Download spaces in which compact sets have countable local bases

PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 48, Number 2, April 1975
SPACES IN WHICHCOMPACT SETS HAVE
COUNTABLE LOCAL BASES
RALPH R. SABELLA
ABSTRACT.
spaces
D -spaces
in which
Among
cient
compact
the results
condition
a question
related
under
posed
sufficient
they
be coconvergent.
spaces;
for
such
paper
neighborhood
spaces
to
such
set
that
of natural
numbers
and
Jlix)
fot all
Ix
72. Using
the notation
j, we define
CGplyJ
there
a space
iCplxJ'D
is an ONA
"U is
P."
CplyJ)
U having
is
that
in stratifiable
is metrizable
lame
if
it there
tnere
is a function
X is a topological
is the open
space,
neighborhood
Sy^i is U-linked
system
of x.
points
{contraconvergent)
whenever \yn ! is (Minkedto
property
N is the
to |xnS if y
S for the set of cluster
coconvergent
some
to
prop-
|Jl(x): x £ X\
Cpíx
to be
a topological
condition.
(ONA)
x £ lj{x, n) = L¡n{x), where
If U is an ONA then the sequence
is a suffi-
developable
a D .-space
of
In relation
metrizability
a nesting-like
U: Xx N-*\J
paper
that
be
implies
assignment
examples
(Dg-spaces).
coconvergent.
shown
that
satisfying
are
bases
in this
are
it is
Coconvergence
in this
local
given
spaces
semimetrizable
it is shown
spaces
countable
F. B. Jones,
a stratification
An open
coconvergent
have
to D.-spaces
which
by
erty
exists
and
sets
P, we shall
of
if Cp\x
|xj.
say
£ U ix )
"X
\
If on X
is
P"
or
If xQ is a limit point of S*„!, then M*„> xQ) ~ ^xk: & = 0,
1, 2, ..-}.
Coconvergence
compact
sets.
Proposition
can be characterized
From
Received
of countably
based
[6] we have
1.
X is coconvergent
that for any compact
which Uit^U):
in terms
set
iff there
K and open
is an
R containing
ONA U on X such
K there
is a k £ N for
x e K^ c Rby the
editors
AMS (MOS) subject
October
classifications
Key words and phrases.
coconvergent,
contraconvergent,
10,
1973
(1970).
and,
in revised
form, January
31, 1974.
Primary 54E35; Secondary
Open neighborhood
assignments,
stratifiable
spaces,
Nagata
(/-linked
spaces.
54D99-
sequences,
Copyright © 1975. American Mathematical
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499
Society
500
r. r. SABELLA
A D -space
is defined
F has
by C. E. Aull
closed
set
U (F)
is an open set containing
a countable
U .{F) C R for some
Definition
1.
k £ N.
local
base
is
in which
\U (F)S, i.e.
for each
n
F, and if F C R, where
Remaining
A space
[l] as a space
with
DQ if each
Aull's
every
n £ N,
R is open,
nomenclature
compact
then
we introduce
set has a countable
local
base.
Without
loss
for a compact
T-., D.-spaces,
countable
of generality
set
as well
[l] and,
A first
hence,
countable
spaces,
countable
are
spaces
is equivalent
Ix^S is U-linked
to the constant
Proposition
2.
X is a D^-space
iff for each
Proof.
K.
of open
\x n \ clusters
Say
sets
with
|i7fe(K)S a local
n, x^ £ UniK)
but
Uk\
does
is an 772£ N such that for all n > m, x
and for some
D0-spaces.
is an ONA
sequence
that
Sx! then
characterization:
compact
K such
Point-
there
of this
containing
base
countable;
K C X there
if x
£ U (k)
is a
for
in K.
X is a DQ-space
If for all
D0-spaces.
are also
is an analogue
local
is first
to one on which
The following
\U^{K)\
a given
DQ-space
to x.
n then
set
second
space
assume
every
converges
sequence
all
Trivially,
as coconvergent
U such that whenever
ixn\
we will always
to be nested.
base
not cluster
4 K. Hence,
for the compact
in K, then
KCX-clix
k > m, Uk{K) C X - cl Sx^: 72> ttzS, which
there
; 72> ttz!,
contradicts
the condi-
tion that xk e UAk).
Conversely,
contained
x £ K, x
assume
in the open
6 Cpjx^S.
the condition
set
to be satisfied.
R, and if x
But
R £jl{x)
If K is compact
£ (J (K) - R for all
and contains
none
n, then
of the
and
for some
x , which
is
a contradiction.
In coconvergent
tion
spaces
1 is equivalent
the characterizing
to the existence
condition
of an ONA
V such
given
that
in Proposifor each
AUn,*0), iUfV„U): x £ Mxk, x0)SS~=1 is a local base for A(x , x ) [6].
As the next proposition
exists
Proposition
each
and following
example
show,
no such
equivalence
for DQ-spaces.
3.
A(x , xQ) has
Proof.
choosing
Let
X be a T ^space.
a countable
The sufficiency
a nested
first
part
countable
local
Then
X is first
countable
The converse
follows
iff
base.
is immediate.
ONA U on X and noting
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that,
by
for any
SPACES IN WHICH COMPACT SETS HAVE COUNTABLE BASES
M*n,
x0)
follows
and each
that
Iß,
k, there
.(A(x
is an
, x0))i?"
nk such
that
. j is a local
xn £ UkixQ)
base
for Mx^,
fot
501
n>
n
It
x ) where
k
S*./(A(V
U £//*,)
X0]) = Uk{x0) U
r =l
The following
which
is an example
of a first
countable,
stratifiable
space
is not a D.-space:
Let X = f(x, y) £ R2: y > OÍ. Define U by
(Ma, b) = |(a, y) £ X: \y- b\< l/n\
¡7n(a, 0) = |(x, y) £ X: y < l/n,
U defines
a first
countable,
x < lS is compact.
Tj
Assume
ii b > 0.
0 < |x - a\ < l/n\
topology
on X.
it has a local
base
U \{a, 0)\.
The set
K = S(x, o): 0 <
JBR(K)S and let
L =
!(x, y): y > OS. For each 72 let Kn = |(x, 0) £ K: L%n ß„(K) = 0 S. It
follows
that
K = U
integer
o
777and an interval
is a ze such
that
.iZa
l¡kic,
.
By Baire's
category
theorem
S(x, O): c _< x _< d\ in which
o) C BmiK)
and
there
are an
Km is dense.
a x 4 c for which
There
(x , 0)
£
tVjc, 0) nKm. Hence, (*, l/2/fe) £ Í7¿(c, O) C Bm(K), i.e. L7 nBmiK)4 0,
contradicting
(x, 0) £ Km.
A space
DQ-spaces
than
is
we require
sets)
The above
provides
have
example
being
compact
normal
the weaker
countable
us with an answer
the countably
X is not a DQ-space.
D^ if it is perfectly
where
closed
Hence
local
stratifiable,
bases,
that
compact
compact
is perfect
and hence
in the negative.
condition
and countably
condition
normality
perfectly
However,
[l].
sets
normal
sufficient?
[2],
we are able
to rim compactness,
i.e.
spaces
each x £ X and U e Tiix) there is a V £ ?I(x) such that
For
(rather
to weaken
in which
for
V C U and bdry V
is compact.
Proposition
compact
set
Proof.
open
4.
is compact
and
rim compact,
it is
D
iff every
is a G...
Let K be compact and l¡nÍK) be a nested
and contains
and open
// X is regular
sets
K, there
V(xj),
...
and contained
is a finite
, V(x^)
with
in R.
Hence,
number
G g of K. If R is
of elements
V(x¿) £ ?í(x¿)
such
B = bdry(U*Lj
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of K, x,,
... , x
that
V{x .)
V(x.))
bdry
is compact.
502
R. R. SABELLA
The sequence
|X - cl Un{K)\
covers
X - U¿_iV(x-)
ar"i,
hence,
B.
It
follows that there is an 72£ N such that B C X - cl U ifd and, therefore, Un{K)
CR.
The converse
Lemma
1.
is true for any
The following
(a)
X is coconvergent.
(b)
X is a D^space,
D.-space
and follows
from Proposition
2.
are equivalent:
each compact
set
K having
a local
base
III {K)\
such that if X, C K2, both compact, then Uk{Ky) C UkiK2) for all k.
(c)
X is
\Uki&{xn,
a space
1
xQ))\
such
in which
that
each
A(x
22 ,
xAU
Ukix) C L/fc(A(xn, x0))
has
a local
base
for each
x £ A(xn,
xQ) awfi
ONA on X.
For each
compact
a// zL
Proof.
Let
U be a nested
K let UniK) = \J\Unix):
x £ XS. By Proposition
for X. It follows immediately
C Un{K2) tot all
Since
A(xn,
xQ) is compact,
paragraph
preceding
noted
in
there
exists
lUiV
(x) : x £ A(xn,
an
ONA
to the existence
that (c) implies
(a).
Lemma
Let
coconvergent
Proof.
Since
Without
loss
sequence
§
that
X
such
that
is a local
follows
base
3,
for
then
^„(X,)
Then
from (b).
the
any
As
condition
A(x
that
, x ),
for A(x , x0) is equivalent
ONA on X.
X is semistratifiable,
It easily
follows
X is developable
of generality
is a development
for all
contradicts
R. W. Heath
D0-space
from this
if it is
an ONA U which
\j
in [5] gives
72 let
X, for if there
are sequences
ix
that if
Ix^S converges
For each
for
72 there
is an ONA V such
to Sx^j, then
we may choose
= \Unix);
is semi-
x £ X!.
is an x £ X and
\ and
to x
|yn!
The
R £
for which
|x!
£ U (x ) - R, it would follow that x £ Cplyn!,
the way in which
which
there
Ix! is V-linked
is (i-linked to |x S and y
able),
(c) trivially
Proposition
X be a T^-space.
and coconvergent.
Jl(x)
which
on
XqMS^j
sequence
stratifiable
such
is a local base
and semistratifiable.
the constant
[4].
V
of a coconvergent
2.
1, \UniK)\
that if Xj C X2, both compact,
72.
each
the
coconvergent
\yA
an example
is not developable.
to a conjecture
by F. B. Jones,
semimetrizable
space
was
chosen.
of a semimetric
The example
viz. a topological
to be a developable
(hence
space
was in answer
property
is that
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semistratifi-
sufficient
it be DQ.
for a
Lemma
2
SPACES IN WHICH COMPACT SETS HAVE COUNTABLE BASES
shows
that
coconvergence
is not known
whether,
is one
in fact,
topological
property
coconvergence
which
is not also
will
503
work.
a necessary
It is
condi-
tion.
Proposition
5.
X is a Moore space
if it is regular,
rim compact
and
semistratifiable.
Proof.
Let
U be a semistratifiable
[>J\U (x); x £ XÎ for any compact
\U, (X)S is a Gg , and using
X.
Now by Lemma
Proposition
ification
6.
The
last
spaces.
But then
We have
is
sufficient
partial
set
also
are applications
7.
X is metrizable
8.
Proof.
for X in X.
of D „-spaces
to stratifiable
space
and coconvergent.
Proposition
of Proposition
3:
not every
7 be weakened,
be DQ to be metrizable?
X is metrizable
X having
a local
if it is stratifiable
base
stratifiable
viz.
We have
7 give
and
D.
is it
only a
with each
\X - (X - X)~ S. (V/e are using
stratification
of the open
If Xj and X2 are compact with
Proposition
implies
in X* and has a local
base
iff it is stratifiable
following
(X - X.)I22 for all 72 and, hence,
Proposition
n XS is a local
of X, is a
[7] we have
an increasing
Proof.
compact-
in
Proposition
denote
and its one point
in X, it is compact
that a stratifiable
compact
for
X is a DQ-space.
Can the conditions
answer
T
base
2, it is developable.
compactification
from the example
D
2 that
4, it is a local
4, X* the one point
!t/„(X)
U (X) =
from Proposition
and by Lemma
compact,
then
two propositions
From
Proposition
space
of Proposition
normal,
If X is compact
\Un{K)\.
the proof
// X is locally
By Proposition
D0-space.
base
X, it follows
7, X is coconvergent,
is perfectly
Proof.
ONA on X. Letting
set
set
\R
! to
R.)
X, C X2, then (X - KA
C
U (X.) C Un {KA,
which byJ Lemma 1 and
¿ *
'221
metrizability.
9.
// X is stratifiable
By Proposition
paracompactness
and rim compact,
5, X is a IVloore space.
which
is sufficient
it is metrizable.
But stratifiability
in Moore spaces
ity.
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for metrizabil-
504
R. R. SABELLA
REFERENCES
1.
C. E. Aull,
Closed
set
countability
axioms,
Ser. A 69 = Indag. Math. 28 (1966), 311-3162. C. J. R. Borges, On stratifiable
Nederl.
Akad.
Wetensch.
Proc.
MR 33 #7973-
spaces,
Pacifie
J. Math. 17 (1966), 1—16-
MR 32 #64093- J. G. Ceder,
(1961), 105-1254.
G. D. Creede,
(1970), 47-54.
5.
6.
of metric
spaces,
Pacific
J. Math.
11
Concerning
semi-stratifiable
spaces,
Pacific
J. Math.
32
MR 40 #8006.
R. W. Heath,
60, Princeton
Some generalizations
MR 24 #A1707.
On certain
Univ. Press,
R. R. Sabella,
first-countable
Princeton,
Convergence
N.J.,
spaces,
Ann.
of Math.
Studies,
no.
1965, pp. 103—113-
properties
of neighboring
sequences,
Proc.
Amer. Math. Soc. 38 (1973), 405-409.
7.-,
Properties
of neighboring
sequences
in stratifiable
spaces,
Proc.
Amer. Math. Soc. (submitted).
DEPARTMENT OF MATHEMATICS, CALIFORNIA STATE UNIVERSITY, NORTHRIDGE,
CALIFORNIA 91324
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