PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 45, Number 3, September
1974
PROPERTIES OF NEARLY-COMPACTSPACES
LARRY L. HERRINGTON
ABSTRACT.
and locally
A general
nearly-compact
product
theorem
spaces
is given
for nearly-compact
along
with
spaces
other
relating
properties.
1. Introduction.
pact
[7] if every
A topological
open cover
of the closures
of which
cover
locally
nearly-compact
closure
is a nearly-compact
this
paper
pact
is to prove
this
result,
spaces
A and
subset
a set
regular-open,
a closed
(open)
if A is closed
each
set
Remark
2.1.
Ta,
containing
Let
the point
operators
collection
\y a\ £ naYa.
on basic
of regular-open
Received
by the editors
AMS(MOS) subject
Keywords
open
sets
and phrases.
set
By noting
(X, T)
if A =
sets
Wa.
that each
of regular-closed
topology
the behavior
\Ua.\"_i
We call
if and only
it follows
in the space
that
sets
regular-open
and
sets.
S = II {Ya: Ya
denoted
by T)
of the closure
there
10, 1973 and, in revised
is
semiregular
(star-open)
Thus
product
it follows
August
space
of regular-open
sets,
classifications
of a set
of T [1, p. 138],
of a collection
the usual
the closure
for a smaller
in (X, T^).
U be a regular-open
of
spaces.
and regular-closed
of a collection
a £ A\ (S has
the prod-
purpose
nearly-com-
of two regular-open
(X, T) star-closed
set is the intersection
has topology
of locally
In a topological
form a base
respectively,
[7] that
A.
the semiregularization
set is the union
shown
to be
whose
for nearly-compact
if A = Int(cl(A))
A of a space
(open),
star-open
interior
spaces.
sets
is said
The primary
will denote
of a set
the intersection
the regular-open
T^ on X, called
been
theorem
cl(A)
the interior
Since
(X, T)
to the product
Throughout,
topology
star-closed
product
pertaining
regular-open
[3, p. 92],
space
the interiors
an open neighborhood
of X. It has
of nearly-compact
A is called
cl(Int(A))
has
to be nearly-com-
subcollection,
is nearly-compact.
a general
a theorem
will denote
2. Product
point
spaces
is obtained.
Int(A)
(X, T) is said
X; a topological
[2] if each
uct of two nearly-compact
Using
space
of X has a finite
exists
in
and
a finite
Ya.)
form, October
and a
23, 197 3.
(1970). Primary 54B10.
Nearly-compact
space,
locally
nearly-compact
space.
Copyright © 1974, American Mathematical Society
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431
432
LARRY L. HERRINGTON
regular-open
set
G = Il
{y°i such that íy°|
T
y
, a e A! and denote
discussion
xU
its topology
that the semiregular
if each
2.2.
cover
4.1 ] that
a space
is compact.
product
Using
theorem
Theorem
this
Proof.
each
2.1.
Assume
It follows
T*.
a finite
subcover
along
Í7].
with Remark
T¿):
the
It was
noted
in [2,
if and only if (X, T*)
2.1, we next give a general
a £ L\\ be any family
IIaYa
Y a is a continuous
if and only
spaces.
if and only if each
that
with the space
(X, T) is nearly-compact
(X, T) is nearly-compact
\(Ya,
from the preceding
associated
Y
of spaces.
is nearly-compact.
is nearly-compact.
open
surjection,
Then
Since
each
[7, Theorem
3.l]
projection
shows
that
Y a is nearly-compact.
naÍYa:
assume
that
Y a has topology
and
a £ AI.
open cover
of the space
ly-compact.
This
Definition
if each
each
T a and
Now let
{U a\ is a star-open
pact
admits
result
Let
Conversely,
T
space
for nearly-compact
II(XY(I z's nearly-compact
P a. IIaYa-*
containing
T..
A topological
regular-open
Theorem
x • • • x (_/
by T„.
topology,
(S, T) is equal to the topology
Remark
x U
e G C U. Now if we let S* = Ila |Ya: Y a has topology
(Ya,
and let
S =
a e A! and 5* = U¿Y a: Y a has topology
S. Using
Remark
cover
in the compact
the proof.
2.1.
is nearly-compact
\U «: U a regular-open
completes
point
Ta)
A topological
in S, ß € ¡\ be a regular-
2.1 and Remark
space
space
S^.
2.2 it follows
Consequently,
(X, T) is called
has an open neighborhood
whose
S is near-
locally
closure
that
nearly-com-
is a nearly-compact
subset of X [2].
[2, Theorem
dorff space,
4.5]
then
to [2, Theorem
shows
that
(X, T%) is locally
4.5] is also
Lemma 2.1.
if (X, T) is a locally
Let
ly nearly-compact
compact.
nearly-compact
We next
show
that
Haus-
the converse
true.
(X, T) be a topological
Hausdorff
space
space.
Then
(X, T) is a local-
if and only if (X, T+) is a locally-compact
Hausdorff space.
Proof.
locally-compact
There
(closure
cl(V)
exists
Only the sufficiency
Hausdorff
a star-open
in (X, Tj)
space
set
proof.
subset
Assume
and let x £ X. Clearly
V in (X, T*)
is a compact
is a nearly-compact
nearly-compact.
requires
subset
containing
in (X, T*).
in (X, T).
Therefore
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that
(X, T+)
is a
(X, T) is Hausdorff.
x such
Since
that
clJ|i(V)=
cl^V)
cl(V),
(X, T) is locally
PROPERTIES
Theorem
Then
2.2.
II Y
Let
\(Y a, Ta):
is locally
nearly-compact
a e AS be a family
nearly-compact
The proof
of Hausdorff
if and only if all the
and at most finitely
Proof.
433
OF NEARLY-COMPACT SPACES
spaces.
Y
are locally
many are not nearly-compact.
is a standard,
well-known
argument
after
noting
Remark
2.1 and Uemma 2.1.
3. General
properties.
pletely
Hausdorff
closed
neighborhoods.
(called
absolutely
nite subfamily
[l, p. 146] if each
closed
ists
space
3.1.
a finite
points
\U
in a compact
sets
3.1.
in I2 along
/
space
'Let
pact
that
2.2]
a similar
need
ex-
). Since
CXj
sets
in X.
family
[4, p. 43].
result
if for
regular-open
are star-closed
is a continuum
subset
, cl(U„
2 a*].
by disjoint
We next
of continua
give
an ex-
not hold in nearly-com-
of the set
of [O, l].
cartesian
Hausdorff
<\fn\.
space,
Then ÍFnl^_j
of / , is disconnected
in / , it is a nearly-compact
K is not a regular-closed
in consideration
Theorem
3.1.
to g be an initial
(a < y implies
though
we give
Let
ordinal.
F aD F
K= I I F
to-
n = 1, 2,
is a descending
K •» f]
family of
F 77 - [O.7 l].
77
It
and not nearly-compact.
K in Example
subset
23b,
semiregular
of / . For each
in l2 suchthat
the set
Even
whose
open
A runs
from [l, Exercise
topology
K, as a subspace
the usual
A x ÍOi, where
It follows
product
subsets
as a subbasis
3.2.
subspace
(a)
a fi-
A C U/,,1there
of a descending
with the complements
connected
we note that
this
//-closed
that
A cU"
I2 =• [O, l] x [O, l] have
Fn = \(x, y) £l2:y
Remark
such
that
subsets
is a nearly-compact
nearly-compact-1
follows
,, such
sx I
//-closed
in general,
pology, T^, is the usual
3,---,let
of X contains
from [7, Corollary
A C X to be an
the intersection
the set of all subsets
p. 147] that
disjoint
spaces.
Example
through
of X has
to be H-closed
if and only if it is an //-closed
X can be separated
that
that
Hausdorff
pact Hausdorff
{U„}"
Cij I
space
known
to show that
covering
It follows
open in X, a eA|
subcollection
[l, p. 138], it follows
ample
a subset
: U
in a Hausdorff
It is well
open
X [6],
to be com-
space.
We define
collection
points
(X, T) is said
if every
cover
(X, T) is said
of distinct
space
X is nearly-compact
Hausdorff
Remark
every
in [l])
closures
space
pair
A Hausdorff
whose
that a Hausdorff
completely
A topological
3.1 is not a nearly-com-
since
it is star-closed.
set in I2 because
Int K = 0.
Also
With
our next theorem.
(X, T) be a nearly-compact
Suppose
that
) of nonempty
Hausdorff
space
and let
\F a: a < co o\ is a descending
star-closed
will be a nonempty nearly-compact
subsets
subset
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in X. Then
of X;
family
434
LARRY L. HERRINGTON
(b) if each
F
is connected
then K will be connected
Proof.
compact
Since
First,
pact
we observe,
Hausdorff
each
space
as we have
subset
in (X, T+).
which
K = ''aFa
is regular-closed,
already
noted,
(X, T) is a nearly-
if and only if (X, T%) is a compact
F a is star-closed
in (X, T^)
in (X, T) and
in (X, T).
in (X, T), it follows
Therefore
implies
that
I I F
that
Hausdorff
each
— K is a nonempty
K is a nonempty
space.
F a is a comcompact
nearly-compact
subset
subset
in
(X, T).
Now assume
regular-closed
fore,
that
each
F a is connected
in (X, T).
[4, 'Lemma
Clearly
2.8 , p. 43] gives
(X, T^.). Now suppose
that
two disjoint
regular-closed
such that
nonempty
and
closed
Fa=
K is
in (X, T*).
There-
and connected
in (X, T).
C,
that
Then
in
there
exist
C? in the subspace
K
in (X, T) and C{ ( i =
C ■ (z = 1, 2) is regular-closed
in (X, T*).
Therefore,
We conclude
that
K is not connected
K is connected
in
the proof.
3.3.
A function
regular-open
sets
are open and a function
regular-open
sets
are open
We say a map
compact
K is regular-closed
is a contradiction.
(X, T). This completes
Remark
sets
in K, it follows
in (X, T) and consequently,
in (X, T^) which
F a is connected
K m f laFa
K is not connected
K = Cj U C2. Since
1, 2) is regular-closed
each
in (X, T) and II
is almost-continuous
if the inverse
is almost-open
if the
images
of
images
of
[8].
/: X -» Y is star-closed
if the images
of star-closed
sets
are closed.
Theorem
3.2.
nearly-compact
closed
subsets
star-closed
in X will
Let
A is an //-closed
is an
f : (X, T„) -* (Y, T) be an almost-continuous
X into a Hausdorff
be H-closed
//-closed
star-closed
A be a star-closed
subset
subset
map.
[5, Theorem
This
subset
connected
in
of star-
f will
be a
it was noted
sets
Our next
in (y,
/: (X, TA -» (Y, T) is almost-con-
T) and consequently
completes
closed.
Therefore,
f : X ■* Y is an almost-continuous
X onto a space
in [5] that
it follows
that
/(//)
/ is a
the proof.
that when
space
X is nearly -compact,
T^.) is continuous,
Y, then
almost-continuous
y will
maps
sur-
be connected.
do not preserve
in general.
theorem
gives
conditions
when connected
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under
map of a
the images
Y. Moreover,
in X. Since
in X. Now since
4] shows
of a connected
However,
Y. Then
subsets
if and only if /: (X, Tn) -» (y,
jection
space
map.
Proof.
tinuous
Let
space
almost-continuous
maps.
sets
are preserved
PROPERTIES OF NEARLY-COMPACT SPACES
Theorem
3.3.
Let
a nearly-compact
each
f: X -» Y be an almost-continuous
space
regular-closed
X into a Hausdorff
connected
set
space
in X will
435
almost-open
Y. Then
map of
the image
be a regular-closed
of
connected
set in Y.
Proof.
that
Let
H be a connected
/(/Y) = F is re guiar-closed
closed.
Suppose
cl (Int (F)).
Now
Since
exists
Then
F there
there
W containing
= 0.Since
3.2 assures
exists
exists
is a regular-open
an open set
/(W)ncl(Int(F))
F.
y £/(//)=
set in X. We first
in Y. Theorem
cl(Int(F))C
Y - cl(Int(F))
regular-closed
x such
us that
a y € F such
an x e H such
set containing
that
show
that
f(x);
F is
that
y í
f(x) = y.
therefore,
there
f(W) C y — cl (Int (F)).
Hence,
x e H = cl (Int (//)), Wn Int (H) 4 0- Therefore,
0 4 f[W C) Int(H)] C fiW) O /(lnt(//))
Cf(W) O Int /(//) C/(W) n cl(lnt(F)).
Thus
/(W) Pi cl (Int (F)) 4 0 which
cl (Int (/(//)))
connected
sets
/(//)
is a regular-closed
in Y. Then
Cj
is a contradiction.
and
there
C 2 in the
is regular-closed
set
exist
Therefore
set
3.4.
space
Let
that
/(//)=
C. U Cy
disjoint
closed
H = [(/" ^CjWn
subset
space
Since
sets
(f~
(C.))
//] U [(/"He.,)) n //].
Hence,
we conclude
in Y.
f: (X, T) -* (Y, TQ) be a continuous
X into a regular
is not
regular-closed
is a contradiction.
connected
/(//)
/(//) =
Y. Then
f(X)
map of a nearly-
will
be a compact
sub-
in Y.
Proof.
if f:(X,
gives
set
which
that
disjoint,
that
C. (i = 1, 2) is regular-closed
of / gives
is a regular-closed
Theorem
compact
such
that
in X suchthat
// is disconnected,
/(//)
/(//)
Y, it follows
in Y. Now the almost-continuity
that
Now assume
two nonempty,
subspace
in
n H and (f~l(C2))r\H
in y.
We conclude
Since
Y is regular,
/: (X, T) -» (Y, TQ) is continuous
T¿)-+(Y, T A is continuous
(X, T+) compact.
in
Y. This
Consequently,
completes
a locally
3.5.
Let
nearly-compact
Hausdorff
space
inverses,
then
point
Hausdorff subspace
of Y.
We first
that
f(X)
of the notion of locally
f(X, T) -* (Y, TA
has nearly-compact
Proof.
it follows
is a compact
sub-
the proof.
Our final theorem is an application
Theorem
if and only
[6, Theorem 2.15]. Now Remark 2.2
observe
that
be a continuous
star-closed
X into a regular
f(X)
will
nearly-compactness.
space
be a locally
map of
Y. If f
compact
/: (X, T) -» (Y, T Q) is a star-closed
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or copyright
restrictions
may apply
to redistribution;
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tinuous
map
if and
only
if /: (X,
T+) -» (Y, TQ) is a closed
continuous
conmap.
LARRY L. HERRINGTON
436
Also
we note
if /-1(y)
f~
(y) is a nearly-compact
is a compact
hypothesis
Lemma
that
subset
2.1 we have
that
into a space
Y. Hence,
following
Remark
(resp.
3.4.
ean)
the result
(resp.
on R having
follows
it follows
ly-compact
The identity
map
to be com-
the
the set
a locally
set
space
R be the reals,
and the set
of R. It follows
1 satisfies
2, p. lOO] and
3.5 is given
For let
Give
Let
(X, T^)
not imply the corresponding
the open intervals
(and consequently
/: (X, T*) -»
space
from [l, Corollary
locally-compact).
is not locally-compact.
Therefore
Hausdorff
(Y, TQ) in Theorem
as a subbasis.
topology
compact.
from the
Now by
9, p. 105].
it does
of (R, T) and topologize
subspace
which
is locally
[l, Proposition
If the space
1/3 < x < 2/31
topology
Consequently,
of a locally-compact
locally-compact),
(X, T) is compact
and
(X, T*)
mapping
the corollary
topology
in (X, T) if and only
that /: (X, T#) ■*(Y, Tn) is a proper map [l, p. 10l].
(Y, T ) is a proper
pact
in (X, T^).
subset
A =\x
and
€ R: x is rational
X = [(J, l]
the subspace
Y = [O, l] with the usual
that
the space
X = [0, l]
nearly-compact)
Hausdorff
1 : X -» Y be the identity
the hypothesis
of Theorem
T the
(Euclidis a near-
space
map of X onto
Y.
3.5, but the space
X
is not locally-compact.
The author
suggestions
is grateful
in preparing
to Professor
Paul
E. Long and the referee
for their
this paper.
REFERENCES
1. N. Bourbaki,
Elements
Paris; Addlson-wesley,
2. D. Carnahan,
of mathematics.
Reading, Mass., 1966.
On locally
nearly-compact
General
topology.
Part
I, Hermann,
MR 34 #5044a.
spaces,
Boll.
Un. Mat. Ital.
(4)
6(1972), 146-153.
3. J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 33 #1824.
4. J. G. Hocking and G. S. Young, Topology, Addison-Wesley,
Reading, Mass.,
1961. MR 23 #A2857.
5. Paul
E. Long
and Donald
A. Carnahan,
Comparing
almost
continuous
functions,
Proc Amer. Math. Soc. 38 (1973), 413-418.
6. J. Porter
and J. Thomas,
On H-closed
Amer. Math Soc. 138(1969), 159-170.
7. M. K. Singal
and Asha Mathur, On nearly-compact
(4) 2(1969), 702-710.
8.
and minimal
Hausdorff
spaces,
Trans.
MR 38 #6544.
spaces,
Boll.
Un. Mat. Ital.
MR 41 # 2628.
M. K. Singal and Asna Rani Singal,
Math. J. 16 (1968), 63-73.
Almost-continuous
mappings,
Yokohoma
MR 41 # 6182.
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ARKANSAS, FAYETTEVILLE,
ARKANSAS 72701
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