Internat. J. Math. & Math. Sci.
VOL. 13 NO.
(1990)55-60
55
ON WEAKER FORMS OF COMPACTNESS LINDELFNESS
AND COUNTABLE COMPACTNESS
D. BABOOLAL, J. BACKHOUSE and R.G. ORI
Department of Mathematics
University of Durban-Westville
Private Bag X54001
Durba n 4000
South Africa
(Received July 6, 1987 and in revised form August 9, 1988)
ABSTRACT.
A theory of e-countable compactness and e-Lindelfness which are weaker
than
concepts
the
developed.
of
countable
compactness
and Lindelofness respectively
is
Amongst other results we show that an e-countably compact space is
pseudocompact, and an example of a space which is pseudocompact but not e-countably
compact with respect to any dense set
is
We also show that every e-
presented.
Lindelof metric space is separable.
KEY WORDS AND PHRASES.
e-countably
e-compact,
compact, e-Lindelof, pseudocompact,
separable.
1980 AMS SUBJECT CLASSIFICATION CODES. 54A05, 54D30, 54G20.
1.
INTRODUCTION.
Using the terminology and notation in
is
compact
iff
its
[3], it
is known that a topological space X
enlargement *X contains only near-standard points, and that a
subset A of a regular Hausdorff space is relatively compact Iff *A contains only near
standard points.
Hechler [I] wanted to know what this condition implied topologically
He was led to the notion of what he called ’e-
in not necessarily regular spaces.
compactness’
which is weaker than the notion of compactness.
It is the purpose of
this paper to develop a theory on the analogous concepts of e-countable compactness
and e-Lindelofness in
particular
we
extend
the
the
spirit
of
well-known
Hechler’s
result
study
that
a
of
e-compactness
countably
pseudocompact to an e-countably compact space is pseudocompact.
compact
[I].
space
In
is
We also show that
the Lindelof condition in the theorem that every Lindelof metric space is separable
can be weakened to e-Lindelof.
56
2.
D. BABOOLAL, J. BACKHOUSE, AND R.G. ORI
PRELIMINARIES.
We begin by recalling Hechler’s definition of e-compactness and e-regularity.
Let D be a dense subspace of a topological space X.
DEFINITION 2.1.
([|I):
(a) X is said to be e-compact with respect to 0 if each
(b) X is e-regular
open cover of X contains a finite subcol[ect[on that covers D.
F there eist
with respect to a dense subset D if for each closed FoX and each p
disjoint open sets U and V such that p
U and F 0Dc V.
By analogy we introduce the
following:
X is e-countably compact with respect to D if every countable
DEFINITION 2.2.
open cover of X has a finite subcollectlon coerl, D.
X is e-Lindelf with respect to D if every open cover of X ha n
DEFINITION 2.3.
countable subcollectlon covering D.
Recall Hechler’s extension of the topology T for X:
be a family of subsets of X, and let T(E)=
Let
}.
flnie union of members of
T().
By K() we shall mean K with
F Is a subset of a
th, extended
topology
Hechler used the construction of the extended topology T() to provide examples
of spaces which are e-compact but not compact (see
3.
{U-FIU,T and
Ill
p. 223.
RESULTS.
The
following two
theorems are analogous to the corresponding theorems
on e-
compactness (see [I]).
Then X() is e-Lindelof iff there exists a
Let X be Lndelof.
dense set D (with respect to X(E)) such that for every E g E, E O D is countable.
PROOF. Suppose X(E) is e-Llndelof with respect to a dense set D. Assume for some
E} is an open cover of X having no
E g E, E D is uncountable. Then {X-(E-{x})
x
THEOREM 3.1.
.
countable ubcover of D, contrary to E() being e-Lindelof
for all E
g
Conversely suppose END is countable for all E
X().
Suppose {U
and D is dense with respect to
A} is an open cover of X.
F
Then
and as X is Lindelof, there is a countable subcollectlon [U
Now
But
c
U
is
showing that X(E) is
THEOREM 3.2.
,m E
.
A} covers X,
i
{xj
U, where
F
U
say, coverlng
i--I
of O.
will cover all except at most countably many points
[Ui F!
{x.j }J-I
Thus E O D is countable
is a countable subcollectlon of
a
countable
subcollection
{Ua -Fa:
of {U
F
a
a
A}
Thus
A] covering
e-Lindelf.
Let X be countably compact.
Then X() is e-countably compact
there exists a dense subset D (with respect to X()) such that for every set E
E O D is finite.
PROOF.
some E
g
,
Suppose X()
is
e-countably
compact with respect
Choose an lnfinlte sequence {x
E N D is infinite.
n e IN}. Then iX
ix
finite subcover of D, contrary
let F
D,
(F-x}): x
to
F
to D.
g
,
Assume
for
n IN#in EO D and
n
is a countable open cover of X with no
X being e-countably compact.
ON WEAKER FORMS OF COMPACTNESS LINDELOFNESS AND COUNTABLE COMPACTNESS
Conversely suppose E 0 D is finite for all E
to
X(E).
Suppose {U i
Fi}i=
E,
E
and D a dense set with respect
is a countable open cover of X(E).
X, and as X is countably compact there is a finite subcover
Now {U
But
k
i
{Ukl}ri=l
covers
{Ui}i.
Then
of X.
{xj }=
Fki}r
covers all except at most finitely many points of D,
i=l
c U where
is a finite subcollection of
Thus
t
{xj}j
is a finite subcollectlon of {U
{Uki- Fkt}ri=I U
{Ui Fi}t=I*
Fi} i= covering
i
57
say.
D, showing X(E)
is e-countably compact.
If X is e-countably compact, then X is pseudocompact.
THEOREM 3.3.
PROOF.
Suppose X is e-countably compact with respect to D. Lec f be a continuous
real valued function of X.
X
{x
Then if G
< n}, {G n n IN} would be an
n
open cover of X having a finite subcover of D.
Clearly then D c G for some n IN.
If(x)
n
By continuity of f, we have f(X)
[-n,n], showing f
c
It is well known
is bounded.
We have the
that a countably compact first countable Hausdorff space is regular.
ollowlng:
THEOREM 3.4.
Every e-countably compact first countable Hausdorff space
is e-
regular (with respect to a dense set D).
PROOF.
p.
Let p
.
X
Since
Hausdorff,
is
GqO Bq
that
F, F closed in X.
E
F there
F’ which,
being
each q
for
{B_}qq
Let
Let B be a countable open neighbourhood base
exists
a
open G and B
q
q
subfamily
E
B
such
be
of
must
B
B’} covers K.
q
is a countable family of open sets covering
Since X is e-countably compact there exists a finite family
Dc(X-F)U
U{HB:
B 8"}- Thus
open and contains p.
We
relates
separability
and Lindelofness introduced above.
is separable.
B B’g
In fact the
Lindel’f
.
V. Let
It is easily verified thatUNV
how
show
now
FNDcU{:
{HBI
F so that {X-F}U
to
"cB’
such that
N{B:B 8"} which is
of
compactness
Lindel’f
metric space
the generalizations
It is well known chat every
condilon can be weakened to e-Lindelofness as the
following result shows.
THEOREM 3.5.
If X is meCrlc and
e-Llndel’f
with respect co a dense set D then X
is separable.
PROOF.
X,
Let F
For each n
there
exists
U {F
n
n
a
IN }.
E
IN let U
n
countable
={S(x, I/n)
set
F
n
Then F is countable.
c
Ix
E
X}.
X such
Since U
n
is an open cover of
that D cu {S(x, I/n): x
F }.
n
D. BABOOLAL, J. BACKHOUSE, AND R.G. ORI
58
F is dense in X:
CLAIM.
.
Now
<
DcU {S(x,I/2N):
d(z,x) < I/2N.
I/N
S(y,
1/2N)O D
Thus d(y,x)
,
Hence S(y,)fl F
We then have:
<
F2N
< 1/2N
d(y,z) + d(z,x)
Find N such thac
O.
D such that d(y,z)
# so there exists z
#
implies that there exists
F2N
>
Let y c X be arbitrary, e
+
Now
F such that
c
I/2N
< I/2N.
I/N
.
<
showing F dense in X.
THEOREM 3.6. For metric spaces the following are equivalent
(a) X is separable
(b) X is 2nd countable
(c) X is Lindelof
(d) X is e-Lindelof (with respect to any dense set)
Since
eve ry
is
space
compact
e-countably
pseudocompact T space is countably compact we also have
4
THEOREM 3.7.
For T
4
pseudocomp act
and
eve ry
spaces, the following are equivalent
(a) X is countably compact
(b) X is e-countably compact (with respect to any dense set)
(c) X is pseudocompact
4.
EXAMPLES.
This is an example of a space which is pseudocompact but not e-countably compact
with respect to any dense set.
Consider Z
+,the
positive integers wlch the relatively prime topology, i.e. with
{U (b) a,b
a
basis B
covers Z
+
+
of Z
that there exists a countable open cover
subcollectlon of
{b + na
,(a,b) I} where U a (b)
Z
+
Z}.
n
such that the closures of no finite
This would then imply that Z
Recall that Z
compact with respect to any dense set
We shall show
+
+
is not e-countably
is pseudocompact
([2], p.83).
then {U3(1) U5(2) U5(3) U5(4) U7(5), U?(6), UII(7) UII(8),...},
Let
{Upi(al)}in.l
and let
U
Case
’.
3(I)
be any finite subcollectfon
Then let
u m (m+,)0u (a i)
so chac
U
’ ’
Case 2:
U3(2) 0U3(1)
#
Pi
3
V i, 2
Z+.
U
@, so that
I
If
PlP2"’’Pn
and note that
i
Pl
3(I)
m
of.
U3(1)
is the only element in
U’ Z+;
#
otherwise let m
’,
then
p2P3...pn
where
pl-3,
and
m.
The n
U (re+l)OU
m
and
U3(m+l) O U3(I)
(in
(ai)
the
@V, 2 < I
case
where
nelghbourhood of m+l meeting no member of
m’
po m
’.
n
m+1) so Chat
If m+l, let
U (m+l)O
m
U3(m+l)
is
a
ON WEAKER FORMS OF COMPACTNESS LINDELOFNESS AND COUNTABLE COMPACTNESS
where
Hence
PolS
a prlme 3
PoUm,(m’+l) O U3(m’+l)
and
Po
3.
Then
3p om
and
3pom +
59
I.
is the required nelghbourhood of m’+l meeting no member of
’.
Llndelf countably compact space
e-Llndel’df, e-countably compact space is
This example is motivated by the result that a
The analogous statement that an
compact.
is
e-
compact is not in general true as the following example shows:
Recall the Novak space (see [2] p. 134):
Lec Z
+
denote the positive integers with the
+
+
Let F be the
of Z
Stone Cech compactiflcatlon
family of all countably infinite subsets of S, well-ordered by the least ordinal r of
c
F} be a collection of subsets of S such that
card (S).
cardinal 2
Let
C
is the unique
where
0
card (PA) <
whenever D < A, and
+ + +
Z which permutes each odd integer
extension to S of the continuous function f: Z
n+l
Then we define
n + (-I)
with its even successor, i.e. f(n)
topology and
discrete
S
the
{PAIA
2c’ PD PA
P U
{PAIA
(PA PA
F}, and then define Novak’s space by
XffiPU Z
+.
+
Also X
X ([2] p. 135), hence X is e-Lindel’f with respect to Z
is countably compact ([2] p. 135), but as X is not compact, X cannot be absolutely
Thus X is not e-compact
closed (as a regular absolutely closed space is compact).
Note that
Cx(Z+)
with respect to any dense set.
REFERENCES
I.
2.
3.
HECHLER, S.H., On a Notion of Weak Compactness in Non-.Regular Spaces, Sc_udies in
Academic Press, Inc.
Topology, (Edited by N.M. Stavrakas, K.R. Allen).
(1975), 215-237.
Counterexamples in Topo!ogy (2nd Ed.
STEEN, L.A. and SEEBACH, J.A., Jr.
Sprtnger-Verlag, New York, Heidelberg, Berlin, (1978).
MACHOVER, M. and RIRSCHFIELD, Lectures on Non-Standard Analysls,Lecture Notes in
Mathematlcs,94 Sprlnger-Verlag, Berlin- New York, 1969.