I nternat. J. Math. & Math. Sci.
Vol. 2 #3 (1979) 481-486
481
ON HAUSDORFF COMPACTIFICATIONS OF NON-LOCALLY COMPACT SPACES
JAMES HATZENBUHLER and DON A. MATTSON
Department of Mathematics
Moorhead State University
Moorhead, Minnesota 56560
(Received December 19, 1978 and in Revised form February 2, 1979)
ABSTRACT.
Let
X
be a completely regular, Hausdorff space and let
be the set of points in
R
is compact.
If
so does the quotient
one for
X-R.
X
X
R
which do not possess compact neighborhoods.
Assume
has a compactification with a countable remainder, then
X/R,
and a countable compactificatlon of
A characterization of when
a countable remainder is obtained.
X/R implies
X/R has a compactification with
Examples show that the above implications
cannot be reversed.
KEY WORDS AND PHRASES. Countable remainders, compactificions non-locally
compact spaces, components of 8X X.
1980 Mathematics Subject
Classification Cod: 54D5.
482
i
J. HATZENBUHLER and D. A. MATTSON
INTRODUCTION.
Let
X
be a completely regular, Hausdorff topological space.
of characterizing when
uX- X
X
has a Hausdorff compactification
X,
The question
where
infinite, has been answered for the locally compact case
is countably
by Magill [2] and for the case when
the Stone-Cech compactification of
8X by Okuyama [4] (where
uX
In case
X).
X
8X
is
is an arbitrary completely
regular space, no such characterization has been given.
The purpose of this
paper is to contribute results toward such a characterization.
Let
hoods.
R
be the set of points in
X
which do not possess compact neighbor-
uX of X, R
Then for all compactifications
Herein we observe that for compact
R,
a countable compactification is that
Cl0ux(OUX
X) X. (See [5].)
a necessary condition for
X/R have one.
X
to have
The main theorem of this
paper characterizes when X/R has a countable compactification.
2.
CRARACTER!ZAT!ON
OF
e,(X/R_).
Throughout this paper all compactifications are Hausdorff compactifications.
Let
N
denote the natural numbers.
a completely regular space
X
tifying
R
to a single point.
X
and if
then a countable compactification of
R
If
is a compact, non-empty subset of
has a coutable compactification
yX,
X/R can be obtained from yX by iden-
It is readily verified that the resulting space
is Hausdorf f.
If
e(X/R)
a countable compactification of
THEOREM i.
X/R, then (X/R)
is a countable compactification of
If
X
X- R.
is also
Thus, we have the following:
is completely regular and
the following conditions implies the next:
(A)
X
(B)
X/R has a countable compactification;
has a countable compactification;
R
is compact, then each of
HAUSDORFF COMPACTIFICATIONS OF SPACES
(C)
483
X- R has a countable compactiflcation.
Examples will be provided to show that none of these implications can be
reversed.
If
Let
(A) no longer implies (C) as in Theorem I.
is non-compact, then
R
be the unit disc in the standard plane with a countable dense subset
X
removed from the boundary.
X
clearly,
R.
The remaining boundary points constitute
X- R,
has a countable compactification but
Then,
the open disc, has
no countable compactification.
(SX- X) R.
Y
Let
THEOREM 2.
and non-empty.
Then the following are equivalent:
(A)
X/R has a countable compactification.
’B)
R
PROOF.
is a
G6-set
Y
in
(A) implies (B).
{Pnln
Take
X/R,
X
into
y(X/R).
y(X/R).
t
has an extension
t
carries
y(X/R),
t
to
t
(cf. Lemma 6.11 [i]).
Y
N
NR X.
N
Since
R_ N, t0[N]
Thus, there is a neighborhood
N
Y
t(z)
t[SX
is any neighborhood of
S
X]
y
G
in
in
8X,
R
of
R
in
X/R.
y(X/R)
X/R.
X- R
8X
choose
z e N
t
Y
that
onto
Since the
is dense in
R
such that
t(x)
t(x)
is compact
N
y
r
R.
in
G X/R.
t0[N
(X
8X
t[R]
r
where
then since
for which
and it follows from the continuity of
y(X/R)
8X
is a neighborhood of
(X/R)
y(X/R)
where
which maps
onto
y e-SX- X,
and
there exists a compact neighborhood
Set
t
Y.
be the canonical mapping
0
X/R] {r},
[y(X/R)
onto
x e R
If
t
is a homeomorphism and
X- R
carries
8X- X
X/R,
y(X/R)
and let
o
We first show that
restriction of
and in
Then
e N}
are components of
R
and components of
is a countable compactificatlon of
of
R compact
be a completely regular Hausdorff space with
Let X
N).
X/R.
If
Then
t(y).
Hence
J. HATZENBUHLER and D. A. MATTSON
484
Kn
Since each
Let
m
Thus
C I,
C
choose
{r}
N}
{Pn
of
N}.
Y
x
C
E
f
are components of
Y.
(B) implies (A).
are clopen in Y
{Vnln
n e
E
N}
N,
t[R] # t(x).
Since
Kn
x
Thus
R}
Wn(x,r)
Wn(X)Ix
(
Wn(xs)lJ
.,p(x)}
Kn}
n
Let
Y
C1
R.
covers
R
Take
{Vnln
N}
is an open cover of
N}
R
which
Let
For each
R.
Y,
E
is compact.
@. Let
(x,r)
e K
n
x R.
there exists a
Wn(x,r).
Now
so that a finite subfamily
Wn(X)={Wn(x,r i) li
R Wn(X), and
Y
,
E
N,
Un
let
$
x
W (x)
n
l,...,p(x)}.
Since
there is a finite subcover
UI,
l,...,n- i}.
and for
{Wn(xj)lJ
l,...,q(n)}.
n > i,
Cn
Then each
Cn
take
R
to a point.
[Y
Then each
{Uili
is a clopen subset of
Let c- be the equivalence relation in
a point and
{Unln
Note that Y
Y, where x
in
must be
I
,...,q(,)).
For each
{Cili
of r
is a clopen subset of
{Y
R.
t C
f
But
so that components of
Cl,
Y satisfying
is an open covering of
l,
W (x)
n
C
are in distinct quasi-components of
clopen neighborhood
{Wn(x,r) r
(Wn(x,ri) li
N}
E
where
This contradicts the fact
We assume that each Kn
Y- Vn.
r
and
Thus,
{Unln
be open subsets of
set
(See [3]).
First we show that there exist sets
such that
Y,
be a component of
i
into the real numbers.
is countable
and
Y.
in
C
N).
Now there exists a continuous InJectlon
C.
I
Y
are open in
K
n
=U {Knln
BX- X
Evidently,
G-set
is a
connected and not a singleton, since
that the image of
E ,N.
n
R and let
be a component of
C
If
C_ C I.
for each
is compact, the sets
Knln
R-{Y
f
t-l(pn),
Kn
Next, let
Y
l,...,n}]
and
8X which identifies each
The projection of
SX onto
8X/e
Un
BX- X
Cn
to
is denoted by
H.
HAUSDORFF COMPACTIFICATIONS OF SPACES
For each
N,
n e
Y into disjoint open sets.
is a partition of
separated by open sets
U
8X/v.
any other point of
8X- Y,
It remains
manner.
Let
mapping
J
8X- Y
X/R.
X/R into
of
J
can be
H[V]
and
are
can be separated from
X-
neigh-
8X/-- in the desired
i
X
and let
8X
in
p
be the
is relation preserving, a continuous
8X/r, is induced such that
J
H
p
is also a closed mapping, hence an embedding of
8X/,%2 as desired.
Cn
have compact
can be embedded in
Since
Y
8X/,%J is a compact Hausdorff space.
it follows that
onto
H[U]
H[Cn]
(Cn,Y- Cn)
Now
and
Cn
Thus,
This shows that
X/R
8X/,#.
in
Evidently,
be the natural embedding of
X
follows that
8X.
in
Since points of
to show that
i
projection of
V
and
8X/,.
disjoint open subsets of
borhoods in
H[C n]
consider the point
485
It
i.
X/R into
This completes the proof.
In [2] Magill shows that a locally compact space X has a countable
compactification if and only if
8X- X
As
has infinitely many components.
an application of the proof of Theorem 2, the following is proven.
COROLLARY 3.
Let
X
be completely regular with
a countable compactification, then
PROOF.
8X
X
onto
Let
8X- X
a(X/R)
X/R.
compact.
X
If
has
has infinitely many components.
be a continuous mapping of
t
R
Since the subspace
a(X/R)
8X
onto
K
(u(X/R)
X/R)
which carries
{t(R)}
is compact and countable, it contains an open countable discrete subspace.
Since
a(X/R)
X/R contains infinitely many components of K, Y must contain
infinitely many components.
The converse of Corollary 3 is false when
Example (A) shows that
Example
to insure that
is not locally compact.
X/R can have a countable compactification, so that
8X- X has infinitely many components, but
tion.
X
X
has no countable compactifica-
(A) also shows that condition (B) of Theorem i is not sufficient
X
has a countable compactification when
R
is compact.
J. HATZENBUHLER and D. A. MATTSON
486
EXAMPLE (A).
it is clear that
X
X.
n e N, L
and, for
I
I x
n
{n}.
For
Furthermore,
R
L0
and
The existence of a continuous surJection from
8X- X
fixed and which carries
S- X
onto
The following example shows that for
R
n Ln’
S
X- R
a singleton, where
S
8X
is a compactification
onto
S
which leaves
guarantees that condition (B)
R non-empty and compact the
cation of (C) by (B) of Theorem i cannot be reversed.
with
X
X/R has a countable compactification.
Hence
of Theorem 2 is satisfied.
X,
be the unit
is not rim compact, xd hence does not have a countable
X
compactlfication (cf. [6]).
of
2,
R
be the closed unit square in
S
I x {0},
L0
interval,
Let
imli-
It suffices to exhibit
has a countable compactification but
X
does not.
EXAMPLE (B).
X
R
[{(x,y)
l-i
(i,0)}.
In the plane R 2
< x < i;
Since
X
i < y <
take
i}{(i,01}1
(,
0),
e N}.
Then
is not rim compact, it has no countable compactlflcation.
However, a countable compactlfication for
points
-n
{(n-- 01In
for each
n e
N,
X- R
is obtained by adjoining the
and taking the one-polnt compactlfication
of the resulting space.
REFERENCES
i.
Gillman, L. and Jerison, M. Rings of continous functions, The University
Series in Higher Math., Princeton, N.J., 1960.
2.
Magill, K. D., Jr. Countable compactifications, Caned. J. Math. 1__8 (1966),
616-620.
3.
Mrowka, S. Continuous functions on countable subspaces, Port. Math.
29 (1970), 177-180L
4.
Okuyama, A. A characterization of a space with countable infinity,
A.M.S. 28 (1971), 595-597.
5.
Rayburn, M. On Hausdorff compactiflcatlons, Pac. J. of Math. 4__4 (1973),
707-714.
6.
Zippin, L. On semicompact spaces, Amer. J. Math.
Pro___c.
5__7 (1935), 327-341.