Pub
.
Ma
.
UAB
Vol
.
30
Ns
1
Maig
1986
CONDITIONS
OF
ANGELIC
TYPE IN
FUNCTION
SPACES
by
Miguel
Ba celó

and Miguel
A
.
Canela
ABSTRACT
:
This
pape
deals
wi h
a
class
o
opological
spaces
in
which
a-compac ness
and
compac ness
coincide
and he
igh ness
c
a
compac
subse
is
less o
equal
han
a,
a
being
an
in ini e
ca dinal
numbe
.
This
class
is
a
na u al
ex ension
o
he
class
o
s ic ly
angelic
spa-
ces,
in oduced
by W
.
Go ae s
.
Su icien
condi ions
a e
gi en
o
a
space
o
con inuous
unc ions
o
belong
o his
class,
and
some
esul s
on
locally
con exspaces
a e
cb ained
as
an
applica ion
.
1
.
In oduc ion
-
The aim
o his
pape
is
o gi e
a
desc ip ion
u
a
class
o
opolo-
gical
spaces in
which
a-compac ness
and
compac ness
coincide,
a
being
an
in ini e
ca dinal
numbe ,
which
will emain
ixed
h oughou his
pape
.
This
class
is
a-p oduc i e
(closed
by
p oduc s
o
amilies
o
ca d
.i iali-
y
<a)
.
The
class
desc ibed
will be
deno ed
by Aa
,
whe e
he "A"
s ands
o ,
angelic
.
Indeed,
o
a
=
a
0
,
ou
class
coincides
wi h
he
class
o
s ic ly
angelic
spaces
in oduced
by W
.
Go ae s
[2]
.
These
spaces
a e
a
subclass o
he
angelic
spaces
s udied
by
P yce
in
[5]
.
The
p ope ies
o
he
class
o
angelic
spaces
can
be
ound
in
In.Sec ion
2
we
ecall
some
opological
no ions
which
will be used
h oughou
his
wo k
.
In
Sec ion
3
we
in oduce
he
class
A
n
,
p o ing
some
'p ope ies
.
The
main
esul
is
he
s abili y
o
p oduc s
o
ca di
nali y
<
a,
and
he
key o
his esul
is
a
heo em
c V
.I
.
Malyhin
[41
.
In
Sec ion
4 we
gi e
su icien
condi ions
o
a
space
o
con i-
nuous
unc ions
o
belong
o
A
, ,
endowed
wi h
he
opology c
poin wise
con e gence
.
In
Sec ion
5
we
es ic
ou sel es
o
locally
con ex spa-
ces,
ob aining
wi h
he
ools
gi en
in
he
p e iousSec ions
some
e-
sul s
appea ed
in
a
o me
pape
by
M
.
Valdi ia
[6]
.
2
.
Some
opological
-
no ions
.-
a
will
deno e
a
ixed in ini eca dinal
numbe
.
A
subse
S
o
a
opological
space
X
(all
he
spaces in ol ed
a e
Hausdo )
is a-
compac
e e y
ne
(xcon ained
in S,
wi h

III
<
a
has
a
clus e
,
poin
x
E
S
.

S

is

ela i ely

a-
compac

i
e e y

ne

(x
i
)
iEI'

wi h

III
<a

has a
clus e

poin

xEX
.

E e y

( ela i ely)

compac

subse

is

( ela i ely)
a-compac ,
bu
no
con e sely
.
A
coun e example
can
be
ob ained
modi-
ying
he
usual
example
o
a
sequen ially
compac
space which
is
no
compac
[1,
1
.2(7)]
.
i
I X is a
opological
space,
he
igh ness
o X,
(X),is
he
mini-
mal
ca dinal numbe
wi h
he
ollowing
p ope y
:
i
S is a
subse
o
X
and
xE
cl(S)

is a
closu e
poin
o
S,

he e
is a
subse

MC
S

wi h
1M1
<
and

x E
cl
(M)
.
.
The

densi y

cha ac e

o

X,

d
(X),
is

he

minimal
ca dinali y
o a
.
dense
subse
o
X
.
The weigh
o X,
w(X),
is
he
mini-
mal
ca dinali y
o
a
basis
o open
subse s
o he
opology
o
X
.
The
weigh

o

X

a

he

poin

x E X,

w(X),

is

he
minimal

ca dinali y
o
a
x

>
basis

o
neighbou hoods
o
x
.
Clea ly
wx
(X)
<
w(X),

bu no
con e sely
.
-
Fo
he e
and
c he
ca dinal
unc ions
o
he
Gene al
Topology,
[3)
can
be
used as
a
s anda d
e e ence
.
3
.
The
class
A
a
.-
The
class
A a
will be
he
class
o
all
opological
spaces
X
sa is-
ying
he
ollowing
condi ions
:
(i)
E e y ela i ely
a-compac
o
Xis
ela i ely
compac
.
(ii)
E e y
compac
subse
c
X
has
igh ness
< a
.
(iii)
I
a
subse
SCX
is
ccmpac
and
d(S)<
a,
hen
w
x
(S)
<
a
o
e e y
xES
.
We gi e
nex
he
p ope ies
o his
class
.
3
.1
.
P oposi ion
:
The
condi ion
(ii)
o
he
p eceding
de ini ion
can
be
eplaced
by
:
(ii)' I
SCX
is
ela i ely
compac
and
xEcl(S),
he e
isane
(xi)iEI
con ained
in S,
wi h

III
<
a
and lim
x
i
=
x,
ob aining
an
equi alen
de ini ion
.
P oo
:

(ii)
'

implies

(ii)
.

Con e sely,

i
(S)
<
a

o
a
compac
subse
SC
X,

we

ake

MC
S

wi h

x
E
cl
(M)

and

1
M1
<
a
.

Then
cl (M)

is

compac

and
has
densi y
cha ac e
<
a,
and,
by
(iii)
he
weigh a
x is < a
.
The e o-
I
X
is a
opological
space, he
igh ness
o
X,
(X),is
he
mini-
mal
ca dinal
numbe
wi h
he
ollowing
p ope y
:
i S is a
subse o
X
and

xE
cl
(S)

is

a

closu e

poin

o

S,

he e

is

a

subse

MC
S

wi h
¡MI
<
and

x
E
cl(M)
.

The

densi y

cha ac e

o

X,

d(X), is

he

minimal
ca dinali y
o
a
dense
subse o
X
.
The weigh
o
X,
w(X),
is
he
mini-
mal
ca dinali y
o
a
basis
o open
subse s
o he
opology o
X
.
The
weigh
o
X
2.
he
poin
xEX,
wx(X), is
he
minimal
ca dinali y
o
a
basis

o
neighbou hoods
o
x
.

Clea ly
wx(X) <
w(X),

bu no
con e sely
.
-
Fo
he e
end
c he
ca dinal
unc ions
o
he
Gene al
Topology,
[3]
can
be used as
a
s anda d
e e ence
.
3
.
The
class
A
a
.-
The
class
A
a
will
be
he
class
o
all
opological
spaces
X
sa is-
ying
he
ollowing
condi ions
:
(i)
E e y
ela i ely
a-compac
o
X is
ela i ely
compac
.
(ii)
E e y
compac
subse
c
X
has
igh ness
<
a
.
(iii)

I

a

subse
S
C
X

is

compac
and
d(S)<
a,

hen
w
x
(S)<
a
o

e e y
XES
.
We
gi e
nex
he
p ope ies
o his
class
.
3
.1
.
P oposi ion
:
The condi ion
(ii)
o
he
p eceding
de ini ion
can
be
eplaced
by
:
(ii)'
I
SCX
is
ela i ely
compac
and
xEcl(S),
he e
isane
(xi)ic-I
con ained
in
S,
wi h

III
<a
and
.
lim
x
i
= x,
ob aining
an
equi alen de ini ion
.
P oo
:

(ii)
'
implies

(ii)
.

Con e sely,

i
(S)
<
a

o
a
compac
subse
SC
X,

we

ake

MC
S

wi h

x
E
cl (M)

and

1
M1
<
a
.

Then

cl
(M)

is

compac

and
has
densi y
cha ac e
< a,
and,
by
(iii)
he
weigh
a xis <a
.
The e o-
52
We
ha e
(i)
.
Suppose

ha

S
C
U
X
j
is

compac ,

and

we

a e
going
o
show ha
¡EI
(S)<
a
.
We
can
suppose,
wi hou
loss
o
gene ali y,
S
=

ni(S)
.
¡El
Acco ding
o a
esul o
V
.I
.
Malyhin
[4,
Theo em
4),
he
ini e
p oduc
o
compac
spaces
wi h
igh ness
<a
has
igh ness
<
a
.
I
x is a
closu e

poin

o
a
subse
D
C
S,

we
can

ind,

o
each

J
C
1

ini e,

a
subse
MJC
J
(D)
wi h
1MJ1
<
a
and
iT
J
(x)E
cl(MJ)
.
We
can
choose
now
M
J
CD
wi h
nj(MJ
)
=
MJ
and
x
E
cl(M)
.
We ha e
(ii)
.
Finally,

i

S C
U
X
1
.

is

co pac

and
d
(S
)
< a
,

hen

d
(i
i
(S)
)
<a,

and
¡El
-
hus
n
(S)
has
weigh
<
a in
e e y poin
.
Keeping
in
mind
he
cons uc-
i

-
ion o
a
basis
o
neighbou hoods o
he
p oduc
opology,
and
e-
calling
III<
a,
i
is
easy
o
see
ha
11
n
1
.(S) has
weigh
< a
a e e-
iE
I
y
poin
.
Q
.E.D
.
and
1
M
J
j<a
.
Ac ually,
4
.
Spaces
o
con inuous
unc ions
.-
M-
U
M
J
J
has
ca dinali y
<a
We
a e
going
o
see
ha
ce ain
a
.ssump ions
on a
opological
space
X
imply
ha
he
space
C(X)
=
C(X,1R)
o
con inuous eal
unc ions
be-
longs
o
he
class
A
a
,
endowed
wi h
he
opology
o
poin wisecon e gen
ce
W
X
.
Indeed,
we ob ain
esul s
analogous
o
hose
ob ained
by
W
.
Go-
ae s
[c]

o he
case
a
=¡{
0
.
4
.1
.
P oposi ion
:.
Le
X
be
a
compac
space
.
Then
(C(X),w
x
)
belongs
o
Aa
.
I is
well-known
ha
e e y
subse
S
C
C(X),
wX
ela i ely
coun a-
compac , is
wX
- ela i ely
compac
and
coun able
igh néss
(e .g
.[1])
.
we
can,
eplacing
i
necessa y
X
by
a
sui able
quo ien ,
P oo
:
bly
To
check
(iii),
suppose
ha
S
sepa a es
he
poin s
o X
.
I S i wX
-compac
and
DCS
is
dense,

wi h

ID
1
<
a,

X

admi s
a
basis
o uni o mi yo
ca dinali y
<
a
,

and
hence
d(X) < a
.
Re e sing
he
a gumen ,
we
ha e
a
se
o
con inuous
eal
unc ions,
o
ca dinali y
<a
,
on S,
which
sepa a es
he
po
.in s
o
S,
and
he e o e
w(S)
<el
.
Q
.E .U
.
4
.2
.

P oposi ion
:

Le
X be a
opological
space
wi h
a
dense
subse
D C X
which
is
ela i ely
a-compac
.
Then
(C(X),w
x
)
belongs
o Aa
.
P oo
:
Condi ions
(i)
and
(ii)
can
be
ob ained
as
in
4
.1
om
known
e-
sul s
(see
[1])
.
To
check
(iii),
we
ema k
ha ,
i S
C
C(X)
is w
X
-co n-
pac ,
we
can
conside
.
he
mapping
01
:
X
-
C(S)
de ined
by
0 1
(x)( )
_
=
(x),

which
is
con inuous
wi h
espec
o
he
opology
w
S
.

Then
0 1
(D)
is
ela i ely
a-compac
in
(C(S),w
S
),
and
i s
closu e
FI
is
a
compac
subse
o
(C(S),w
S
)
.
Now
m2
:
S 1
C(H),
de ined
in
an
analogous
way,
is
injec i e
and
con inuous,
and
he e o e
a
homeomo phism,
and,
using
4
.1,
we
a e
done
.
Q
.E
.D
.
4
.3
.
P oposi ion
:
Le
X,Y
be
opological
spaces
.
a)

I
X

admi s

a

dense

subse
D
C
X
such ha

(C(D),w
D
)

is

in
A,,,

hen
(C(X),W
X
)
is
in A a
.
b)

I
X

U

X
i
,

whe e

(C(Xi),wx
.)

is

in

Aa
and

111<
a,

hen

(C
(X),wX)
iEI

1
is
in Aa
c) I
4)
:
Y
-
"
X
is
con inuous
and
su jec i e,
and
(C(Y),w
y
)
is in
A
a
,
hen
(C(X),w
x
)
is in Al
.
P oo
:
Fo
a)
Take
he
es ic ion
map
C(X)
-
C(D)
and
epply
3
.3
.
Fo
b),
cons uc
an
injec ion
C(X)
-
HC(X
.)
in a
na u al
way and
apply
iEI
1
3
.4,
ollowed
by 3
.3
.
Fo
c),
ake
he
mapping
0*
:
C(X)
-
C(Y)
de ined
by
~*( )
=
o~
and use
3
.3
.
Q
.E
.D
.
4 .4
.
Co olla y
:
Le
X
be
a
opological
space
wi h
a
amily
(X
i
)
iEI
o
ela i ely
a
-compac
subse s

such

ha

i s

union

is

dense

and

~11<
a
.
Then
(C(X),W
X
)
is in Aa
.
Finally,
we
ha e
he
ollowing
esul ,
which
is a
na u al
ex ension
o
a
heo em
o
D .H
.
F emlin
([1,3
.5],
[2,
P oposi ion
9])
:
4
.5
.
P oposi ion
:
Le
X
be
a
opological
space,
and
Z a
me ic
space,
and
suppose
ha
(C(X),w
x
)
is in
á
.
Then
(C(X,Z),wx)
is
in
A
a
.
P oo
:
The
a gumen gi en
in [2)
o he
coun able
case
can
be
used
.
Q
.E
.U
.
5
.
Applica ions
o
locally
con ex spaces
.-
The
esul s
which
ha e
been s a ed
he e
ha e
a
pu ely
opological
na u e
.
Ne e heless,
some
pa icula
cases
ha e been
p o ed
by
o he
me hods,
o
ins an e
o he
case o
a
weak opology on
a
locally
con-
ex
case
.
I
X,Y a e
eal
locally
con ex
spaces,
he
space
L(X,Y) o
con i-
nuous
linea ope a o s
is
a
subspace o
he
space
C(X,Y) o
con inuous
unc ions,

closed
wi h
espec
o
wX
.

The

opology
induced
by

w X

on
L(X,Y)
is
usually
called simple
opology
.
F om
4
.5
and
°
.2,
we
ob ain
di ec ly
:
5
.1
.
P oposi ion
:
Le
X,Y
be
locally
con ex
spaces,
X
being
he
union
o
a
amily
(Di)iEI
o
ela i ely
a-compac
subse s,
wi h
111<
a,
and
Y
me izáble
.

Then
L(X,Y),
endowed
.
wi h
he
simple
opology,
is in Al
.
We
can
conside ,
in
pa icula ,
he
case
in
which
Y
=IR and
X
is
he
dual
E',
endowed
wi h
he
weak opology
a(E',E)
.
We
ob ain
hus
he
ollowing esul
:
5
.2
.
Co olla y
(M
.
Valdi ia
[6])
:
Le E be
a
locally
con ex
space,

E'
being
he
union
o
a
amily
(D
i
)
¡
EI
o
a(E',E)- ela i ely
a-compac
subse s,
wi h
I
II
<a
.
Then
e e y
weakly
( ela i ely)
a-compac subse
o
E
is
weak-
ly
( ela i ely)
compac
.
NOTE
:
In
[6],
his
esul
is
s a ed
unde
an
ex a
assump ion,
he
con-
exi y o
he
D1
.'s,
bu
his
assump ion
is
supe luous
.
Using
5
.1
and
3
.3,
we ha e
:
5
.3
.
Co olla y
(M
.
Valdi ia
(6Í)
:
Le
E
be
a
ec o
space
and
T
and
T'
wo
locally
con ex
opologies
on
E,
T
ine
han
T'
.
Suppose
ha
T'
admi s
a
ze o-neighbou hood
basis
o
ca dinali y
<
a
,
and
deno e
by
E'
_
_
(E,T)'.I
A
C
Eis
a(E,E')-( ela i ely)a-compac ,
A
is
a(E,E')-( ela i-
ely)
compac
.
FINAL
NOTE
:
We ha e
desc ibed
a
class
o
spaces
in
which compac ness
and
a-compac ness
a e he
same,
wi h
good
s abili y
p ope ies,
which
allows
us
o
ob ain he
esul s
o his
Sec ion
.
I
can
be
ema ked
ha almos
all
is
he
same
i
we
eplace
condi ion
(iii)
o
he
de ini ion
by
he
ollowing

s onge
condi ion
:

I
S
CX

is
compac
and

d(S)<a,

hen
W(S)
~
a
.

Ne e heless,

he

class

ob ained

in

his

way

will
.b
e
mo e
es ic ed,
because
he e
a e
sepa able,
i s
coun able
compac
spaces
which
a e no
me izable
(e .g
.
he
Helly
compac )
.
REFERENCES
1
.
K
.
Flo e
:
Weakly
compac
se s
.
Sp inge ,
Be lin-Heidelbe g-New
Yo k,
1980
.
2
.

W
.
Go ae s
:
A
p oduc i e
class
o
angelic
spaces
.
J
.
London
Ma h
.
Soc
.
22
(1980),
355-364
.
3
.

I
.
Juhasz
:
Ca dinal
unc ions
in
Topology
.
Ma h
.
Cen e
T ac s
34,
Ams e dam,1971
.
4
.
V
.I
.
Malyhin
:
The
igh ness
and
Suslin
numbe
in
exp
X
and
in a
p oduc
o spaces
.
So ie Ma h
.
Dokl
.
13
(1972),
496=499
.
5
.
J
.D
.
P yce
:
A
de ice o
R
.J
.
Whi ley's
applied
o
poin wise
com-
pac ness
in
spaces
o
con inous
unc ions
.
P oc
.
London Ma h
.
Soc
.
(3)
23
(1971),
532-546
.
6
.
M
.
Valdi ia
:
Some c i e ia
o
weak
compac ness
.
J
.
eine
angew
.
Ma h
.
225
(1972),
165-169
.
Rebu
el
15
d'ac
ubne
del
1985
MiguelBa celó
EscolaTécnica
Supe io
d'Enginye s
Indus ialsde
Ba celona
.
A inguda
Diagonal
Ba celona
SPAIN
.
MiguelA
.
Canela
Depa amen
de
Teo ia
de
.Funcions,
Uni e si a
d
Uni e si a
de
Ba celona
G an
Via,
585
08007
Ba celona
SPAIN
.