved in 54. Corollary 1.3 is proved in 55, and some
applications are obtained in 56. Finally, 57 is devoted to examples.
Embed X in a completely metrizable space X' , and let D' be a GJ-subset
of X' x Y such that D' n ( X x Y) = D . Since X' and Y are completely
metrizable, so is X' x Y and hence so is D' . Also P(x) E F ( D 1 ) for every
x E X . Hence @ : X + F ( D ' ) satisfies all the hypotheses of Theorem 1.1 (with
Y replaced by D' and y, by @ ).
Now let g be a selection for V I A . Define g : X
D' by g ( x ) = ( x , g ( x ) ) .
Then g is a selection for @I A , so, by Theorem 1.1 applied to @ , this g extends
to a selection j' for $51 U for some open U 1A ; moreover, if y,(x) is C" for
every x E X\A , then @ ( x )(which is homeomorphic to y,(x) ) is also C" for
X E X\A, s o o n e c a n take U = X . Now define f : X + Y by f = n 2 0 f ,
where n,: X x Y
Y is the projection. This f is the required extension of
g.
It should be remarked that the validity of part (a) of Theorem 1.2 for metrizable X was already established in [3, Theorem 9.21.
-+
-+
Throughout this section, X is a topological space and (Y , d ) a metric space.
Y
If y,: X -+ 2 , then Gry, (the graphof y,) denotes { ( x , y ) E X x Y: y E
~ ( x ). )The half-open interval ( 0 , I] will be denoted by J .
Lemma 3.1. Suppose 2 is a collection of open rectangles U = U' x U" in
X x Y. Define u: U 2 + J by
U(X , y) =
sup{r
E
J : {x) x B,(y) c U for some U
' The converse is false; see Example 7.1.
E
21,
A THEOREM ON CONTINUOUS SELECTIONS
and define y : U Z!+ F ( J ) by y ( x , y ) = (0 , u ( x , Y )I . Then:
( a ) y is 1.s.c.
( b ) ~f ( x , y ) , ( x , y') E U Z!,then l u ( x ,Y ) - u ( x ,yl)l 5 d ( y Y ' ) .
( c ) If ( x ,y ) E U Z ! and ( x ,y') $ U Z ! , then u ( x ,Y ) 5 d ( y Y ' ) .
Proof. Routine verification.
Lemma 3.2. Let y, : X -+ 2 be equi- L C n . Then for each i E w there exists a
collection q.of open rectangles U = U' x U" in X x Y covering Gr a, such
that, if u j : U q.+ J is as in Lemma 3.1 , then
whenever ( x , y ) E U q.and r < u i ( x , y ) .
Proof. Fix i E w . By Lemma 1.4, each ( x * , y * ) E Gr a, has a neighborhood
) every x E ( 7 ' .
U = U ' x U" in X x Y such that u U n r ( x )+ B , 1 2 i ( y * ) n y , ( xfor
W e may suppose that diam U" < 1 / 2 i , so that U" n p ( x ) 4 B l l i ( y )n y,(x)
whenever ( x , y ) E U . Letting q. be the collection o f all such U , it follows
from the definition o f u , that q.has the required property.
Lemma 3.3. Let a,: X + 2 Y . For each i E w , let q.be a collection of open
rectangles in X x Y covering Gr y, , and let ui : U q.-+ J and y j : q.
F ( J ) be as in Lemma 3.1. Let E = n , ( u q ) , let Z = J ~ dejine
,
y : E -+
9 ( Z ) by y ( x , y ) =
y , ( x . y ) , and dejne 6 : X + 2 Y x Z by
u
-+
ni
Then:
( a ) y is 1.s.c.
( b ) If a, is l.s.c., so is 6 .
( c ) If x E X , and i f a,(x)E F ( E ( x ) ), then 6 ( x )E F ( Y x Z ) .
( d ) 6 ( x ) is homeomorphic to y,(x) x Z for all x E X .
( e ) If x E X , and if a,(x) is C" , then 6 ( x ) is C n .
( f ) If y, is equi- L C n , and i f the collections q are chosen as in Lemma
3.2, then { 6 ( x ) :x E X ) is e q u i - L C n .
( g ) If A c X is paracompact 2 , then for every selection g for y,lA there
exists a selection g' for 61A such that n,og' = g (where n, : Y x Z + Y
is the projection).
Proof.
( a ) Each y , is 1.s.c. by 3.1 ( a ) ,so y is also 1.s.c.
( b ) From ( a )and the definition o f 6 .
( c ) Suppose ( y ( j ) z, ( j ) )-+ ( y , z ) in Y x Z , with ( y ( j ) z, ( j ) )E 6 ( x ) for
all j , and let us show that ( y , z ) E 6 ( x ) .
It suffices if A is normal and countably paracompact.
E. MICHAEL
240
First, let us check that y E q(x) . Suppose not. Then y 4 E(x) , because
y (j)E q(x) and q(x) is relatively closed in E(x) . Hence (x , y ) 4 U FZ, for
some i , so ui(x , y (j))-+ 0 for this i by 3.1(c); hence zi(j) -+ 0 (because
z,(j) E yi(x , y ( j ) ) ), and thus zi(j) + zi (because zi # 0 since z E Z ), which
is impossible because z(j) z .
Next, let us show that z E y ( x , y) . For all i we have zi(j) 5 ui(x, y ( j ) ) +
u i ( x ,y ) (by 3.l(b)) and z,(j) z, , so zi 5 ui(x ,y) and thus zi E ~ ( xy),.
Hence z E y ( x ,y ) .
(d) This follows from the fact that, by 3.l(b), the restriction uil({x) x q(x))
is continuous for all x E X and all i .
(e) This follows from (d), since the product of two Cn spaces is Cn.
(f) Let (y* , z*) E Y x Z , and let W x L be a neighborhood of (y* , z*) in
Y x Z . We will show that there is a neighborhood V x H of (y* , z*)
in Y x Z such that (V x H ) n B(x) 4 ( W x L) n B(x) for all x E X .
For 6 > 0 , define N,(z*) = {z E Z:lzi - zfl < 6 for i 5 116). Pick
y > 0 such that N3,(z*) c L . We may assume that L = N,,(z*) and that
W c B,(Y*)
Let us begin by showing that, if x E X , if y , y' E W n p ( x ) , and if N,(z*) n
y ( x , y) # 0 , then L n y ( x , y') # 0 . Indeed, pick z E N,(z*)ny(x,y ) . Now
y , Y' E W c B,(y*), so d(y , y') < 2y and hence lui(x, y) - u i ( x , yt)l < 27
for all i by 3.1 (b). Thus there is a z' E y ( x , y') such that IZ; - zil < 2y for
all i . Now Iz; - z;l < 2y + y = 3y for i 5 l/3y, so z' E N,,(z*) c L . Hence
z' E L n y ( x , y') .
Next, let us show that there is a neighborhood V x K of (y* , z*) in Y x Z
suchthat V c W and, if x E X and (VxK)nB(x) # 0 , then Vnp(x) 4 Wn
q(x) . Indeed, pick j E w such that B ~ ~ , ( Yc* W
) , and let e = min(fz; , $) .
Let V = B,(y*), and let K = {z E Z:zj > 2e). To sees that this works,
suppose x E X and (V x K) n 8(x) # 0 , and pick (y , z) E (V x K) n 19(x).
Then y E q ( x ) , d(y ,y*) < e , and 2e < zj Iu j ( x , y ) , so, by the property of
%, in Lemma 3.2,
-+
-+
v n P(X) c B2,(y) n P(X) 4 B,/,(Y)n r(x) c ' 2 / j ( ~ * ) n ~ ( xc) W n r(x)
and hence V n q(x) 4 W n q (x).
Now let V and K be as in the previous paragraph, and define H = K n
N,(z') . That completes the construction of V and H , and it remains to verify
that (V x H ) n 8(x) 4 (W x L) n 8(x) for all x E X . Since that is clear if
(V x H ) n B(x) = 0 , we may suppose that (V x H ) n B(x) # 0 . We must
show that, for m 5 n , every continuous g : S m -+ (V x H ) n 8(x) extends to
a continuous f :Bm+' -+ ( W x L) n 8(x) . In the following construction of f ,
we let rcI : Y x Z -+ Y and rc, : Y x Z Z denote the projections.
-+
Note that this will be shown for all ( y * , z * ) E Y x Z , although the definition of { O ( x ) :x E
X ) being equi- LCn only requires it for ( y * . z * ) E U { O ( x ) :x E X ) .
A THEOREM ON CONTINUOUS SELECTIONS
24 1
Since (V x H ) n 8 ( x ) # 0 , we have (V x K ) n 8 ( x ) # 0 , so V n p ( x ) 4
W n p ( x ) by the way V and K were chosen. Hence R , o g : S" + V n p(x)
extends to a continuous h : B"" -t Wnp(x) . Now observe that ( w x N , ( z * ) ) ~
8(x) # 0 (since W 1 V and NY(z*)1H ), SO NY(z*)n y ( x ,y ) # 0 for some
y E W n p ( x ) , and hence L n y ( x ,y') # 0 for all y' E W n p ( x ) by the result
established three paragraphs back. In particular, L n y ( x , h(t)) # 0 for all
t E Bm+', SO we can define a : Bm+' -,F ( L ) by
Note that a is 1.s.c. because y is 1.s.c. (by (a)) and because L is open in Z .
Also each a ( t ) is a convex subset of Z , so each a ( t ) is cmand {a(t): t E
Bm+') is equi- LC" . Since L is completely metrizable, it follows that a
satisfies all the hypotheses of Theorem 1.1 (with Y replaced by L ). Observe
next that n20g is a selection for alSm, for if t E Smthen g ( t ) E (V x H ) n 8 ( x )
and hence n2(g(t))E H n y ( x , x,(g(t))) = H n y ( x ,h(t)) c a ( t ) . Thus, by
Theorem 1.1, n2 o g extends to a selection k for' a . We can now define
f : B"" + ( W x L ) n 8 ( x ) by f (t) = (h(t) , k(t)) , and this f is the required
extension of g .
(g) Define {: A -,F ( Z ) by {(x) = y ( x , g ( x ) ) . It will suffice to show that
{ has a selection h , for then the function g': A -, Y x Z defined by
g'(x) = ( g ( x ) , h (x)) has the required properties. Define ti: A -,F ( J )
by {,(x) = y,( x , g ( x ) ) . Then {(x) =
ti(x) (because y (x , y ) =
y,(x ,y ) ), so it suffices to show that each 5, has a selection. Now
yi is 1.s.c. by 3.1(a), and hence so is {, . Since {,(x) is a half open
interval (0 , ui(x , g(x))], it follows from [ 1, Theorem 1.41 that ti has
a selection.
ni
n,
Let X , Y , p : X -, 2' and D c X x Y be as in the hypotheses of Theorem
1.1 as modified in Theorem 1.2. Since D is a G, in X x Y , we have D = Di
with each D, open in X x Y . Since p is equi- LCn, we can choose collections
of open rectangles in X x Y as in Lemma 3.2; since Gr p c D ,we can choose
these q.sothat Uq. c D,. Let E c X x Y , Z = ( 0 , 1 l w , y : E + F ( Z )
and 8 : X -, 2YxZbe as in Lemma 3.3. Clearly E c D .
Let us check that 8 satisfies the hypotheses of Theorem 1.1, with Y replaced
by Y x Z and 0, by 8 . First, since Y and Z are completely metrizable, so
is Y x Z . Next, 8 is 1.s.c. by 3.3(b). Since @ ( x )E F ( D ) and thus p ( x ) E
F ( D ( x ) ) for all x E X , and since Gr yl c E c D , we have q ( x ) E F ( E ( x ) ) ,
and hence 8(x) E F ( Y x Z ) by 3.3(c). Finally, {B(x): x E X ) is equi- LC"
by 3.3(f).
Now let A c X be closed with dimx X\A 5 n + 1 , and let g be a selection for VIA. We must show that g can be extended as in the conclusion of
ni
242
E. MICHAEL
Theorem 1.1. By 3.3(g), there is a selection g' for 8lA such that n, o g' = g .
By Theorem 1.1 applied to 8 (see the previous paragraph), g' extends to a
selection f' for 81U for some open U 1 A in X ; if q ( x ) is C n for all
x E X\A , then so is O(x) by 3.3(e), and we can therefore take U = X . Letting f = n, o f' , we see that f is the required extension of g .
The following lemma, whose simple verification is omitted, records a useful property of equi- LCn maps. For the definition of q, , see the paragraph
preceding Corollary 1.3.
Lemma 5.1. Let q : X -, 9 ( Y ) be equi- LC" , and let D
Then q, is also equi- L C n .
c X x Y be open.
Proof of Corollary 1.3. Since D is open in X x Y , qD is also l.s.c., and qD
is equi- LCn by Lemma 5.1. Clearly @,(x) E F ( D ) for every x E X . We
can therefore apply Theorem 1.2, with A = 0 and with q replaced by q, , to
conclude that q, has a selection.
In this section, we apply Corollary 1.3 to improve two results in [5]. Our first
theorem sharpens [5, Theorem 3.11.
Theorem 6.1. Let X be paracompact, Y a Banach space, and q : X -, F ( Y )
1.s.c. with each q ( x ) a convex subset of Y . Let E c X x Y be closed, define
X' = {x E X : q ( x ) n E ( x ) # 01,and suppose that
dim X < dim q ( x ) - dim conv(q(x) n E ( x ) )
for all x E X I . Then q has a selection f such that f ( x ) 4 E ( x ) for all
XEX.
Proof. Let D = (Xx Y)\E . We must prove that q, has a selection, which we
do by showing that q and D satisfy the assumptions of Corollary 1.3 with n
defined by dim X = n + 1 .
Clearly D is open in X x Y . Since the intersection of each open ball in
Y with each q ( x ) is convex and thus C n , we see that {q(x): x E X) must
be equi- L C n , and hence q is equi- LCn by Corollary 1.5. Finally, q, (x)
is nonempty and C n for all x E X ; this is clear if x 4 X' (because then
qD(x) = q ( x ) ), and when x E X' it follows from the inequality in the statement
of our theorem and from [5, Lemma 2.11 with K = q ( x ) .
By contrast, {(o(x): x E X ) being equi- LCn does not imply that {(oD(x):x E X) is
equi- LCn ; see Example 7.2.
Theorem 3.1 of [5] makes the stronger (and less simple) assumption that E is the graph of a
continuous (with respect to the HausdortT metric on 9 ( Y ) ) function y l : X + 9 ( Y ) .
A THEOREM ON CONTINUOUS SELECTIONS
243
The following generalization of Theorem 6.1 sharpens [5, Theorem 3.31, and
implies, in particular, that Theorem 6.1 remains true if E is only an F, (rather
than closed) in Y .
Theorem 6.2. Let X , Y and y, be as in Theorem 6.1 . Let Ei c X x Y
( i = 1 , 2 , ... ) be closed, dejine
= {x E X : ~ ( xn)E,(x) # 0), and suppose
that
dim X < dim y, (x) - dim conv( y, (x) n E, (x))
XI
for all x E x,! and all i . Then y, has a selection f such that f (x) 4 E,(x)
for all x E X and all i .
Proof. This result can be derived from Theorem 6.1 just as [5, Theorem 3.31
was derived from [5, Theorem 3.11. We omit the details.
The following examples are based on the trivial observation that, if y , : X -,
2* , if each ~ ( x is) L C n (in the usual sense), and if X is discrete, then y, is
equi- L C n .
Example 7.1. A 1.s.c. y,: X + F ( Y ) such that y, is equi- L C n for all n but
{y,(x): x E X ) is not equi- L C 0 .
Proof. Let X be the discrete space of positive integers, let Y = R , and let
p ( n ) = (0, l l n ) for all n E X . This y, has the required properties.
Our second example is referred to in Footnote 4.
Example 7.2. A 1.s.c. y,: X + F ( Y ) and an open D c X x Y such that
{ ~ ( x: x) E X ) is equi- L C n for all n but {y,, (x): x E X ) is not equi- L C 0 .
Proof. Let X and Y be as in Example 7.1, define y, : X + F ( Y ) by y,(x) =
[0, l l n ] , and let D = { ( n , y ) E X x Y , y # 1 / 2 n ) . This y, and D have the
required properties.
1. C. H. Dowker, On countably paracompact spaces, Canad. J. Math. 3 (1951), 219-224.
2. E. Michael, Continuous selections I, Ann. of Math. (2) 63 (1956), 361-382.
Continuous selections 11, Ann. of Math. (2) 64 (1956), 562-580.
3. ,
4.
,
5. ,
A selection theorem, Proc. Amer. Math. Soc. 17 (1966), 1404-1406.
Continuous selections avoiding a set, Topology Appl. 28 (1988), 195-213.