Publicacions
Ma emá iques,
Vol
32
(1988),
123-127
.
Abs ac
EXPONENTIAL
LAW
FOR
UNIFORMLY
CONTINUOUS
PROPER
MAPS
R
.
AYALA,
E
.
DOMINGUEZ,
A
.
QUINTERO
The
pu pose
o
his
no e
is
o
p o e
he
exponen ial
law
o
uni o mly
con inuous
p ope
maps
.
Le
X
be a
egula
space
and
Y
a
locally
compac
egula
space
.
I is
well
known
ha
he
spaces o
con inuousmaps
C(X
x
Y,
Z)
and
C(X,
C(Y
Z))
a e
homeomo phic
conside ing
he
compac -open
opology
.
This
p ope y
has
impo an
consequences
in
he
s udy
o
he
pa h-componen s
o
he
unc ion
spaces
and
in
Homo opy
Theo y
:
The
exponen ial
law
o
he
uni o mly
con-
inuous
p ope
maps
has
simila
consequences
in
some
pa icula
cases
.
All he
spaces
we
conside ,
unless
o he wise
men ioned,
a e
me ic spaces
.
A
p ope
map
will
.
b
e a
con inuous
map
:
X
-+
Y
such
ha
o
e e y
compac
subspace
K
o Y,
-1(K)
is
á
compac
in
X
.
To
abb e ia e,-
we
will
say
ha
is
a
p-map
.
A
u-map
is
a
uni o mly,cgn inuous
map,
and
a
up-map
will
be
a
uni o mly
con inuous
p-map
.
A
up-isomo phism
l
a
homeomo phism
such
ha
and
-1
a e
u-maps
.,
By
C(X,
Y),

C
p
(X,
Y)
and
C

p
(X,
Y)
we
will
deno e he
se s o
con inuous
maps,
p-maps
and
up-maps
be ween
X
and Y,
espec i ely
.
Wi h
Cú
p
(X,
Y)
we
will
ep esen he
space
o
up-maps
wi h
he
opology
o
uni o m
con e gen e
.
In
his
no e
we
p o e
ha
he
up-maps
ollow
he
exponen ial
law
i
X
is
compac
;
ha
is,
he
unc o s
X
x
(-)
and
Cú
p
(X,
-)
a e
adjoin
.
We
also
p o e
ha
i
X
is
no
compac
hese
unc o s
a e
no
gene ally
adjoin
.
A
up-homo opy
(p-homo opy)
be ween
up-maps
(p-maps
is
a
homo opy
which
is
a
up-map
(p-map
.
Wi h
[-, -], [-,
-]
P
,
and
[-,
-]

p
we
will
ep-
esen
he
se s
o
homo opy,
p-homo opy
and
up-homo opy
espec i ely
.
Also,
he
co esponding
homo opy
classes
will
be
deno ed
by [
],
[
]
p
and
[
]up
.
R"
will
s and
o
he
n-dimensional
Euclidean
space,
and
I
o
he
uni
in e al
[0,1]
wi h
ha
dis an e
.
The
Euclidean
no m
will
be
ep esen ed
by

-
~,
and
he
dis an e
o a
me ic
space
by
d(-,
-)
.
This
wo k
has
been
suppo ed
in
pa
by
CAICYT
g an
0812-84
12
4

R
.
AYALA,
E
.
DOMINGUEZ,
A
.
QUINTERO
Theo em
.
Le
X,
Y,
Z
me ic
spaces
.
We
can
de ine
an
injec i e
map
ID
:
Cup
(X
x
Y,
Z)
-)
Cup(X,
Cü
p
(Y,
Z))
as
1(
)
(x) (y)
=
(x,
y)
.
I
X
is
compac ,
hen
1
is
on o
.
P oo
:
I is
easy o
see
ha
:
(a)
Fo
each
x
E
X
he
map
oD( )(x)
is
a
up-map,
because
i is
he
composi-
ion
o
wo
up-maps
.
(b)
oD
( )
is
a u-map,
because
is
a
u-map
.
(c)
Le
us
see
ha
4>( )
is
a
p-map
.
Gi en
a
compac
subspace
K
C
Cú
p
(Y,Z),
we
only ha e
o
p o e
ha
any
sequence
{xn}
in
1( )
-1
(K)
has
a
clus e
poin
.
Le
{z,,}
be
a
subsequence
o
{xj
such
ha
{-D( )
(z
n
)}
is
con e gen
;
le
B
E
K
be
he
limi
poin
.
Then,
o
each
yo
E
Y
he
sequence
{
(zn,
yo)}
con e ges
o B(yo)
.
Consequen ly,
H
=
{
(zn,
yo)
;
n
E
N}U{e(yo)}
is
a
compac
subspace
o
Z
.
This
implies ha
he
sequence
{x,,}
has
a clus e
poin
.
Hence
-P is
well de ined
and
i is
an
injec i e
map
.
Since
X
is
compac ,
each
con inuous
map
de ined
on
X
is
also
uni o mly
con inuous
.
Then,
gi en
a
con inuous
map
g
:
X
-)
Cú
p
(Y,
Z)
i
is
enough
o
show
ha
:
X
x
Y->
Z
de ined
as
(x,
y)
=
g
(x)
(y)
is
a
up-map
.
To
p o e
ha
is
a
u-map,
le
,,
:
Y
-+
Z
and
,
:
X
-->
Z
he
maps
de ined
by
=
(y)
=
(x,
y)
=
(x)
o
each
couple
(x,
y)
.
Acco ding
o
[1,
X
.2
.1
.2] i
su ices
o
show
ha
he
se s
H={
.,
;xEX}
and
C={ ,
;yEY}
a e
uni o mly
equicon inuous
.
Bu
H
=
g
(X)
is
a
compac
subse o
Cú
p
(Y,
Z),
hence
i is
uni o mly
equicon inuous
by
he
heo em
o
Ascoli
(see
[1,
.X
.2 .5
.2])
.
Since
g
is
a up-map,
i
can
be
easily
shown
ha
C
is
uni o mly equicon inuous
se
.
I
emains
o
show
ha
is
a
p-map
.
Le
K
be
a
compac
subse
o
Z
.
I
M
=
U{g(x)-'(K)
;
xE
X},
i is
easy
o
check
ha
-1
(K)
C
X
x
M
.
I
su ices
o
p o e
ha
M
is
a
compac
subse
o
Y
.
Gi en a
sequence
{y,,}
in
M,
he e
is
a
sequence
{x,,}
C
X
such
ha
g
(x
,, )
(y,,)
E
K,
o
each
n
E
N
.
Because
X
and
K
a e
compac ,
we
can assume
ha
{xn}
and
{g(xn)(yn)}
con e ge
o
x
o
E
X
and
zo
E
K
espec i ely
.
Then,
{g(xn)} con e ges
o
g(x
o
),
and
i
is
ob ious
ha
o
each
e
>
0
he e
exis s
n
o
such
ha
d(zo,g(xo)(yn))
<
e i
n
>_
no
.
The e o e,
K
=
{g(xo)(yn)
;
n
E
N}
U
{zo}
UNIFORMILY
CONTINUOUS
PROPER
MAPS

12
5
is
compac
.
Since
g(x0)
is
p ope ,
g(xo)
-1
(K)
is
also
compac
and
{y,,}
C
g(xo)
-1
(K)
implies
ha
{y,,}
has
a
clus e
poin
.
We
conclude
ha
M
is
compac
.
The
ollowing
s a emen
is
easily
p o ed
:
Co olla y
.
Le
X,
Y,
Z
be
me ic spaces
.
Mo eo e ,
suppose
ha
X
is
com-
pac
.
Then,
1)
<D
:
Cúp
(X
x, Y,
Z)
-->
Cup
(X,
Cúp
(Y,
Z))
is
up-isomo phism
.
2)
The
se
o
pa h-componen s
o
Cú
p (Y,
Z)
is
in
bijec i e
co espondence
wi h
[Y,
Z] up
.
3)
1
induces
a
bijec i e
co espondence
1
:
[X
x
Y,
Z]up
-i
[X,
Cú
p (Y,
Z)]
.
Rema ks
andExamples
.
I
we
conside only
p ope
maps,
he
na u al
mapD
:
C
p
(X
Z
Y,
Z)
---)
C
p
(X,
C
p (Y,
Z))
whe e
C
p
(Y,
Z)
is
endowed
wi h
he
compac -open
opology,
is
easily
checked
o
be
well
de ined
and
injec i e
.
How-
e e ,
se e al
oubles
appea
:
(1)
The
compac ness
o
X
is
necessa y
in
o de
o
p o e
ha
1
is
on o,
e en
i
C
p (Y,
Z)
is
endowed
wi h
he
uni o m
con e gence
opology
.
The
ollowing
example shows
his
ac
:
Le
g
:
R
->
C
p
(I,
R)
=
C(I,
R)
be
he
map
gi en
by
g(x)
( )
=
x -(1- )x3
.
I
is
easy o
check
he
con inui y
o
g
.
In
o de
o
p o e
ha
g
is
p ope ,
we
ake
a
compac
K
C
C(I,R)
and
a
sequence
{x

}
C
g
-1
(K)
.
Then
he e
exis s
a
subsequence
{z
} o
{x

},
such
ha
{g(zn)} con e ges
o
B
E
K
.
In
pa icula ,
limg(z

)(1)
=
limz

=
B(1)
and
we
conclude
ha
g
E
Cp(R,C(I,R»
.
Bu
he
con inuous
map
(x,
)
=
g(x)
( ) is
no
p ope
because
«1'
)
21
,
)
E
-1
(0)
o
each
E
[0,1)
.
(2)
Al hough
X
is
compac ,
we
canno
ensu e
ha
1D is
on o
i
we
conside
he
compac -open
opology
on
C
p
(Y,
Z)
:
Le
g
:
I
-)C
p
(R,
R)
gi en
by
g(0) (x)
=
go
(x)
=
x and
g( )
(x)
=
g
(x)
_
{
?
x
+
11

i
0
1/2
<
x
1
/
2
_

(0
<
<
1)
.
I is
clea
ha
g
E
C
p
(R,
R)
o
each
E
I
.
The
con inui y
o
í
--->
g
ollows
om
he
ac
ha lim

=
o
in
I
implies ha
{g
}
con e ges uni o mly
on
he
compac
subse s
o
Z
o
g
o
.
Bu
( ,
x)
--
g
(x)
is
no
a
p ope
map
because
( ,1l )
E
-1
(0)
o
each
E
(0,1]
.
(3)
The
p oo
o
he
Theo em
assu es
ha
1
is
on o
i
we
conside
he
uni o m
con e gence
opology,
u,
on
C
p
(Y,
Z) and
we
assume
he
compac ness
o
X
.
Bu
in
such
si ua ion,
1
:C
p
(X
xY,Z)
--+C(X,CP
(Y,
z»
12
6

R
.
AYALA,
E
.
DOMINGUEZ,
A
.
QUINTERO
is
no
well de ined
as
shows
he
nex
example
:
Le
:
I
x
R
->
R
3
be
he
map
( ,
x)
=
( ,
x, x)
.
This
map
is
a
p-map,
and
i
1(
)
was
con inuous
and
,,
-i
o,
gi en
E
>
0
he e
would
exis
n
o
E
N
such
ha
i
n
>
no
x
I I
n
-
0

1
:51
( n
,
x, n x)
-
( o'
x,
o
x)

I
<
E
o
each
x
E
R
.
Taking

x
1
la ge
enough would
yield
he
con adic ion
E
<
x
I I
n

-
o

I
<
E
.
The
nex
p oposi ion
shows
ha
he e
is
no
any
possible
duali y
up-
iso-
mo phismwhen
X
is
no
compac
.
P oposi ion
.
Le
Z
be he
open
in e al
(-1,1)
.
The e
exis s
no
up-homo-
opy
equi alen e
be ween
C
úp
(R
2
,
Z)
and
CúP
(R,
CúP
(R, Z))
.
We
will
need
he
ollowing
lemma
:
Lemma
.

The
majo
.1
:
[Rn,
Z]uP
->
[Rn,
Z]
P
gi en
by
A([
]++p)
=
[
]
p
is
bijec i e
.
1
P oo
.
I
,
g
:
Rn
->
Z
a e
p-homo opic
up-maps,
he
homo opy
H
:
Rn
x
I
->
Z
gi en
by
H(x,
)
=
(x)
+
(1- )g(x)
is
a
u-map
.
Now,we
a e
going
o
show
ha
H
is
p ope
:
Le
K
C
Z
be
a
compac
subse
and
{ n
=
(xn,
'
,»
a
seiluence
in
H
-1
(K)
.
We
may
assume
ha
{ n }
con e ges
o
o
.
I
we
suppose
ha
{ n
}has
no'
clus e
poin s
we
ha e
lim
x
n
=
oo
.
Because
and
g
a e
p homo opic
we
ge lim
(x
n
)
=
limg(x
n
)
E
{-1,1}
.
I
his
common
limi s
is
1
and
U
is
an
euclidean
neighbou hood
o
1
missing
K,
he e
exis s
n
o
such
ha
(xn),g(xn)
E
U
o
each
n
>_
n
o
.
In
pa icula ,
H(xn, n)
E
U

(n
>_
no)
con adic s
he
assump ion
{(xn, n)}
C
H
-1
(K)
.
So,
A
is
an
injec i e
map
.
In
o de
o
p o e
ha
A
is
on o
we
ecall
ha
[Rn,
Z]p
=
[Sn
-1
,
S
°
],
and
i s
elemen s
a e
he
p-classes o
he
maps
g_
1
,
g
1
:
Rn
-a
Z
gi en
by
g
;
( )
=
(
j/(1+
1
1)

(j
=
-1,1)
i
n
>
2,
o
h( ),
1
h( )
1,
-h( ),
-
1
h( )
i
n
=
1,
whe e
h( )
=
(2/7 )
a c an( )
.
This
ollows,
o
ins an e,
om
he
embedding
heo ems
o
Edwa ds-Has ings,
see
[2,
6
.2
.7]
.
Now,
a
is
on o
because
all
he
ep esen a i es
a e
up-maps
.
P oo
o P oposi ion
4
:
I
su ices
o
p o e
ha
hose
spaces
ha e
no
he
same
numbe
o
pa h-componen e
.
As a
consequence
o
Co olla y
2
.2)
he
pa h-
componen s
o
Cú
P
(R
2
,
Z)
a e
in bijec i e
co espondence
wi h
[R
2
,
Z]up
.
Bu
[R
2
,
Z]up
-
[R2,
Z]p,
by
lemma
5,
and
he
la e
se
has
wo
elemen s
.
Now,we
a e
going
o
show
ha
Cúp
(R, Cúp (R,
Z))
has
a
leas
ou
pa h-
componen s
.
As
aboye,
Cúp
(R,
Z)
has
ou
pa h-componen e,
and
hey
a e
he
componen e
o
go
(x)
=
(2/7 )
a c an(x),

91
=
-
90,
92
=I
g
o
I
and
g
3
=
=-
1
go
1 .

Since
Cúp
(R,
Z)
is
me izable
and
g
o is
a u-map,

o
:
R
-~
Cú
P
(R,
Z)
de ined
by
o
( )
(x)
=
g
o(
+
x)
is
a
u-map
.

Also,
by
using
he
UNIFORMILY
CONTINUOUS
PROPER
MAPS

127
heo em
o
Ascoli
i is
easy
o
check
ha
o is
a
p-map
.
So,
we
ha e
go
a
up-map
o
such
ha
o
(R)
lies
in
he
pa h-componen
o g
o
.
In
a
simila
way,
we
ge
up-maps
i
wi h
i
(R)
lying
in
he
pa h-componen
o
gi
(i
=
1,
2,
3)
.
In
pa icula ,
i
and
i
a e
no
up-homo opic
(0
<_ i
=,
4 j
<
3)
.
We
conclude,
applying
co olla y
2
.2)
again,
ha
he
up-maps
{
;}o<
;<3
de ine
ou
dis inc
pa h-componen e
.
Rema k
.
I
he
me ic
on
Z
is
no
bounded,
lemma
5
is
alse
.
Indeed,
o
each
pai o eal
numbe s
a
l
,
a
2
>
0,
he
up-maps
l
,
2
:
R

)
R
gi en
by
i ( )
=
a

(i
=
1,
2),
a e
p-homo opic,
bu
no
up-homo opic
:
I
H
:
R
xI
---~
R
is
a
up-homo opy
be ween
i
and
2
,
by
[3,
111
.10]
he e
would
exis
e
>
0
such ha
H
(x, )
-
H
(y,
')
1<
max{e
1
(x,
)
-
(y,
')
1,
e}
o
each
couple
(x,
), (y,
')
E
R
x
I
.
The e o e,
la,
-a2
1
x=1
i
(x)
-
2(x)
j=j
H(x,
0)
-
H(x,1)
1
:5E
and
aking
x
la ge
enough
he
abo e
inequali y
would
yield
he
con adic ion
e<x1
al-a2
j<
c
.
In
ac ,
we
ha e
p o ed
ha
ca d
[R,
R]up
>
ca d
R
.
Re e ences
1
.

N
.
BOURBAKI,
in
"Gene al
Topology,"
He mann,1966
.
2
.

D
.
A
.
EDWARDS,
H
.
M
.
HASTINGS,
Cech
and
S een od
Homo opy
The-
o y
wi h
applica ions
o
Geome ic
Topology,
Lec
.
No es
542
Sp inge
(1976)
.
3
.

J
.
R
.
ISBELL,
Uni o m
spaces,
Ma h
.
Su eys,
12
AMS
(1964)
.
R
.
Ayala
:
Dp o
.
d
e
Geome ía
y
Topología
Facul ad de
Ma emá icas
41012-Se illa,
SPAIN
.
E
.
Dominguez
:
Dp o
.
d
e
Ma emá icas
Facul ad de
Ciencias
Ciudad
Uni e si a ia
50009-Za agoza,
SPAIN
.
A
.
Quin e o
:
Dp o
.
d
e
Geome ía
y
Topología
Facul ad
de
Ma emá icas
41012-Se illa,
SPAIN
.
Rebu
el
30
de
Jung
de
1987