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Carnegie Mellon University
Research Showcase @ CMU
Department of Mathematical Sciences
Mellon College of Science
1968
Paracompact subspaces
Richard A. Alo
Carnegie Mellon University
Harvey L. Shapiro
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PARACOMPACT SUBSPACES
Richard A. Alo
and
Harvey L. Shapiro
Report 68-9
March, 1968
PARACOMPACT SUBSPACES
by
Richard A. Alo and Harvey L. Shapiro
1.
Introduction.
The definition of paracompactness for a topological space
(X,3) states that every open cover of the space have a locally
finite open refinement.
space
S
When applying the definition to a sub-
of (X,3), the phrases
f
open cover1 and ! open refinement1,
of course, refer to the relative topology on
also is in reference to open sets in
S.
S.
Locally finite
Immediate questions
arise when these phrases are used in reference to the topology
for the whole space
S
X.
That is, one can consider a subspace
of (X,3) to be STRONGLY PARACOMPACT if every cover of
by members of
members of
?
3.
has a locally finite in
X
S
refinement by
(In [1], such subspaces were called a-paracompact.)
It is clear that every strongly paracompact subspace of a
topological space is paracompact.
not equivalent.
However, the two notions are
This follows from the example of a completely
regular, non-normal space due to Niemytzki.
subset of the plane
y >_ 0.
(x,0) of
Let
R.
S
R x R
Let
X
be that
consisting of the points (x,y) for
be that subset of
X
consisting of the points
The relative product topology
3
on
to include also as neighborhoods of the points in
X
S
is enlarged
the
e-spheres tangent to the points (x,0) together with ((x,0)} for
all
G > 0.
That is, the sets
K(x,€) = { (x,0) U { (u,v)eS : (u - x ) 2 + (v - e ) 2 < e2}}
are also included for all
e > 0.
With this new topology
S
is
2
a closed discrete subspace of
However, D
X
and hence is paracompact.
is not strongly paracompact.
is an open in
X
cover of
S
For the cover (K(x,]/2))
that has no refinement and' is not
locally finite in
X.
point (x,0) in
will meet an infinite number of the sets
K(x,l/2).
S
For, any
3
neighborhood
N
of a
As our main theorem will show this subspace
not strongly paracompact since it is not
S
P-embedded in
is
X.
In particular, it will be shown that for normal Hausdorff
spaces a subspace is strongly paracompact if and only if it is
paracompact and
topological space
P-embedded.
X
A subset
S
is
P-EMBEDDED in a
if every continuous pseudometric on
extends to a continuous pseudometric on
X.
S
This concept was
studied in [3] and [5]. Its relationship to paracompactness was
studied in [4] and [6]. The notion of
the notion of
P-embedding generalizes
C-rembedding where extensions of continuous real
valued functions are required.
The importance of
P-embedding can be readily realized from
the fact that it plays the same role in collectionwise normal
spaces as
C-embedding plays in normal spaces.
In particular
a space is collectionwise normal if and only if every closed
subset is
P-embedded (see [3]). Hence the notion of strongly
paracompact, from the results in this paper, will have applications in such spaces.
The notion of strongly paracompact can also be generalized
by making use of infinite cardinal numbers
7.
have already been applied to paracompactness and
Such techniques
P-embedding.
Hence we will relate this new notion of 'strongly
y-paracompact'
y
to the notions of
7-paracompact
and
P -embedded.
Throughout
this paper emphasis will be placed on the notions in the definitions of paracompactness that deal with the topologies of the
subspace and of the space.
Hence we will refer to, for example,
the local finiteness of a cover as being either locally finite
in
S (the subspace) or locally finite in
X.
will mean a topological space with topology
2.
Definitions and Basic Results.
topological space (X,2) and let
number.
The relative topology on
The subset
2
S
is
refinement by metnbers of
y
3g.
if every cover by members of
a locally finite in
special case of
X
y = /\
then
and such that
pseudometric topology
if every
be a subset of the
be an infinite cardinal
S
will be denoted by
has a locally finite in
3
of cardinality at most
G
3^.
2.
continuous extension to
S
S
y
has
In the
7-paracompact is the usual notion
G
of
S
is dense in
The subset
S
d
on
S
is
7-
whose cardinality is
S
is
relative to the
P -EMBEDDED in
y-separable continuous pseudometric on
The subspace
3 .
The subset is STRONGLY y-PARACOMPACT
A pseudometric
SEPARABLE if there is a subset
X
«T.
refinement by members of
of countably paracompact.
y
S
X
7-PARACOMPACT if every cover by members of
of cardinality at most
at most
y
Let
Also (X,?) or
S
has a
X.
is PARACOMPACT (respectively STRONGLY
PARACOMPACT) if every cover of
?) has a locally finite in
S
by members of
S (respectively in
? o (respectively
X) refinement
4
by members of
2
(respectively
3").
every continuous pseudometric on
pseudometric on
X.
S
It is
P-EMBEDDED if
extends to a continuous
This is equivalent to saying that
S
is
paracompact (respectively strongly paracompact, respectively
P-embedded) if and only if
S
is
y-paracompact
(respectively
y
strongly
y-paracompact, respectively
cardinal numbers
S
y.
P -embedded) for all
By an extension
is meant a cover of
X
U*
of a cover
such that the trace of
U
U*
on
on
S
is
The definitions of other terms used in this paper can be
found in [3]. The following results will be needed in the main
part of this paper.
Theorem 2.1 (see [3]). If
y
S
is a subspace of
X
and if
is an infinite cardinal number then the following statements
are equivalent:
S
(2)
Every locally finite in
members of
of
P y -embedded in
(1)
3
is
3^
of power at most
X.
S
y
normal cover of
has a refinement
that can be extended to a locally finite in
cover of
X
(3)
by members of
members of
3
normal cover of
^
X
by
by members
normal
3^.
Every locally finite in
by members of
S
S
of power at most
normal open cover of
y
has a refinement by
that can be extended to a locally finite in
X
by members of
S
X
3.
Theorem 2,2 (see for example [1]).
If
X
is a topological
space then the following statements are equivalent:
5
(1)
X
is normal.
(2)
Every locally finite open cover of
Lemma 2.3 (Tukey, see [7]). Let
by members of
and let
^g
of a subset
lr = (Va) n
S
3.
n S C U
for each
a
U = (U )
be a cover
of a topological space
V
on
S
refines
locally finite open cover to = (W )
a
is normal.
be a locally finite open cover of
such that the trace of
W
X
X.
of
X
X
Then there is a
X
such that
aeIm
Main Results.
Theorem 3,1.
If
X
is a topological space and if
S
is
S
is
y
a normal
y-paracompact
strongly
y-paracompact.
Proof.
Le*t
U*
cardinality at most
U
of
U*
on
S
P -embedded subset of
be a cover of
y.
Since
S
S
is
X9 then
by members of
3> with
y-paracompact, the trace
has a locally finite in
S
refinement \s
by members of ^ c . By Lemma 2.3, \s can be assumed to have
cardinality at most y. Normality of S implies that \s is
normal in S b y Theorem 2.2. Moreover, since S is Pyembedded in
X, Theorem 2.1 states that
extends to a locally finite in
members of
3.
X
topology
Hence
on
X.
Theorem 3.2.
S
refinement of
S
has a refinement that
normal cover
Now the family {W PI V :
a locally finite in
3
X
V
WeU>*
U
is strongly
U>* of
and
X
by
VeU*} is
by members of the
y-paracompact.
If (X,3) is a normal topological space and if
is a closed subset of
X, then the following statements are
equivalent:
y
(1)
(2)
S
S
is y-paracompact and P -embedded in
is strongly y-paracompact.
proof«
3.1.
S
S
is normal, (2) follows from (1) by Theorem
On the other hand, if
is y-paracompact.
y
is
S
Since
P -embedded in
cover of
S
then a cover
at most
y
S
is strongly
X.
In fact if
of
S
U
3g
of cardinality at most
by members of
By (2) , there is a locally finite in
U*.
3
by members of
3
locally finite in
finite in
S
X
to
of
to*
on
X
S
G
y
to*
of
U 0 SeU.
S
such that
to*
by
refines
S.
G = to* U (XNS} is a locally
The family
X
by members of
X
3
to
U
It is clear that
cover of
whose trace with
is normal, G
to
is
and
S
is
is normal.
P -
X.
Corollary 3.3.
space and if
cover
for each
is the refinement of
is the required extension of
embedded in
Ue^
needed in 2.1.
is to. By 2.2 and the fact that
Hence
y
«T with cardinality
with cardinality at most
Then the trace
S
is a locally finite in
is obtained by selecting one
members of
y-paracompact then
Theorem 2.1 is now used to show that
by members of
U*
X.
S
If
X
is a normal Hausdorff topological
is a subset of
X, then the following statements
are equivalent:
(1)
S
is strongly paracompact.
(2)
S
is paracompact and
Proof.
X.
In [1 , Corollary 4]it is shown that a strongly
paracompact subset is closed.
(2) #
P-embedded in
Thus 3.2 shows that (1) implies
Since paracompact Hausdorff spaces are normal, Theorem 3.1
shows that (2) implies (1). This completes the proof.
7
Corollary 3,4,
If
is a topological space and if
S
is a paracompact Hausdorff
P-embedded subset of
S
is strongly paracompact in
X,
Corollary 3,5,
X
is a topological space and if S
v/
is a normal Hausdorff countably paracompact P -embedded subset
of
X 5 then
S
If
X, then
X
is strongly countably paracompact.
In [2] , it was shown that ja normal space
X
jjs
7-collectionwise
y
normal if and only if every closed subset is P -embedded in X.
In particular^ it was shown that X jus normal if and only if
PWo -embedded in
every closed subset is
X.
Using these results
it is now possible to obtain the following three corollaries.
Corollary 3,6,
If
S
is a closed subset of a normal
7-collectionwise normal space
if and only if ' S
is strongly
Corollary 3,7.
space
X
then
X, then
S
If
S
S
is
7-paracompact
7-paracompact.
is a closed subset of a normal
is countably paracompact if and only if
S
is strongly countably paracompact.
Corollary 3,8,
normal space
If
X, then
S
S
is a closed subset of a collectionwise
is paracompact if and only if
S
is
S
and
strongly paracompact.
Suppose
S
F
is a strongly
is a subset of a topological
7-paracompact subset of
X.
give some sufficient conditions for
paracompact in
It is now possible to
F
to be strongly
7-
X,
Theorem 3,9,
Let
F
be a strongly
7-paracompact subset
of the closed subspace
S
of the topological space
X.
If
8
there is an open set
is strongly
Let
U*
cardinality at most
F
of
S
y.
G
members of
3
3^.
U*
S
in
X
F
U
of
U*
U
F.
S
From
X.
G
on
S
3
is a cover
has a locally
#c.
The trace \s
refinement of
c
S
Clearly
with
and
V
G
U*
by
open in
X
is locally finite
is closed.
If
of the closed subspace
in
then
by members of
by members of
is open in
Corollary 3.10.
G
F
By assumption
and covers
since
F c G ^ S
X.
is a locally finite in
it follows that V
X
such that
The trace
refinement
\s* on
in
X
be a cover of
by members of
finite in
in
y-paracompact in
Proof.
of
G
such that
F
S
F
c
is a strongly paracompact subset
of (X,,3) and if there is an open set
G
c
S
then
F
is strongly paracompact
X.
Corollary 3.11.
If
F
is a paracompact Hausdorff
embedded subset of the closed subspace
is an open set
G
in
X
strongly paracompact in
Proof.
such that
F ^ G c: s
then
F
is
X.
S.
strongly paracompact in
Hence by Corollary 3.10
X
and if
F
S
is
is a closed paracompact
is a closed subset of
contained in the interior of
X.
F
Y.
Corollary 3.12 (see [1]). If
compact in
of (X,3) and if there
By Corollary 3.4, F is a strongly paracompact subset
of the closed subspace
subset of
S
P-
M, then
F
X
that is
is strongly para-
9
REFERENCES
1.
C. E. Aull, Paracompact Subsets, Proc. of the Second Prague
Topological Symposium, 1966.
2.
T. E. Gantner, Extensions of Pseudometrics and Uniformities,
Thesis, Purdue University, 1966.
3.
H. L. Shapiro, Extensions of Pseudometrics, Canad. J. Math.,
18 (1966), 981-998.
4.
, Closed Maps and Paracompact Spaces, Canad.
J. Math,, to appear.
5.
, A Note on Extending Uniformly Continuous
Pseudometrics, Bulletin de la Societe Mathematigue de
Belqicrue, 18 (1966), 439-441.
6.
, A Note on Closed Maps and Paracompact Spaces,
to appear.
7.
J. W. Tukey, Convergence and Uniformity in Topology,
Princeton, Princeton
Press, 1940.
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